| // Copyright 2020 Google LLC |
| // |
| // This source code is licensed under the BSD-style license found in the |
| // LICENSE file in the root directory of this source tree. |
| |
| $assert ELEMENTS_TILE % 4 == 0 |
| $assert ELEMENTS_TILE >= 4 |
| $SIMD_TILE = ELEMENTS_TILE // 4 |
| $ABC = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ" |
| $VMULADDQ_F32 = "vfmaq_f32" if FMA else "vmlaq_f32" |
| #include <assert.h> |
| |
| #include <arm_neon.h> |
| |
| #include <xnnpack/common.h> |
| #include <xnnpack/raddstoreexpminusmax.h> |
| |
| |
| extern XNN_INTERNAL const float xnn_table_exp2_k_over_64[64]; |
| |
| void xnn_f32_raddstoreexpminusmax_ukernel__${"neonfma" if FMA else "neon"}_lut64_p2_x${ELEMENTS_TILE}${"" if ACCUMULATORS == 1 else "_acc%d" % ACCUMULATORS}( |
| size_t elements, |
| const float* input, |
| float* output, |
| float* sum, |
| float max) XNN_DISABLE_TSAN |
| { |
| assert(elements % sizeof(float) == 0); |
| |
| const float32x4_t vmagic_bias = vmovq_n_f32(0x1.800000p23f); |
| // The smallest x for which expf(x) is normalized. |
| const float32x4_t vdenorm_cutoff = vmovq_n_f32(-0x1.5D589Ep6f); |
| const float32x4_t vlog2e_x64 = vmovq_n_f32(0x1.715476p6f); |
| $if FMA: |
| const float32x4_t vminus_ln2_o64_hi = vmovq_n_f32(-0x1.62e43p-7f); |
| const float32x4_t vminus_ln2_o64_lo = vmovq_n_f32(0x1.05c61p-35f); |
| $else: |
| // Last 13 bits are zeroes |
| const float32x4_t vminus_ln2_o64_hi = vmovq_n_f32(-0x1.630000p-7f); |
| const float32x4_t vminus_ln2_o64_lo = vmovq_n_f32(0x1.BD0106p-19f); |
| |
| const float32x4_t vc2 = vmovq_n_f32(0x1.FFFF0Ap-2f); |
| |
| const int32x4_t vindex_mask = vmovq_n_s32(INT32_C(0x3F)); |
| |
| const float32x4_t vi_max = vdupq_n_f32(max); |
| |
| $if ELEMENTS_TILE > 4: |
| $for K in range(ACCUMULATORS): |
| float32x4_t vacc${K} = vmovq_n_f32(0.0f); |
| for (; elements >= ${ELEMENTS_TILE} * sizeof(float); elements -= ${ELEMENTS_TILE} * sizeof(float)) { |
| // Load ${ELEMENTS_TILE} (${SIMD_TILE}x4) inputs at a time. |
| $for N in range(0, ELEMENTS_TILE, 4): |
| const float32x4_t vi${ABC[N:N+4]} = vld1q_f32(input); input += 4; |
| |
| // Subtract maximum input x := i - i_max. This implies x <= 0. |
| $for N in range(0, ELEMENTS_TILE, 4): |
| const float32x4_t vx${ABC[N:N+4]} = vsubq_f32(vi${ABC[N:N+4]}, vi_max); |
| |
| // Compute reduced argument n := round(x * 64 / log(2)). |
| // We do it by adding a large number (magic bias), which cause rounding of the result to an integer, then subtracing |
| // the large number back. The first addition is combined with multiplication by log2e into a single FMA instruction. |
| // The trick with adding large number is valid only within certain bounds (|x * 64 / log(2)| <= 2**22, i.e. |
| // |x| <= 0x1.62E43p+15 = 45426.09375), but that is acceptable, because inputs outside of [-87.336540, 0.0] |
