GTR1 distribution

Average Error: 0.5 → 0.5

Time: 16.6s

Precision: binary32

Cost: 16512

\[\left(0 \leq cosTheta \land cosTheta \leq 1\right) \land \left(0.0001 \leq \alpha \land \alpha \leq 1\right)\]

Math

\[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

↓

\[\frac{\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha \cdot 2}}{\pi}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/
  (- (* alpha alpha) 1.0)
  (*
   (* PI (log (* alpha alpha)))
   (+ 1.0 (* (* (- (* alpha alpha) 1.0) cosTheta) cosTheta)))))

↓

(FPCore (cosTheta alpha)
 :precision binary32
 (/
  (/ (/ (fma alpha alpha -1.0) (* (log alpha) 2.0)) PI)
  (fma (fma alpha alpha -1.0) (* cosTheta cosTheta) 1.0)))

C

float code(float cosTheta, float alpha) {
	return ((alpha * alpha) - 1.0f) / ((((float) M_PI) * logf((alpha * alpha))) * (1.0f + ((((alpha * alpha) - 1.0f) * cosTheta) * cosTheta)));
}

↓

float code(float cosTheta, float alpha) {
	return ((fmaf(alpha, alpha, -1.0f) / (logf(alpha) * 2.0f)) / ((float) M_PI)) / fmaf(fmaf(alpha, alpha, -1.0f), (cosTheta * cosTheta), 1.0f);
}

Julia

function code(cosTheta, alpha)
	return Float32(Float32(Float32(alpha * alpha) - Float32(1.0)) / Float32(Float32(Float32(pi) * log(Float32(alpha * alpha))) * Float32(Float32(1.0) + Float32(Float32(Float32(Float32(alpha * alpha) - Float32(1.0)) * cosTheta) * cosTheta))))
end

↓

function code(cosTheta, alpha)
	return Float32(Float32(Float32(fma(alpha, alpha, Float32(-1.0)) / Float32(log(alpha) * Float32(2.0))) / Float32(pi)) / fma(fma(alpha, alpha, Float32(-1.0)), Float32(cosTheta * cosTheta), Float32(1.0)))
end

