HairBSDF, sample_f, cosTheta

Average Error: 0.2 → 0.1

Time: 16.2s

Precision: binary32

Cost: 13152

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

Math

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

↓

\[\mathsf{fma}\left(v, \log \left(e^{\frac{-2}{v}} - u \cdot \mathsf{expm1}\left(\frac{-2}{v}\right)\right), 1\right) \]

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 (- (exp (/ -2.0 v)) (* u (expm1 (/ -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((expf((-2.0f / v)) - (u * expm1f((-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(exp(Float32(Float32(-2.0) / v)) - Float32(u * expm1(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): 3 points increase in error, 0 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)): 9 points increase in error, 9 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 u around 0 0.1

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

4. Simplified 0.1

   \[\leadsto \mathsf{fma}\left(v, \log \color{blue}{\left(e^{\frac{-2}{v}} - u \cdot \mathsf{expm1}\left(\frac{-2}{v}\right)\right)}, 1\right) \]
   Proof
   (-.f32 (exp.f32 (/.f32 -2 v)) (*.f32 u (expm1.f32 (/.f32 -2 v)))): 0 points increase in error, 0 points decrease in error
   (-.f32 (exp.f32 (/.f32 -2 v)) (*.f32 u (Rewrite<= expm1-def_binary32 (-.f32 (exp.f32 (/.f32 -2 v)) 1)))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> cancel-sign-sub-inv_binary32 (+.f32 (exp.f32 (/.f32 -2 v)) (*.f32 (neg.f32 u) (-.f32 (exp.f32 (/.f32 -2 v)) 1)))): 0 points increase in error, 0 points decrease in error
   (+.f32 (exp.f32 (/.f32 -2 v)) (Rewrite<= distribute-lft-neg-in_binary32 (neg.f32 (*.f32 u (-.f32 (exp.f32 (/.f32 -2 v)) 1))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (exp.f32 (/.f32 -2 v)) (neg.f32 (Rewrite=> *-commutative_binary32 (*.f32 (-.f32 (exp.f32 (/.f32 -2 v)) 1) u)))): 0 points increase in error, 0 points decrease in error
   (+.f32 (exp.f32 (/.f32 -2 v)) (Rewrite=> distribute-lft-neg-in_binary32 (*.f32 (neg.f32 (-.f32 (exp.f32 (/.f32 -2 v)) 1)) u))): 0 points increase in error, 0 points decrease in error
   (+.f32 (exp.f32 (/.f32 -2 v)) (*.f32 (neg.f32 (Rewrite=> sub-neg_binary32 (+.f32 (exp.f32 (/.f32 -2 v)) (neg.f32 1)))) u)): 0 points increase in error, 0 points decrease in error
   (+.f32 (exp.f32 (/.f32 -2 v)) (*.f32 (neg.f32 (+.f32 (exp.f32 (/.f32 -2 v)) (Rewrite=> metadata-eval -1))) u)): 0 points increase in error, 0 points decrease in error
   (+.f32 (exp.f32 (/.f32 -2 v)) (*.f32 (Rewrite=> distribute-neg-in_binary32 (+.f32 (neg.f32 (exp.f32 (/.f32 -2 v))) (neg.f32 -1))) u)): 0 points increase in error, 0 points decrease in error
   (+.f32 (exp.f32 (/.f32 -2 v)) (*.f32 (+.f32 (Rewrite<= mul-1-neg_binary32 (*.f32 -1 (exp.f32 (/.f32 -2 v)))) (neg.f32 -1)) u)): 0 points increase in error, 0 points decrease in error
   (+.f32 (exp.f32 (/.f32 -2 v)) (*.f32 (+.f32 (*.f32 -1 (exp.f32 (/.f32 -2 v))) (Rewrite=> metadata-eval 1)) u)): 0 points increase in error, 0 points decrease in error
   (+.f32 (exp.f32 (/.f32 -2 v)) (*.f32 (Rewrite<= +-commutative_binary32 (+.f32 1 (*.f32 -1 (exp.f32 (/.f32 -2 v))))) u)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-commutative_binary32 (+.f32 (*.f32 (+.f32 1 (*.f32 -1 (exp.f32 (/.f32 -2 v)))) u) (exp.f32 (/.f32 -2 v)))): 0 points increase in error, 0 points decrease in error

5. Final simplification 0.1

   \[\leadsto \mathsf{fma}\left(v, \log \left(e^{\frac{-2}{v}} - u \cdot \mathsf{expm1}\left(\frac{-2}{v}\right)\right), 1\right) \]

Alternatives

Alternative 1
Error	0.2
Cost	13088

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

Alternative 2
Error	0.1
Cost	9952

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

Alternative 3
Error	0.2
Cost	9952

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

Alternative 4
Error	0.2
Cost	6816

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

Alternative 5
Error	2.7
Cost	3876

\[\begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(v \cdot \mathsf{expm1}\left(\frac{2}{v}\right) + \frac{-2 + \frac{-4}{v}}{\frac{v}{u}}\right) + -1\\ \end{array} \]

Alternative 6
Error	2.8
Cost	3556

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

Alternative 7
Error	2.9
Cost	804

\[\begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -2 \cdot \left(1 - u\right)\right) + \frac{0.5}{v} \cdot \left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot -4 + 4\right)\right)\\ \end{array} \]

Alternative 8
Error	2.9
Cost	548

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

Alternative 9
Error	3.0
Cost	420

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

Alternative 10
Error	4.2
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2022313 
(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))))))))