Average Error: 0.2 → 0.2
Time: 10.3s
Precision: binary32
\[10^{-5} \leq u \land u \leq 1 \land 0 \leq v \land v \leq 109.746574\]
\[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
↓
\[\sqrt[3]{{\left(1 + v \cdot \log \left(u + {\left(e^{-2}\right)}^{\left(\frac{1}{v}\right)} \cdot \left(1 - u\right)\right)\right)}^{3}} \]
1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)
↓
\sqrt[3]{{\left(1 + v \cdot \log \left(u + {\left(e^{-2}\right)}^{\left(\frac{1}{v}\right)} \cdot \left(1 - u\right)\right)\right)}^{3}}

(FPCore (u v)
 :precision binary32
 (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))
↓
(FPCore (u v)
 :precision binary32
 (cbrt
  (pow
   (+ 1.0 (* v (log (+ u (* (pow (exp -2.0) (/ 1.0 v)) (- 1.0 u))))))
   3.0)))
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 cbrtf(powf((1.0f + (v * logf(u + (powf(expf(-2.0f), (1.0f / v)) * (1.0f - u))))), 3.0f));
}

Error

Bits error versus u

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

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

  3. Applied add-cbrt-cube_binary320.2

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

  4. Simplified0.2

    \[\leadsto \sqrt[3]{\color{blue}{{\left(v \cdot \log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right) + 1\right)}^{3}}} \]
  5. Using strategy rm

  6. Applied div-inv_binary320.2

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

  7. Applied exp-prod_binary320.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2021204 
(FPCore (u v)
  :name "HairBSDF, sample_f, cosTheta"
  :precision binary32
  :pre (and (<= 1e-5 u 1.0) (<= 0.0 v 109.746574))
  (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))