| // result in denormalized or underflown expf(x). We fixup the result for such inputs at the very end of the |
| // algorithm. |
| $for N in range(0, ELEMENTS_TILE, 4): |
| float32x4_t vn${ABC[N:N+4]} = ${VMULADDQ_F32}(vmagic_bias, vx${ABC[N:N+4]}, vlog2e_x64); |
| |
| // Create a floating-point number s (scale) such that s := 2**(n / 64) for such inputs that expf(x) is normalized, |
| // i.e. -87.33642 <= x <= 0.0. As n has 6 fractional bits, we split s == 2**(n / 64) = 2**e * 2**(n / 64 - e), where |
| // e := int(n / 64). We create s in two steps: |
| // 1. Fetch 2**(n / 64 - e) = 2**(n % 64) from the table using the 6 low bits of n, as integer. Note that the |
| // fetched values are in the [1.0, 2.0) range, i.e. their floating-point exponent is 0. |
| // 2. Adjust fecthed value by addition of e to its floating-point exponent. The result is always a normalized |
| // number, because for -87.33642 <= x <= 0.0 (inputs for which expf(x) is normalized) we have -126 <= e <= 0, |
| // and thus the adjusted exponent is not lower than -126. |
| // |
| // Extract e from bits 6:14 of n and shift it into bits 23:31 (position of floating-point exponent). |
| $for N in range(0, ELEMENTS_TILE, 4): |
| const int32x4_t ve${ABC[N:N+4]} = vshlq_n_s32(vbicq_s32(vreinterpretq_s32_f32(vn${ABC[N:N+4]}), vmovq_n_s32(INT32_C(0x3F))), 17); |
| |
| // Use bits 0:6 bits of n, as integer, as an index for table lookup of l := 2**(n % 64). |
| $for N in range(0, ELEMENTS_TILE, 4): |
| const uint64x2_t vidx${ABC[N:N+4]} = vreinterpretq_u64_s32(vandq_s32(vreinterpretq_s32_f32(vn${ABC[N:N+4]}), vindex_mask)); |
| const uint64_t vidx${ABC[N:N+2]} = vgetq_lane_u64(vidx${ABC[N:N+4]}, 0); |
| const uint64_t vidx${ABC[N+2:N+4]} = vgetq_lane_u64(vidx${ABC[N:N+4]}, 1); |
| |
| $for N in range(0, ELEMENTS_TILE, 4): |
| float32x2_t vl${ABC[N:N+2]} = vld1_dup_f32(&xnn_table_exp2_k_over_64[(uint32_t) vidx${ABC[N:N+2]}]); |
| float32x2_t vl${ABC[N+2:N+4]} = vld1_dup_f32(&xnn_table_exp2_k_over_64[(uint32_t) vidx${ABC[N+2:N+4]}]); |
| |
| $for N in range(0, ELEMENTS_TILE, 4): |
| vl${ABC[N:N+2]} = vld1_lane_f32(&xnn_table_exp2_k_over_64[(uint32_t) (vidx${ABC[N:N+2]} >> 32)], vl${ABC[N:N+2]}, 1); |
| vl${ABC[N+2:N+4]} = vld1_lane_f32(&xnn_table_exp2_k_over_64[(uint32_t) (vidx${ABC[N+2:N+4]} >> 32)], vl${ABC[N+2:N+4]}, 1); |
| const float32x4_t vl${ABC[N:N+4]} = vcombine_f32(vl${ABC[N:N+2]}, vl${ABC[N+2:N+4]}); |
| |
| // Adjust exponent of the value l fetched from the table to get the final s value. |
| $for N in range(0, ELEMENTS_TILE, 4): |
| const float32x4_t vs${ABC[N:N+4]} = vreinterpretq_f32_s32(vaddq_s32(vreinterpretq_s32_f32(vl${ABC[N:N+4]}), ve${ABC[N:N+4]})); |
| |
| // Subtract the large number back to get final n := round(x * 64 / log(2)) as a floating-point number. |
| $for N in range(0, ELEMENTS_TILE, 4): |
| vn${ABC[N:N+4]} = vsubq_f32(vn${ABC[N:N+4]}, vmagic_bias); |
| |
