HairBSDF, sample_f, cosTheta

Average Error: 0.51% → 0.52%

Time: 14.5s

Precision: binary32

Cost: 16288

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

Math

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

↓

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

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) (pow E (/ -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), powf(((float) M_E), (-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), (Float32(exp(1)) ^ 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 0.52

   \[\leadsto \color{blue}{\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right), 1\right)} \]
   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 [=>] 0.52	\[ \mathsf{fma}\left(v, \log \color{blue}{\left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)}, 1\right) \]

3. Applied egg-rr 0.52

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

4. Final simplification 0.52

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

Alternatives

Alternative 1
Error	0.51%
Cost	10016

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

Alternative 2
Error	0.53%
Cost	9952

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

Alternative 3
Error	0.5%
Cost	9952

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

Alternative 4
Error	0.51%
Cost	6816

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

Alternative 5
Error	0.83%
Cost	6752

\[1 + v \cdot \mathsf{log1p}\left(\left(1 - u\right) \cdot \mathsf{expm1}\left(\frac{-2}{v}\right)\right) \]

Alternative 6
Error	8.95%
Cost	3748

\[\begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 + v \cdot \left(\frac{-2}{v} + u \cdot \left(e^{\frac{2}{v}} + -1\right)\right)\\ \end{array} \]

Alternative 7
Error	8.92%
Cost	3684

\[\begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + v \cdot \left(u \cdot \left(-1 + \frac{1}{e^{\frac{-2}{v}}}\right)\right)\\ \end{array} \]

Alternative 8
Error	8.92%
Cost	3556

\[\begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \mathsf{expm1}\left(\frac{2}{v}\right) \cdot \left(v \cdot u\right)\\ \end{array} \]

Alternative 9
Error	94.18%
Cost	32

\[-1 \]

Alternative 10
Error	13.07%
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2023089 
(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))))))))