Average Error: 0.2 → 0.2
Time: 16.2s
Precision: binary32
Cost: 13216
\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]
\[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
↓
\[1 + v \cdot \left(3 \cdot \log \left(\sqrt[3]{\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)}\right)\right) \]
(FPCore (u v)
 :precision binary32
 (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))
↓
(FPCore (u v)
 :precision binary32
 (+ 1.0 (* v (* 3.0 (log (cbrt (fma (- 1.0 u) (exp (/ -2.0 v)) u)))))))
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 + (v * (3.0f * logf(cbrtf(fmaf((1.0f - u), expf((-2.0f / v)), u)))));
}

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(v * Float32(Float32(3.0) * log(cbrt(fma(Float32(Float32(1.0) - u), exp(Float32(Float32(-2.0) / v)), u))))))
end

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

Error

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

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

  3. Final simplification0.2

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

Alternatives

Alternative 1
Error0.2
Cost10080
\[1 + v \cdot \left(3 \cdot \left(0.3333333333333333 \cdot \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)\right)\right) \]
Alternative 2
Error0.2
Cost6944
\[1 + v \cdot \left(3 \cdot \left(0.3333333333333333 \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\right)\right) \]
Alternative 3
Error0.2
Cost6816
\[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
Alternative 4
Error4.3
Cost32
\[1 \]

Error

Reproduce

herbie shell --seed 2022297 
(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))))))))