| // 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 >= 1 |
| $ABC = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ" |
| #include <assert.h> |
| |
| #include <xnnpack/common.h> |
| #include <xnnpack/raddstoreexpminusmax.h> |
| |
| #include <fp16/bitcasts.h> |
| |
| |
| // Note redefine as uint32[] to avoid redundant bitcasts. |
| extern XNN_INTERNAL const uint32_t xnn_table_exp2_k_over_64[64]; |
| |
| void xnn_f32_raddstoreexpminusmax_ukernel__scalar_lut64_p2_x${ELEMENTS_TILE}${"" if ACCUMULATORS == 1 else "_acc%d" % ACCUMULATORS}( |
| size_t elements, |
| const float* input, |
| float* output, |
| float* sum, |
| float vi_max) |
| { |
| assert(elements % sizeof(float) == 0); |
| |
| const float vmagic_bias = 0x1.800000p23f; |
| // The smallest x for which expf(x) is normalized. |
| const float vdenorm_cutoff = -0x1.5D589Ep6f; |
| const float vlog2e_x64 = 0x1.715476p6f; |
| // Last 13 bits are zeroes |
| const float vminus_ln2_o64_hi = -0x1.630000p-7f; |
| const float vminus_ln2_o64_lo = 0x1.BD0106p-19f; |
| |
| const float vc2 = 0x1.FFFF0Ap-2f; |
| |
| const uint32_t vindex_mask = UINT32_C(0x3F); |
| |
| $if ELEMENTS_TILE > 1: |
| $for K in range(ACCUMULATORS): |
| float vacc${K} = 0.0f; |
| for (; elements >= ${ELEMENTS_TILE} * sizeof(float); elements -= ${ELEMENTS_TILE} * sizeof(float)) { |
| // Load ${ELEMENTS_TILE} inputs at a time. |
| $for N in range(ELEMENTS_TILE): |
| const float vi${N} = input[${N}]; |
| input += ${ELEMENTS_TILE}; |
| |
| // Subtract maximum input x := i - i_max. This implies x <= 0. |
| $for N in range(ELEMENTS_TILE): |
| const float vx${N} = vi${N} - 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. |
| $for N in range(ELEMENTS_TILE): |
| float vn${N} = vx${N} * vlog2e_x64 + vmagic_bias; |
| |
| // 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). |
| $for N in range(ELEMENTS_TILE): |
| const uint32_t ve${N} = (fp32_to_bits(vn${N}) & UINT32_C(0xFFFFFFC0)) << 17; |
| |
| // Use bits 0:6 bits of n, as integer, as an index for table lookup of l := 2**(n % 64). |
| $for N in range(ELEMENTS_TILE): |
| const uint32_t vidx${N} = fp32_to_bits(vn${N}) & vindex_mask; |
| // Adjust exponent of the value l fetched from the table to get the final s value. |
| $for N in range(ELEMENTS_TILE): |
| const float vs${N} = fp32_from_bits(xnn_table_exp2_k_over_64[vidx${N}] + ve${N}); |
| |
| // Subtract the large number back to get final n := round(x * 64 / log(2)) as a floating-point number. |
| $for N in range(ELEMENTS_TILE): |
| vn${N} -= 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. |
| $for N in range(ELEMENTS_TILE): |
| float vt${N} = vn${N} * vminus_ln2_o64_hi + vx${N}; |
| |
| $for N in range(ELEMENTS_TILE): |
| vt${N} = vn${N} * vminus_ln2_o64_lo + vt${N}; |
| |
| // Compute degree-2 polynomial approximation for exp(t) on [-log(2)/128, log(2)/128]. |
| $for N in range(ELEMENTS_TILE): |
| float vp${N} = vt${N} * vc2; |
| |
| $for N in range(ELEMENTS_TILE): |
| vp${N} = vp${N} * vt${N} + vt${N}; |
| |
| // 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 |
| $for N in range(ELEMENTS_TILE): |
| float vf${N} = vp${N} * vs${N} + vs${N}; |
| |
| // For inputs below denormal cutoff, replace output with +0.0f. |
| // Note that for NaN inputs, comparison result is false, and outputs are left unchanged. |
| $for N in range(ELEMENTS_TILE): |
| if XNN_UNPREDICTABLE(vx${N} < vdenorm_cutoff) { |
| vf${N} = 0.0f; |
| } |
| |
| // Store ${ELEMENTS_TILE} outputs at a time. |
| $for N in range(ELEMENTS_TILE): |
| output[${N}] = vf${N}; |
| output += ${ELEMENTS_TILE}; |
| |
| // Accumulate computed exponents. |
| $for N in range(ELEMENTS_TILE): |
| vacc${N % ACCUMULATORS} += vf${N}; |
| } |
| $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} += vacc${A + ACC_SLICE}; |
| $ACC_SLICE *= 2 |
| |
| float vacc = vacc0; |
| $else: |
| float vacc = 0.0f; |
| for (; elements >= sizeof(float); elements -= sizeof(float)) { |
| // Load 1 input at a time. |
| const float vi = *input++; |
| |
| // Subtract maximum input x := i - i_max. This implies x <= 0. |
| const float vx = 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. |
| float vn = vx * vlog2e_x64 + vmagic_bias; |
| |
| // 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 uint32_t ve = (fp32_to_bits(vn) & UINT32_C(0xFFFFFFC0)) << 17; |
| |
| // Use bits 0:6 bits of n, as integer, as an index for table lookup of l := 2**(n % 64). |
| const uint32_t vidx = fp32_to_bits(vn) & vindex_mask; |
| // Adjust exponent of the value l fetched from the table to get the final s value. |
| const float vs = fp32_from_bits(xnn_table_exp2_k_over_64[vidx] + ve); |
| |
| // Subtract the large number back to get final n := round(x * 64 / log(2)) as a floating-point number. |
| 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. |
| float vt = vn * vminus_ln2_o64_hi + vx; |
| vt = vn * vminus_ln2_o64_lo + vt; |
| |
| // Compute degree-2 polynomial approximation for exp(t) on [-log(2)/128, log(2)/128]. |
| float vp = vt * vc2; |
| vp = vp * vt + vt; |
| |
| // 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 |
| float vf = vp * vs + vs; |
| |
| // For inputs below denormal cutoff, replace output with +0.0f. |
| // Note that for NaN inputs, comparison result is false, and outputs are left unchanged. |
| if XNN_UNPREDICTABLE(vx < vdenorm_cutoff) { |
| vf = 0.0f; |
| } |
| |
| // Store 1 output at a time. |
| *output++ = vf; |
| |
| // Accumulate computed exponents. |
| vacc += vf; |
| } |
| *sum = vacc; |
| } |