Average Error: 0.2 → 0.2
Time: 5.9s
Precision: binary32
\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]
\[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
↓
\[\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, \sqrt[3]{e^{\frac{-2}{v}}} \cdot e^{\frac{-1.3333333333333333}{v}}, u\right)\right), 1\right) \]
1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)
↓
\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, \sqrt[3]{e^{\frac{-2}{v}}} \cdot e^{\frac{-1.3333333333333333}{v}}, u\right)\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
 (fma
  v
  (log
   (fma
    (- 1.0 u)
    (* (cbrt (exp (/ -2.0 v))) (exp (/ -1.3333333333333333 v)))
    u))
  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 fmaf(v, logf(fmaf((1.0f - u), (cbrtf(expf((-2.0f / v))) * expf((-1.3333333333333333f / v))), u)), 1.0f);
}

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. Simplified0.1

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

  3. Applied egg-rr0.2

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

  4. Taylor expanded in v around 0 0.2

    \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, \sqrt[3]{e^{\frac{-2}{v}}} \cdot \color{blue}{{\left(e^{\frac{-0.6666666666666666}{v}}\right)}^{2}}, u\right)\right), 1\right) \]

  5. Simplified0.2

    \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, \sqrt[3]{e^{\frac{-2}{v}}} \cdot \color{blue}{e^{\frac{-1.3333333333333333}{v}}}, u\right)\right), 1\right) \]

  6. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, \sqrt[3]{e^{\frac{-2}{v}}} \cdot e^{\frac{-1.3333333333333333}{v}}, u\right)\right), 1\right) \]

Reproduce

herbie shell --seed 2022127 
(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))))))))