| // Compute reduced argument t := x - n * log(2) / 64. |
| // Use Cody-Waite range reduction method (note the two constants representing log(2) / 64) to improve accuracy. |
| $for N in range(0, ELEMENTS_TILE, 4): |
| float32x4_t vt${ABC[N:N+4]} = ${VMULADDQ_F32}(vx${ABC[N:N+4]}, vn${ABC[N:N+4]}, vminus_ln2_o64_hi); |
| |
| $for N in range(0, ELEMENTS_TILE, 4): |
| vt${ABC[N:N+4]} = ${VMULADDQ_F32}(vt${ABC[N:N+4]}, vn${ABC[N:N+4]}, vminus_ln2_o64_lo); |
| |
| // Compute degree-2 polynomial approximation for exp(t) on [-log(2)/128, log(2)/128]. |
| $for N in range(0, ELEMENTS_TILE, 4): |
| float32x4_t vp${ABC[N:N+4]} = vmulq_f32(vt${ABC[N:N+4]}, vc2); |
| |
| $for N in range(0, ELEMENTS_TILE, 4): |
| vp${ABC[N:N+4]} = ${VMULADDQ_F32}(vt${ABC[N:N+4]}, vt${ABC[N:N+4]}, vp${ABC[N:N+4]}); |
| |
| // Reconstruct the final f value: |
| // f = s * (1 + t * (1 + t * c2)) |
| // = s * (1 + t + t * (t * c2)) |
| // = s + s * (t + t * (t * c2)) |
| // = s + s * p |
| $for N in range(0, ELEMENTS_TILE, 4): |
| float32x4_t vf${ABC[N:N+4]} = ${VMULADDQ_F32}(vs${ABC[N:N+4]}, vs${ABC[N:N+4]}, vp${ABC[N:N+4]}); |
| |
| // For inputs below denormal cutoff, replace output with +0.0f. |
| // Note that for NaN inputs, comparison result is false, and outputs are left unchanged. |
| $for N in range(0, ELEMENTS_TILE, 4): |
| vf${ABC[N:N+4]} = vreinterpretq_f32_u32(vbicq_u32(vreinterpretq_u32_f32(vf${ABC[N:N+4]}), vcltq_f32(vx${ABC[N:N+4]}, vdenorm_cutoff))); |
| |
| // Store ${ELEMENTS_TILE} (${SIMD_TILE}x4) outputs at a time. |
| $for N in range(0, ELEMENTS_TILE, 4): |
| vst1q_f32(output, vf${ABC[N:N+4]}); output += 4; |
| |
| // Accumulate computed exponents. |
| $for N in range(0, ELEMENTS_TILE, 4): |
| vacc${N % ACCUMULATORS} = vaddq_f32(vacc${N % ACCUMULATORS}, vf${ABC[N:N+4]}); |
| } |
| $if ACCUMULATORS > 1: |
| // Add up all accumulators to vacc0 |
| $ACC_SLICE = 1 |
| $while ACC_SLICE < ACCUMULATORS: |
| $for A in range(0, ACCUMULATORS, ACC_SLICE * 2): |
| $if A + ACC_SLICE < ACCUMULATORS: |
| vacc${A} = vaddq_f32(vacc${A}, vacc${A + ACC_SLICE}); |
| $ACC_SLICE *= 2 |
| |
| float32x4_t vacc = vacc0; |
| $else: |
| float32x4_t vacc = vmovq_n_f32(0.0f); |
| for (; elements >= 4 * sizeof(float); elements -= 4 * sizeof(float)) { |
| // Load 4 inputs at a time. |
| const float32x4_t vi = vld1q_f32(input); input += 4; |
| |
| // Subtract maximum input x := i - i_max. This implies x <= 0. |
| const float32x4_t vx = vsubq_f32(vi, vi_max); |
| |
| // Compute reduced argument n := round(x * 64 / log(2)). |
| // We do it by adding a large number (magic bias), which cause rounding of the result to an integer, then subtracing |
| // the large number back. The first addition is combined with multiplication by log2e into a single FMA instruction. |
| // The trick with adding large number is valid only within certain bounds (|x * 64 / log(2)| <= 2**22, i.e. |
