GTR1 distribution

Average Accuracy: 98.5% → 98.5%

Time: 15.3s

Precision: binary32

Cost: 10240

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

Math

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

↓

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

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
 (/
  (/ (/ (* -0.5 (fma alpha alpha -1.0)) (log alpha)) PI)
  (- -1.0 (* cosTheta (- (* cosTheta (* alpha alpha)) cosTheta)))))

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 (((-0.5f * fmaf(alpha, alpha, -1.0f)) / logf(alpha)) / ((float) M_PI)) / (-1.0f - (cosTheta * ((cosTheta * (alpha * alpha)) - cosTheta)));
}

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(Float32(Float32(-0.5) * fma(alpha, alpha, Float32(-1.0))) / log(alpha)) / Float32(pi)) / Float32(Float32(-1.0) - Float32(cosTheta * Float32(Float32(cosTheta * Float32(alpha * alpha)) - cosTheta))))
end

Derivation

1. Initial program 98.5%

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

2. Simplified 98.5%

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

   [Start] 98.5	\[ \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
   associate-/r* [=>] 98.5	\[ \color{blue}{\frac{\frac{\alpha \cdot \alpha - 1}{\pi \cdot \log \left(\alpha \cdot \alpha\right)}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta}} \]
   difference-of-sqr-1 [=>] 98.1	\[ \frac{\frac{\color{blue}{\left(\alpha + 1\right) \cdot \left(\alpha - 1\right)}}{\pi \cdot \log \left(\alpha \cdot \alpha\right)}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]
   *-commutative [=>] 98.1	\[ \frac{\frac{\color{blue}{\left(\alpha - 1\right) \cdot \left(\alpha + 1\right)}}{\pi \cdot \log \left(\alpha \cdot \alpha\right)}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]
   *-lft-identity [<=] 98.1	\[ \frac{\frac{\left(\alpha - 1\right) \cdot \left(\alpha + 1\right)}{\color{blue}{1 \cdot \left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)}}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]
   times-frac [=>] 98.0	\[ \frac{\color{blue}{\frac{\alpha - 1}{1} \cdot \frac{\alpha + 1}{\pi \cdot \log \left(\alpha \cdot \alpha\right)}}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]

3. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha}}{\pi} \cdot \frac{1}{-1 - \mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot \left(cosTheta \cdot cosTheta\right)}} \]

4. Simplified 98.5%

   \[\leadsto \color{blue}{\frac{\frac{\frac{-0.5 \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha}}{\pi}}{-1 - cosTheta \cdot \left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right)\right)}} \]
   Proof

   [Start] 98.5	\[ \frac{-0.5 \cdot \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha}}{\pi} \cdot \frac{1}{-1 - \mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot \left(cosTheta \cdot cosTheta\right)} \]
   *-commutative [<=] 98.5	\[ \color{blue}{\frac{1}{-1 - \mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot \left(cosTheta \cdot cosTheta\right)} \cdot \frac{-0.5 \cdot \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha}}{\pi}} \]
   associate-*l/ [=>] 98.5	\[ \color{blue}{\frac{1 \cdot \frac{-0.5 \cdot \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha}}{\pi}}{-1 - \mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot \left(cosTheta \cdot cosTheta\right)}} \]
   *-lft-identity [=>] 98.5	\[ \frac{\color{blue}{\frac{-0.5 \cdot \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha}}{\pi}}}{-1 - \mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot \left(cosTheta \cdot cosTheta\right)} \]
   associate-*r/ [=>] 98.5	\[ \frac{\frac{\color{blue}{\frac{-0.5 \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha}}}{\pi}}{-1 - \mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot \left(cosTheta \cdot cosTheta\right)} \]
   *-commutative [=>] 98.5	\[ \frac{\frac{\frac{-0.5 \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha}}{\pi}}{-1 - \color{blue}{\left(cosTheta \cdot cosTheta\right) \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right)}} \]
   associate-*l* [=>] 98.5	\[ \frac{\frac{\frac{-0.5 \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha}}{\pi}}{-1 - \color{blue}{cosTheta \cdot \left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right)\right)}} \]

5. Applied egg-rr 98.5%

   \[\leadsto \frac{\frac{\frac{-0.5 \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha}}{\pi}}{-1 - cosTheta \cdot \color{blue}{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta + -1 \cdot cosTheta\right)}} \]

6. Final simplification 98.5%

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

Alternatives

Alternative 1
Accuracy	98.7%
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
Accuracy	98.5%
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
Accuracy	97.6%
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 4
Accuracy	95.2%
Cost	6720

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

Alternative 5
Accuracy	66.2%
Cost	6528

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

Alternative 6
Accuracy	66.2%
Cost	6528

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

Reproduce

herbie shell --seed 2023129 
(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)))))