HairBSDF, sample_f, cosTheta

Average Error: 0.51% → 0.5%

Time: 18.1s

Precision: binary32

Cost: 13088

\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]

Math

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

↓

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

FPCore

(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) (exp (/ -2.0 v)) u)) 1.0))

C

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), expf((-2.0f / v)), u)), 1.0f);
}

Julia

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)
	return fma(v, log(fma(Float32(Float32(1.0) - u), exp(Float32(Float32(-2.0) / v)), u)), Float32(1.0))
end

Derivation

1. Initial program 0.51

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

2. Simplified 0.5

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

   [Start] 0.51	\[ 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   +-commutative [=>] 0.51	\[ \color{blue}{v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) + 1} \]
   fma-def [=>] 0.5	\[ \color{blue}{\mathsf{fma}\left(v, \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right), 1\right)} \]
   +-commutative [=>] 0.5	\[ \mathsf{fma}\left(v, \log \color{blue}{\left(\left(1 - u\right) \cdot e^{\frac{-2}{v}} + u\right)}, 1\right) \]
   fma-def [=>] 0.5	\[ \mathsf{fma}\left(v, \log \color{blue}{\left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}, 1\right) \]

3. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.52%
Cost	9952

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

Alternative 2
Error	0.51%
Cost	6816

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

Alternative 3
Error	4.2%
Cost	6688

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

Alternative 4
Error	2.71%
Cost	6564

\[\begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;\mathsf{fma}\left(v, \log u, 1\right)\\ \mathbf{else}:\\ \;\;\;\;-1 - v \cdot \left(u \cdot \left(1 + \frac{-1}{e^{\frac{-2}{v}}}\right)\right)\\ \end{array} \]

Alternative 5
Error	2.71%
Cost	3684

\[\begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;-1 - v \cdot \left(u \cdot \left(1 + \frac{-1}{e^{\frac{-2}{v}}}\right)\right)\\ \end{array} \]

Alternative 6
Error	2.71%
Cost	3556

\[\begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\frac{2}{v}\right) \cdot \left(v \cdot u\right) + -1\\ \end{array} \]

Alternative 7
Error	3.04%
Cost	3428

\[\begin{array}{l} t_0 := 1 - u \cdot u\\ \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;1 + v \cdot \log u\\ \mathbf{else}:\\ \;\;\;\;1 + \left(2 \cdot \left(u + -1\right) + 0.5 \cdot \frac{\left(1 - u\right) \cdot 4 + -4 \cdot \frac{\frac{t_0 \cdot t_0}{1 + u}}{1 + u}}{v}\right)\\ \end{array} \]

Alternative 8
Error	9.41%
Cost	1316

\[\begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(2 \cdot \left(u + -1\right) + 0.5 \cdot \frac{\left(1 - u\right) \cdot 4 + -4 \cdot \left(\frac{1}{1 + u} + \left(u \cdot \left(u + -1\right) - \frac{u}{\frac{1 + u}{u}}\right)\right)}{v}\right)\\ \end{array} \]

Alternative 9
Error	9.41%
Cost	1316

\[\begin{array}{l} t_0 := 1 - u \cdot u\\ \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(2 \cdot \left(u + -1\right) + 0.5 \cdot \frac{\left(1 - u\right) \cdot 4 + -4 \cdot \frac{\frac{t_0 \cdot t_0}{1 + u}}{1 + u}}{v}\right)\\ \end{array} \]

Alternative 10
Error	9.41%
Cost	1124

\[\begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(2 \cdot \left(u + -1\right) + 0.5 \cdot \frac{\left(1 - u\right) \cdot 4 + 4 \cdot \frac{u + -1}{\frac{1 + u}{1 - u \cdot u}}}{v}\right)\\ \end{array} \]

Alternative 11
Error	9.41%
Cost	932

\[\begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(2 \cdot \left(u + -1\right) + 0.5 \cdot \frac{\left(1 - u\right) \cdot 4 + -4 \cdot \left(1 + u \cdot \left(u + -2\right)\right)}{v}\right)\\ \end{array} \]

Alternative 12
Error	9.56%
Cost	868

\[\begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(2 \cdot \left(u + -1\right) + 0.5 \cdot \frac{\left(1 - u\right) \cdot 4 + -4 \cdot \left(1 + u \cdot -2\right)}{v}\right)\\ \end{array} \]

Alternative 13
Error	9.56%
Cost	548

\[\begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(2 \cdot \left(u + -1\right) - 0.5 \cdot \frac{u \cdot -4}{v}\right)\\ \end{array} \]

Alternative 14
Error	9.56%
Cost	484

\[\begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{\frac{1}{u + \frac{u}{v}}} + -1\\ \end{array} \]

Alternative 15
Error	9.56%
Cost	356

\[\begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(u + \frac{u}{v}\right) + -1\\ \end{array} \]

Alternative 16
Error	10.19%
Cost	228

\[\begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 - u \cdot -2\\ \end{array} \]

Alternative 17
Error	10.86%
Cost	100

\[\begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 18
Error	94%
Cost	32

\[-1 \]

Reproduce

herbie shell --seed 2023088 
(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))))))))