| // |x| <= 0x1.62E43p+15 = 45426.09375), but that is acceptable, because inputs outside of [-87.336540, 0.0] |
| // result in denormalized or underflown expf(x). We fixup the result for such inputs at the very end of the |
| // algorithm. |
| float32x4_t vn = ${VMULADDQ_F32}(vmagic_bias, vx, vlog2e_x64); |
| |
| // Create a floating-point number s (scale) such that s := 2**(n / 64) for such inputs that expf(x) is normalized, |
| // i.e. -87.33642 <= x <= 0.0. As n has 6 fractional bits, we split s == 2**(n / 64) = 2**e * 2**(n / 64 - e), where |
| // e := int(n / 64). We create s in two steps: |
| // 1. Fetch 2**(n / 64 - e) = 2**(n % 64) from the table using the 6 low bits of n, as integer. Note that the |
| // fetched values are in the [1.0, 2.0) range, i.e. their floating-point exponent is 0. |
| // 2. Adjust fecthed value by addition of e to its floating-point exponent. The result is always a normalized |
| // number, because for -87.33642 <= x <= 0.0 (inputs for which expf(x) is normalized) we have -126 <= e <= 0, |
| // and thus the adjusted exponent is not lower than -126. |
| // |
| // Extract e from bits 6:14 of n and shift it into bits 23:31 (position of floating-point exponent). |
| const int32x4_t ve = vshlq_n_s32(vbicq_s32(vreinterpretq_s32_f32(vn), vmovq_n_s32(INT32_C(0x3F))), 17); |
| |
| // Use bits 0:6 bits of n, as integer, as an index for table lookup of l := 2**(n % 64). |
| const uint64x2_t vidx = vreinterpretq_u64_s32(vandq_s32(vreinterpretq_s32_f32(vn), vindex_mask)); |
| const uint64_t vidx_lo = vgetq_lane_u64(vidx, 0); |
| const uint64_t vidx_hi = vgetq_lane_u64(vidx, 1); |
| float32x2_t vl_lo = vld1_dup_f32(&xnn_table_exp2_k_over_64[(uint32_t) vidx_lo]); |
| float32x2_t vl_hi = vld1_dup_f32(&xnn_table_exp2_k_over_64[(uint32_t) vidx_hi]); |
| vl_lo = vld1_lane_f32(&xnn_table_exp2_k_over_64[(uint32_t) (vidx_lo >> 32)], vl_lo, 1); |
| vl_hi = vld1_lane_f32(&xnn_table_exp2_k_over_64[(uint32_t) (vidx_hi >> 32)], vl_hi, 1); |
| const float32x4_t vl = vcombine_f32(vl_lo, vl_hi); |
| // Adjust exponent of the value l fetched from the table to get the final s value. |
| const float32x4_t vs = vreinterpretq_f32_s32(vaddq_s32(vreinterpretq_s32_f32(vl), ve)); |
| |
| // Subtract the large number back to get final n := round(x * 64 / log(2)) as a floating-point number. |
| vn = vsubq_f32(vn, vmagic_bias); |
| |
| // Compute reduced argument t := x - n * log(2) / 64. |
| // Use Cody-Waite range reduction method (note the two constants representing log(2) / 64) to improve accuracy. |
| float32x4_t vt = ${VMULADDQ_F32}(vx, vn, vminus_ln2_o64_hi); |
| vt = ${VMULADDQ_F32}(vt, vn, vminus_ln2_o64_lo); |
| |
| // Compute degree-2 polynomial approximation for exp(t) on [-log(2)/128, log(2)/128]. |
| float32x4_t vp = vmulq_f32(vt, vc2); |
| vp = ${VMULADDQ_F32}(vt, vt, vp); |
| |
