math_fp32.bend checks

The Cephes expf polynomial after its constant and linear terms.

The 32-bit word of a U32, read as an F32.

exp(r)-1-r near |r| <= ln(2)/2. Chebyshev coefficients, rounded to F32; paired evaluation shortens the dependency chain without contracting products.

b=RN(1+q). Recover the small term lost in that sum before adding the reduced exponential. Writing (1-b)+q keeps the eight lanes together under the default vectorizer; q-(b-1) makes it mix unrelated addition trees.

Above 18 SiLU rounds to x. Pass x through, including positive infinity; the negative tail uses the clamped input before the final underflow.

u=-value, n approximates u/ln(2) to the nearest integer, r=u-n*ln(2). The exponent word makes H=2^(64-n), so sigmoid(x)=H/(exp(r)+2^-n)*2^-64. H remains normal; only the last multiplication underflows. The tiny denominator addend q may be floored to the least normal F32 without changing its rounded sum.

At x>=18 sigmoid rounds to 1. Below -110 both sigmoid and SiLU round to zero; multiplying the clamped x before the final scaling preserves -0 for SiLU, including at -infinity. NaNs pass the clamp and propagate.

Sign restoration and the tiny interval are word selections. For |x|<2^-12, |x-tanh(x)|<|x|^3/3 is below half an F32 ulp, so return x, including -0.

At |x|>=10, 1-tanh(|x|)<2*exp(-20)<2^-25. Clamping before arithmetic also keeps infinities away from intermediate products; NaNs propagate.

Chebyshev interpolation of (tanh(sqrt(z))/sqrt(z)-1)/z, |t|<=0.75. The coefficients are F32 values.

One additional term for the expm1 correction in the tanh quotient.

Split 2^n at +64 for nonnegative inputs and -64 for negative inputs. The first product is always normal and exact; only the last product can underflow. The exponent words come directly from the biased integer n.

FastTwoSum retains the low bits of 1+r before adding the quadratic tail.

For s near r/(2+r), d = r-(2+r)*s gives the exact real identity log(1+r) = 2*atanh(s) + log(1+d/(1+s)). The odd series ends at s^11. Split r and s into 12-bit heads so their head product is exact; retain the low products in d. (1-s)*(1+s^2+s^4) approximates 1/(1+s). FastTwoSum retains the low bits when adding the exponent's ln(2) part.

The exponent carry at mantissa word 0x3504f4 selects the same reduced interval as halving m above sqrt(2), without a floating-point selection.

Scaling a subnormal by 2^23 is exact. Zero, negatives, infinities and NaNs are selected explicitly, after normalizing the finite positive input.