HairBSDF, sample_f, cosTheta

Average Error: 0.2 → 0.2

Time: 14.6s

Precision: binary32

Cost: 9952

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

Math

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

↓

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

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 (+ u (* (- 1.0 u) (exp (/ -2.0 v))))) 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((u + ((1.0f - u) * expf((-2.0f / v))))), 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(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v))))), Float32(1.0))
end

Derivation

1. Initial program 0.2

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

2. Simplified 0.2

   \[\leadsto \color{blue}{\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right), 1\right)} \]
   Proof
   (fma.f32 v (log.f32 (fma.f32 (-.f32 1 u) (exp.f32 (/.f32 -2 v)) u)) 1): 0 points increase in error, 0 points decrease in error
   (fma.f32 v (log.f32 (Rewrite<= fma-def_binary32 (+.f32 (*.f32 (-.f32 1 u) (exp.f32 (/.f32 -2 v))) u))) 1): 0 points increase in error, 2 points decrease in error
   (fma.f32 v (log.f32 (Rewrite<= +-commutative_binary32 (+.f32 u (*.f32 (-.f32 1 u) (exp.f32 (/.f32 -2 v)))))) 1): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary32 (+.f32 (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 1 u) (exp.f32 (/.f32 -2 v)))))) 1)): 8 points increase in error, 3 points decrease in error
   (Rewrite<= +-commutative_binary32 (+.f32 1 (*.f32 v (log.f32 (+.f32 u (*.f32 (-.f32 1 u) (exp.f32 (/.f32 -2 v)))))))): 0 points increase in error, 0 points decrease in error

3. Taylor expanded in v around 0 0.2

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

4. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.2
Cost	6816

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

Alternative 2
Error	2.9
Cost	3556

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

Alternative 3
Error	3.0
Cost	676

\[\begin{array}{l} \mathbf{if}\;v \leq 0.4000000059604645:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;v \cdot \left(\frac{u}{v} \cdot \left(2 + \left(\frac{2}{v} + \frac{1.3333333333333333}{v \cdot v}\right)\right)\right) + -1\\ \end{array} \]

Alternative 4
Error	3.1
Cost	356

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

Alternative 5
Error	3.3
Cost	228

\[\begin{array}{l} \mathbf{if}\;v \leq 0.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot 2 + -1\\ \end{array} \]

Alternative 6
Error	30.1
Cost	32

\[-1 \]

Alternative 7
Error	4.2
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2022294 
(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))))))))