Average Error: 0.2 → 0.2
Time: 15.9s
Precision: binary32
\[10^{-05} \leq u \land u \leq 1 \land 0 \leq v \land v \leq 109.746574\]
\[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\]
↓
\[1 + \left(v \cdot \log \left(\sqrt{u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right) + v \cdot \log \left(\sqrt{u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)\right)\]
1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)
↓
1 + \left(v \cdot \log \left(\sqrt{u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right) + v \cdot \log \left(\sqrt{u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\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 (log (sqrt (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))))
   (* v (log (sqrt (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))))
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 * logf(sqrtf(u + ((1.0f - u) * expf(-2.0f / v))))) + (v * logf(sqrtf(u + ((1.0f - u) * expf(-2.0f / v))))));
}

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-sqr-sqrt_binary320.2

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

  4. Applied log-prod_binary320.2

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

  5. Applied distribute-rgt-in_binary320.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2021173 
(FPCore (u v)
  :name "HairBSDF, sample_f, cosTheta"
  :precision binary32
  :pre (and (<= 1e-05 u 1.0) (<= 0.0 v 109.746574))
  (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))