Average Error: 0.2 → 0.2
Time: 4.1s
Precision: binary32
\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]
\[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
↓
\[1 + \left(\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, {\left({\left(e^{\frac{-2}{v}}\right)}^{3}\right)}^{0.3333333333333333}, u\right)\right), 1\right) + -1\right) \]
(FPCore (u v)
 :precision binary32
 (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))
↓
(FPCore (u v)
 :precision binary32
 (+
  1.0
  (+
   (fma
    v
    (log (fma (- 1.0 u) (pow (pow (exp (/ -2.0 v)) 3.0) 0.3333333333333333) u))
    1.0)
   -1.0)))
float code(float u, float v) {
	return 1.0f + (v * logf((u + ((1.0f - u) * expf((-2.0f / v))))));
}
↓
float code(float u, float v) {
	return 1.0f + (fmaf(v, logf(fmaf((1.0f - u), powf(powf(expf((-2.0f / v)), 3.0f), 0.3333333333333333f), u)), 1.0f) + -1.0f);
}

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))))
end
↓
function code(u, v)
	return Float32(Float32(1.0) + Float32(fma(v, log(fma(Float32(Float32(1.0) - u), ((exp(Float32(Float32(-2.0) / v)) ^ Float32(3.0)) ^ Float32(0.3333333333333333)), u)), Float32(1.0)) + Float32(-1.0)))
end

1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)
↓
1 + \left(\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, {\left({\left(e^{\frac{-2}{v}}\right)}^{3}\right)}^{0.3333333333333333}, u\right)\right), 1\right) + -1\right)

Error

Bits error versus u

Bits error versus v

Derivation

  1. Initial program 0.2

    \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]

  2. Applied egg-rr0.2

    \[\leadsto 1 + \color{blue}{\left(\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right), 1\right) - 1\right)} \]

  3. Applied egg-rr0.2

    \[\leadsto 1 + \left(\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, \color{blue}{{\left({\left(e^{\frac{-2}{v}}\right)}^{3}\right)}^{0.3333333333333333}}, u\right)\right), 1\right) - 1\right) \]

  4. Final simplification0.2

    \[\leadsto 1 + \left(\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, {\left({\left(e^{\frac{-2}{v}}\right)}^{3}\right)}^{0.3333333333333333}, u\right)\right), 1\right) + -1\right) \]

Reproduce

herbie shell --seed 2022162 
(FPCore (u v)
  :name "HairBSDF, sample_f, cosTheta"
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0)) (and (<= 0.0 v) (<= v 109.746574)))
  (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))