Average Error: 0.2 → 0.2
Time: 5.8s
Precision: binary32
\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]
\[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
↓
\[\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, {\left(e^{\frac{\sqrt[3]{-2} \cdot \sqrt[3]{-2}}{\sqrt{v}}}\right)}^{\left(\frac{\sqrt[3]{-2}}{\sqrt{v}}\right)}, u\right)\right), 1\right) \]
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, {\left(e^{\frac{\sqrt[3]{-2} \cdot \sqrt[3]{-2}}{\sqrt{v}}}\right)}^{\left(\frac{\sqrt[3]{-2}}{\sqrt{v}}\right)}, 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)
    (pow
     (exp (/ (* (cbrt -2.0) (cbrt -2.0)) (sqrt v)))
     (/ (cbrt -2.0) (sqrt 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), powf(expf((cbrtf(-2.0f) * cbrtf(-2.0f)) / sqrtf(v)), (cbrtf(-2.0f) / sqrtf(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. Simplified0.2

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

  3. Applied add-sqr-sqrt_binary320.2

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

  4. Applied add-cube-cbrt_binary320.2

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

  5. Applied times-frac_binary320.2

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

  6. Applied exp-prod_binary320.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2022081 
(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))))))))