Derivation

1. Initial program 0.5

   \[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

2. Simplified 0.5

   \[\leadsto \color{blue}{\frac{\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha \cdot 2}}{\pi}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)}} \]
   Proof
   (/.f32 (/.f32 (/.f32 (fma.f32 alpha alpha -1) (*.f32 (log.f32 alpha) 2)) (PI.f32)) (fma.f32 (fma.f32 alpha alpha -1) (*.f32 cosTheta cosTheta) 1)): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 (/.f32 (fma.f32 alpha alpha (Rewrite<= metadata-eval (neg.f32 1))) (*.f32 (log.f32 alpha) 2)) (PI.f32)) (fma.f32 (fma.f32 alpha alpha -1) (*.f32 cosTheta cosTheta) 1)): 6 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 (/.f32 (Rewrite<= fma-neg_binary32 (-.f32 (*.f32 alpha alpha) 1)) (*.f32 (log.f32 alpha) 2)) (PI.f32)) (fma.f32 (fma.f32 alpha alpha -1) (*.f32 cosTheta cosTheta) 1)): 0 points increase in error, 6 points decrease in error
   (/.f32 (/.f32 (/.f32 (Rewrite<= /-rgt-identity_binary32 (/.f32 (-.f32 (*.f32 alpha alpha) 1) 1)) (*.f32 (log.f32 alpha) 2)) (PI.f32)) (fma.f32 (fma.f32 alpha alpha -1) (*.f32 cosTheta cosTheta) 1)): 6 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 (/.f32 (/.f32 (-.f32 (*.f32 alpha alpha) 1) 1) (Rewrite<= *-commutative_binary32 (*.f32 2 (log.f32 alpha)))) (PI.f32)) (fma.f32 (fma.f32 alpha alpha -1) (*.f32 cosTheta cosTheta) 1)): 6 points increase in error, 6 points decrease in error
   (/.f32 (/.f32 (/.f32 (/.f32 (-.f32 (*.f32 alpha alpha) 1) 1) (Rewrite<= count-2_binary32 (+.f32 (log.f32 alpha) (log.f32 alpha)))) (PI.f32)) (fma.f32 (fma.f32 alpha alpha -1) (*.f32 cosTheta cosTheta) 1)): 0 points increase in error, 6 points decrease in error
   (/.f32 (/.f32 (/.f32 (/.f32 (-.f32 (*.f32 alpha alpha) 1) 1) (Rewrite<= log-prod_binary32 (log.f32 (*.f32 alpha alpha)))) (PI.f32)) (fma.f32 (fma.f32 alpha alpha -1) (*.f32 cosTheta cosTheta) 1)): 16 points increase in error, 0 points decrease in error
   (/.f32 (Rewrite<= associate-/r*_binary32 (/.f32 (/.f32 (-.f32 (*.f32 alpha alpha) 1) 1) (*.f32 (log.f32 (*.f32 alpha alpha)) (PI.f32)))) (fma.f32 (fma.f32 alpha alpha -1) (*.f32 cosTheta cosTheta) 1)): 0 points increase in error, 16 points decrease in error
   (/.f32 (/.f32 (/.f32 (-.f32 (*.f32 alpha alpha) 1) 1) (Rewrite<= *-commutative_binary32 (*.f32 (PI.f32) (log.f32 (*.f32 alpha alpha))))) (fma.f32 (fma.f32 alpha alpha -1) (*.f32 cosTheta cosTheta) 1)): 6 points increase in error, 0 points decrease in error
   (/.f32 (Rewrite<= associate-/r*_binary32 (/.f32 (-.f32 (*.f32 alpha alpha) 1) (*.f32 1 (*.f32 (PI.f32) (log.f32 (*.f32 alpha alpha)))))) (fma.f32 (fma.f32 alpha alpha -1) (*.f32 cosTheta cosTheta) 1)): 0 points increase in error, 6 points decrease in error
   (/.f32 (/.f32 (Rewrite=> difference-of-sqr-1_binary32 (*.f32 (+.f32 alpha 1) (-.f32 alpha 1))) (*.f32 1 (*.f32 (PI.f32) (log.f32 (*.f32 alpha alpha))))) (fma.f32 (fma.f32 alpha alpha -1) (*.f32 cosTheta cosTheta) 1)): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 (Rewrite=> *-commutative_binary32 (*.f32 (-.f32 alpha 1) (+.f32 alpha 1))) (*.f32 1 (*.f32 (PI.f32) (log.f32 (*.f32 alpha alpha))))) (fma.f32 (fma.f32 alpha alpha -1) (*.f32 cosTheta cosTheta) 1)): 10 points increase in error, 0 points decrease in error
   (/.f32 (Rewrite=> times-frac_binary32 (*.f32 (/.f32 (-.f32 alpha 1) 1) (/.f32 (+.f32 alpha 1) (*.f32 (PI.f32) (log.f32 (*.f32 alpha alpha)))))) (fma.f32 (fma.f32 alpha alpha -1) (*.f32 cosTheta cosTheta) 1)): 0 points increase in error, 0 points decrease in error
   (/.f32 (*.f32 (/.f32 (-.f32 alpha 1) 1) (/.f32 (+.f32 alpha 1) (*.f32 (PI.f32) (log.f32 (*.f32 alpha alpha))))) (fma.f32 (fma.f32 alpha alpha (Rewrite<= metadata-eval (neg.f32 1))) (*.f32 cosTheta cosTheta) 1)): 0 points increase in error, 10 points decrease in error
   (/.f32 (*.f32 (/.f32 (-.f32 alpha 1) 1) (/.f32 (+.f32 alpha 1) (*.f32 (PI.f32) (log.f32 (*.f32 alpha alpha))))) (fma.f32 (Rewrite<= fma-neg_binary32 (-.f32 (*.f32 alpha alpha) 1)) (*.f32 cosTheta cosTheta) 1)): 7 points increase in error, 0 points decrease in error
   (/.f32 (*.f32 (/.f32 (-.f32 alpha 1) 1) (/.f32 (+.f32 alpha 1) (*.f32 (PI.f32) (log.f32 (*.f32 alpha alpha))))) (Rewrite<= fma-def_binary32 (+.f32 (*.f32 (-.f32 (*.f32 alpha alpha) 1) (*.f32 cosTheta cosTheta)) 1))): 16 points increase in error, 2 points decrease in error
   (/.f32 (*.f32 (/.f32 (-.f32 alpha 1) 1) (/.f32 (+.f32 alpha 1) (*.f32 (PI.f32) (log.f32 (*.f32 alpha alpha))))) (+.f32 (Rewrite<= associate-*l*_binary32 (*.f32 (*.f32 (-.f32 (*.f32 alpha alpha) 1) cosTheta) cosTheta)) 1)): 0 points increase in error, 16 points decrease in error
   (/.f32 (*.f32 (/.f32 (-.f32 alpha 1) 1) (/.f32 (+.f32 alpha 1) (*.f32 (PI.f32) (log.f32 (*.f32 alpha alpha))))) (Rewrite<= +-commutative_binary32 (+.f32 1 (*.f32 (*.f32 (-.f32 (*.f32 alpha alpha) 1) cosTheta) cosTheta)))): 10 points increase in error, 0 points decrease in error
   (/.f32 (Rewrite<= times-frac_binary32 (/.f32 (*.f32 (-.f32 alpha 1) (+.f32 alpha 1)) (*.f32 1 (*.f32 (PI.f32) (log.f32 (*.f32 alpha alpha)))))) (+.f32 1 (*.f32 (*.f32 (-.f32 (*.f32 alpha alpha) 1) cosTheta) cosTheta))): 6 points increase in error, 10 points decrease in error
   (/.f32 (/.f32 (Rewrite<= *-commutative_binary32 (*.f32 (+.f32 alpha 1) (-.f32 alpha 1))) (*.f32 1 (*.f32 (PI.f32) (log.f32 (*.f32 alpha alpha))))) (+.f32 1 (*.f32 (*.f32 (-.f32 (*.f32 alpha alpha) 1) cosTheta) cosTheta))): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 (Rewrite<= difference-of-sqr-1_binary32 (-.f32 (*.f32 alpha alpha) 1)) (*.f32 1 (*.f32 (PI.f32) (log.f32 (*.f32 alpha alpha))))) (+.f32 1 (*.f32 (*.f32 (-.f32 (*.f32 alpha alpha) 1) cosTheta) cosTheta))): 17 points increase in error, 6 points decrease in error
   (/.f32 (/.f32 (-.f32 (*.f32 alpha alpha) 1) (Rewrite=> *-lft-identity_binary32 (*.f32 (PI.f32) (log.f32 (*.f32 alpha alpha))))) (+.f32 1 (*.f32 (*.f32 (-.f32 (*.f32 alpha alpha) 1) cosTheta) cosTheta))): 0 points increase in error, 13 points decrease in error
   (Rewrite<= associate-/r*_binary32 (/.f32 (-.f32 (*.f32 alpha alpha) 1) (*.f32 (*.f32 (PI.f32) (log.f32 (*.f32 alpha alpha))) (+.f32 1 (*.f32 (*.f32 (-.f32 (*.f32 alpha alpha) 1) cosTheta) cosTheta))))): 7 points increase in error, 4 points decrease in error