| // Reconstruct the final f value: |
| // f = s * (1 + t * (1 + t * c2)) |
| // = s * (1 + t + t * (t * c2)) |
| // = s + s * (t + t * (t * c2)) |
| // = s + s * p |
| float32x4_t vf = ${VMULADDQ_F32}(vs, vs, vp); |
| |
| // For inputs below denormal cutoff, replace output with +0.0f. |
| // Note that for NaN inputs, comparison result is false, and outputs are left unchanged. |
| vf = vreinterpretq_f32_u32(vbicq_u32(vreinterpretq_u32_f32(vf), vcltq_f32(vx, vdenorm_cutoff))); |
| |
| // Store 4 outputs at a time. |
| vst1q_f32(output, vf); output += 4; |
| |
| // Accumulate computed exponents. |
| vacc = vaddq_f32(vacc, vf); |
| } |
| #if XNN_ARCH_ARM64 |
| float vacc_lo = vaddvq_f32(vacc); |
| #else |
| float32x2_t vacc_lo = vadd_f32(vget_high_f32(vacc), vget_low_f32(vacc)); |
| #endif |
| if (elements != 0) { |
| assert(elements >= 1 * sizeof(float)); |
| assert(elements <= 3 * sizeof(float)); |
| // Load 4 inputs at a time. |
| const float32x4_t vi = vld1q_f32(input); input += 4; |
| |
| // Subtract maximum input x := i - i_max. This implies x <= 0. |
| const float32x4_t vx = vsubq_f32(vi, vi_max); |
| |
| // Compute reduced argument n := round(x * 64 / log(2)). |
| // We do it by adding a large number (magic bias), which cause rounding of the result to an integer, then subtracing |
| // the large number back. The first addition is combined with multiplication by log2e into a single FMA instruction. |
| // The trick with adding large number is valid only within certain bounds (|x * 64 / log(2)| <= 2**22, i.e. |
| // |x| <= 0x1.62E43p+15 = 45426.09375), but that is acceptable, because inputs outside of [-87.336540, 0.0] |
| // result in denormalized or underflown expf(x). We fixup the result for such inputs at the very end of the |
| // algorithm. |
| float32x4_t vn = ${VMULADDQ_F32}(vmagic_bias, vx, vlog2e_x64); |
| |
| // Create a floating-point number s (scale) such that s := 2**(n / 64) for such inputs that expf(x) is normalized, |
| // i.e. -87.33642 <= x <= 0.0. As n has 6 fractional bits, we split s == 2**(n / 64) = 2**e * 2**(n / 64 - e), where |
| // e := int(n / 64). We create s in two steps: |
| // 1. Fetch 2**(n / 64 - e) = 2**(n % 64) from the table using the 6 low bits of n, as integer. Note that the |
| // fetched values are in the [1.0, 2.0) range, i.e. their floating-point exponent is 0. |
| // 2. Adjust fecthed value by addition of e to its floating-point exponent. The result is always a normalized |
| // number, because for -87.33642 <= x <= 0.0 (inputs for which expf(x) is normalized) we have -126 <= e <= 0, |
| // and thus the adjusted exponent is not lower than -126. |
| // |
| // Extract e from bits 6:14 of n and shift it into bits 23:31 (position of floating-point exponent). |
| const int32x4_t ve = vshlq_n_s32(vbicq_s32(vreinterpretq_s32_f32(vn), vmovq_n_s32(INT32_C(0x3F))), 17); |
| |
