Average Error: 0.1 → 0.2
Time: 5.0s
Precision: binary32
\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]
\[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
↓
\[\begin{array}{l} t_0 := e^{\frac{-2}{v}}\\ \sqrt[3]{\mathsf{fma}\left(v, \log \left(\left(t_0 + u\right) - t_0 \cdot u\right), 1\right)} \cdot \sqrt[3]{{\left(\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(t_0, 1 - u, u\right)\right), 1\right)\right)}^{2}} \end{array} \]
(FPCore (u v)
 :precision binary32
 (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))
↓
(FPCore (u v)
 :precision binary32
 (let* ((t_0 (exp (/ -2.0 v))))
   (*
    (cbrt (fma v (log (- (+ t_0 u) (* t_0 u))) 1.0))
    (cbrt (pow (fma v (log (fma t_0 (- 1.0 u) u)) 1.0) 2.0)))))
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) {
	float t_0 = expf((-2.0f / v));
	return cbrtf(fmaf(v, logf(((t_0 + u) - (t_0 * u))), 1.0f)) * cbrtf(powf(fmaf(v, logf(fmaf(t_0, (1.0f - u), u)), 1.0f), 2.0f));
}

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)
	t_0 = exp(Float32(Float32(-2.0) / v))
	return Float32(cbrt(fma(v, log(Float32(Float32(t_0 + u) - Float32(t_0 * u))), Float32(1.0))) * cbrt((fma(v, log(fma(t_0, Float32(Float32(1.0) - u), u)), Float32(1.0)) ^ Float32(2.0))))
end

1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)
↓
\begin{array}{l}
t_0 := e^{\frac{-2}{v}}\\
\sqrt[3]{\mathsf{fma}\left(v, \log \left(\left(t_0 + u\right) - t_0 \cdot u\right), 1\right)} \cdot \sqrt[3]{{\left(\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(t_0, 1 - u, u\right)\right), 1\right)\right)}^{2}}
\end{array}

Error

Bits error versus u

Bits error versus v

Derivation

  1. Initial program 0.1

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

  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

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

  4. Taylor expanded in v around 0 0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2022165 
(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))))))))