3. Final simplification 0.5

   \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha \cdot 2}}{\pi}}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)} \]

Alternatives

Alternative 1
Error	0.4
Cost	10272

\[\begin{array}{l} t_0 := -1 + \alpha \cdot \alpha\\ \frac{t_0}{\log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right) \cdot \left(1 + cosTheta \cdot \left(cosTheta \cdot t_0\right)\right)} \end{array} \]

Alternative 2
Error	0.5
Cost	7104

\[\begin{array}{l} t_0 := -1 + \alpha \cdot \alpha\\ \frac{t_0}{\left(1 + cosTheta \cdot \left(cosTheta \cdot t_0\right)\right) \cdot \left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)} \end{array} \]

Alternative 3
Error	0.8
Cost	6912

\[\frac{1 - \alpha \cdot \alpha}{\left(\log \alpha \cdot \left(\pi \cdot -2\right)\right) \cdot \left(1 - cosTheta \cdot cosTheta\right)} \]

Alternative 4
Error	0.8
Cost	6912

\[\frac{-1 + \alpha \cdot \alpha}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 - cosTheta \cdot cosTheta\right)} \]

Alternative 5
Error	1.6
Cost	6784

\[\left(\alpha + -1\right) \cdot \frac{\frac{\alpha + 1}{\pi}}{\log \alpha \cdot 2} \]

Alternative 6
Error	10.7
Cost	6720

\[\frac{-0.5}{\log \alpha \cdot \left(\pi \cdot \left(1 - cosTheta \cdot cosTheta\right)\right)} \]

Alternative 7
Error	10.7
Cost	6720

\[\frac{-0.5}{\left(1 - cosTheta \cdot cosTheta\right) \cdot \left(\log \alpha \cdot \pi\right)} \]

Alternative 8
Error	1.5
Cost	6720

\[\frac{1 - \alpha \cdot \alpha}{\log \alpha \cdot \left(\pi \cdot -2\right)} \]

Alternative 9
Error	11.0
Cost	6592

\[\frac{-0.5}{\log \alpha} \cdot \frac{1}{\pi} \]

Alternative 10
Error	11.1
Cost	6528

\[\frac{-0.5}{\log \alpha \cdot \pi} \]

Alternative 11
Error	11.0
Cost	6528

\[\frac{\frac{-0.5}{\pi}}{\log \alpha} \]

Alternative 12
Error	29.9
Cost	32

\[0 \]

Reproduce

herbie shell --seed 2022343 
(FPCore (cosTheta alpha)
  :name "GTR1 distribution"
  :precision binary32
  :pre (and (and (<= 0.0 cosTheta) (<= cosTheta 1.0)) (and (<= 0.0001 alpha) (<= alpha 1.0)))
  (/ (- (* alpha alpha) 1.0) (* (* PI (log (* alpha alpha))) (+ 1.0 (* (* (- (* alpha alpha) 1.0) cosTheta) cosTheta)))))