| // Use bits 0:6 bits of n, as integer, as an index for table lookup of l := 2**(n % 64). |
| const uint64x2_t vidx = vreinterpretq_u64_s32(vandq_s32(vreinterpretq_s32_f32(vn), vindex_mask)); |
| const uint64_t vidx_lo = vgetq_lane_u64(vidx, 0); |
| const uint64_t vidx_hi = vgetq_lane_u64(vidx, 1); |
| float32x2_t vl_lo = vld1_dup_f32(&xnn_table_exp2_k_over_64[(uint32_t) vidx_lo]); |
| float32x2_t vl_hi = vld1_dup_f32(&xnn_table_exp2_k_over_64[(uint32_t) vidx_hi]); |
| vl_lo = vld1_lane_f32(&xnn_table_exp2_k_over_64[(uint32_t) (vidx_lo >> 32)], vl_lo, 1); |
| vl_hi = vld1_lane_f32(&xnn_table_exp2_k_over_64[(uint32_t) (vidx_hi >> 32)], vl_hi, 1); |
| const float32x4_t vl = vcombine_f32(vl_lo, vl_hi); |
| // Adjust exponent of the value l fetched from the table to get the final s value. |
| const float32x4_t vs = vreinterpretq_f32_s32(vaddq_s32(vreinterpretq_s32_f32(vl), ve)); |
| |
| // Subtract the large number back to get final n := round(x * 64 / log(2)) as a floating-point number. |
| vn = vsubq_f32(vn, vmagic_bias); |
| |
| // Compute reduced argument t := x - n * log(2) / 64. |
| // Use Cody-Waite range reduction method (note the two constants representing log(2) / 64) to improve accuracy. |
| float32x4_t vt = ${VMULADDQ_F32}(vx, vn, vminus_ln2_o64_hi); |
| vt = ${VMULADDQ_F32}(vt, vn, vminus_ln2_o64_lo); |
| |
| // Compute degree-2 polynomial approximation for exp(t) on [-log(2)/128, log(2)/128]. |
| float32x4_t vp = vmulq_f32(vt, vc2); |
| vp = ${VMULADDQ_F32}(vt, vt, vp); |
| |
| // Reconstruct the final f value: |
| // f = s * (1 + t * (1 + t * c2)) |
| // = s * (1 + t + t * (t * c2)) |
| // = s + s * (t + t * (t * c2)) |
| // = s + s * p |
| float32x4_t vf = ${VMULADDQ_F32}(vs, vs, vp); |
| |
| // For inputs below denormal cutoff, replace output with +0.0f. |
| // Note that for NaN inputs, comparison result is false, and outputs are left unchanged. |
| vf = vreinterpretq_f32_u32(vbicq_u32(vreinterpretq_u32_f32(vf), vcltq_f32(vx, vdenorm_cutoff))); |
| |
| float32x2_t vf_lo = vget_low_f32(vf); |
| if (elements & (2 * sizeof(float))) { |
| // Store 2 outputs at a time. |
| vst1_f32(output, vf_lo); output += 2; |
| |
| // Accumulate 2 computed exponents. |
| #if XNN_ARCH_ARM64 |
| vacc_lo += vaddv_f32(vf_lo); |
| #else |
| vacc_lo = vadd_f32(vacc_lo, vf_lo); |
| #endif |
| |
| vf_lo = vget_high_f32(vf); |
| } |
| if (elements & (1 * sizeof(float))) { |
| // Store 1 output at a time. |
| vst1_lane_f32(output, vf_lo, 0); |
| |
| // Accumulate 1 computed exponent. |
| #if XNN_ARCH_ARM64 |
| vacc_lo += vget_lane_f32(vf_lo, 0); |
| #else |
| vacc_lo = vadd_f32(vacc_lo, vreinterpret_f32_u64(vshl_n_u64(vreinterpret_u64_f32(vf_lo), 32))); |
| #endif |
| } |
| } |
| // Reduce 4 elements in the SIMD register |
| #if XNN_ARCH_ARM64 |
| *sum = vacc_lo; |
| #else |
| vst1_lane_f32(sum, vpadd_f32(vacc_lo, vacc_lo), 0); |
| #endif |
| } |