GTR1 distribution

Average Error: 0.5 → 0.5

Time: 16.7s

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

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

↓

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

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)
  (+ (* (* cosTheta cosTheta) (+ -1.0 (* alpha alpha))) 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)) / (((cosTheta * cosTheta) * (-1.0f + (alpha * alpha))) + 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)) / Float32(Float32(Float32(cosTheta * cosTheta) * Float32(Float32(-1.0) + Float32(alpha * alpha))) + 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

   [Start] 0.5	\[ \frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
   associate-/r* [=>] 0.5	\[ \color{blue}{\frac{\frac{\alpha \cdot \alpha - 1}{\pi \cdot \log \left(\alpha \cdot \alpha\right)}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta}} \]
   difference-of-sqr-1 [=>] 0.6	\[ \frac{\frac{\color{blue}{\left(\alpha + 1\right) \cdot \left(\alpha - 1\right)}}{\pi \cdot \log \left(\alpha \cdot \alpha\right)}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]
   *-commutative [=>] 0.6	\[ \frac{\frac{\color{blue}{\left(\alpha - 1\right) \cdot \left(\alpha + 1\right)}}{\pi \cdot \log \left(\alpha \cdot \alpha\right)}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]
   *-lft-identity [<=] 0.6	\[ \frac{\frac{\left(\alpha - 1\right) \cdot \left(\alpha + 1\right)}{\color{blue}{1 \cdot \left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)}}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]
   times-frac [=>] 0.6	\[ \frac{\color{blue}{\frac{\alpha - 1}{1} \cdot \frac{\alpha + 1}{\pi \cdot \log \left(\alpha \cdot \alpha\right)}}}{1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \]

3. Applied egg-rr 0.5

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

4. Applied egg-rr 0.5

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

5. Applied egg-rr 0.5

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

6. Taylor expanded in cosTheta around 0 0.5

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

7. Simplified 0.5

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

   [Start] 0.5	\[ \frac{\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha \cdot 2}}{\pi}}{\left({\alpha}^{2} - 1\right) \cdot {cosTheta}^{2} + 1} \]
   sub-neg [=>] 0.5	\[ \frac{\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha \cdot 2}}{\pi}}{\color{blue}{\left({\alpha}^{2} + \left(-1\right)\right)} \cdot {cosTheta}^{2} + 1} \]
   unpow2 [=>] 0.5	\[ \frac{\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha \cdot 2}}{\pi}}{\left(\color{blue}{\alpha \cdot \alpha} + \left(-1\right)\right) \cdot {cosTheta}^{2} + 1} \]
   metadata-eval [=>] 0.5	\[ \frac{\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha \cdot 2}}{\pi}}{\left(\alpha \cdot \alpha + \color{blue}{-1}\right) \cdot {cosTheta}^{2} + 1} \]
   unpow2 [=>] 0.5	\[ \frac{\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \alpha \cdot 2}}{\pi}}{\left(\alpha \cdot \alpha + -1\right) \cdot \color{blue}{\left(cosTheta \cdot cosTheta\right)} + 1} \]

8. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.5
Cost	7104

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

Alternative 2
Error	0.5
Cost	7104

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

Alternative 3
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 4
Error	1.7
Cost	6784

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

Alternative 5
Error	1.7
Cost	6784

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

Alternative 6
Error	1.7
Cost	6784

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

Alternative 7
Error	1.7
Cost	6784

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

Alternative 8
Error	10.6
Cost	6720

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

Alternative 9
Error	10.6
Cost	6720

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

Alternative 10
Error	10.6
Cost	6720

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

Alternative 11
Error	10.6
Cost	6720

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

Alternative 12
Error	10.6
Cost	6720

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

Alternative 13
Error	11.1
Cost	6528

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

Alternative 14
Error	11.1
Cost	6528

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

Reproduce

herbie shell --seed 2023073 
(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)))))