Average Error: 0.5 → 0.5
Time: 16.8s
Precision: binary32
Cost: 23296
\[\left(0 \leq cosTheta \land cosTheta \leq 1\right) \land \left(0.0001 \leq \alpha \land \alpha \leq 1\right)\]
\[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
↓
\[\begin{array}{l} t_0 := \mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot \left(cosTheta \cdot cosTheta\right)\\ \frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right)}}{1 - {t_0}^{2}} \cdot \left(1 - t_0\right) \end{array} \]
(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
 (let* ((t_0 (* (fma alpha alpha -1.0) (* cosTheta cosTheta))))
   (*
    (/
     (/ (fma alpha alpha -1.0) (log (pow (* alpha alpha) PI)))
     (- 1.0 (pow t_0 2.0)))
    (- 1.0 t_0))))
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) {
	float t_0 = fmaf(alpha, alpha, -1.0f) * (cosTheta * cosTheta);
	return ((fmaf(alpha, alpha, -1.0f) / logf(powf((alpha * alpha), ((float) M_PI)))) / (1.0f - powf(t_0, 2.0f))) * (1.0f - t_0);
}

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)
	t_0 = Float32(fma(alpha, alpha, Float32(-1.0)) * Float32(cosTheta * cosTheta))
	return Float32(Float32(Float32(fma(alpha, alpha, Float32(-1.0)) / log((Float32(alpha * alpha) ^ Float32(pi)))) / Float32(Float32(1.0) - (t_0 ^ Float32(2.0)))) * Float32(Float32(1.0) - t_0))
end

\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)}
↓
\begin{array}{l}
t_0 := \mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot \left(cosTheta \cdot cosTheta\right)\\
\frac{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right)}}{1 - {t_0}^{2}} \cdot \left(1 - t_0\right)
\end{array}

Error

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. Applied egg-rr0.5

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

  3. Final simplification0.5

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

Alternatives

Alternative 1
Error0.4
Cost10272
\[\begin{array}{l} t_0 := -1 + \alpha \cdot \alpha\\ \frac{t_0}{\log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right) \cdot \left(1 + cosTheta \cdot \left(cosTheta \cdot t_0\right)\right)} \end{array} \]
Alternative 2
Error0.5
Cost10240
\[\frac{-1 + \alpha \cdot \alpha}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot \left(cosTheta \cdot cosTheta\right)\right)} \]
Alternative 3
Error0.5
Cost7104
\[\begin{array}{l} t_0 := -1 + \alpha \cdot \alpha\\ \frac{t_0}{\left(1 + cosTheta \cdot \left(cosTheta \cdot t_0\right)\right) \cdot \left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right)} \end{array} \]
Alternative 4
Error0.5
Cost7104
\[\frac{-1 + \alpha \cdot \alpha}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + cosTheta \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta - cosTheta\right)\right)} \]
Alternative 5
Error0.8
Cost6912
\[\frac{-1 + \alpha \cdot \alpha}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 - cosTheta \cdot cosTheta\right)} \]
Alternative 6
Error1.7
Cost6784
\[0.5 \cdot \left(\frac{\alpha + 1}{\pi \cdot \log \alpha} \cdot \left(\alpha + -1\right)\right) \]
Alternative 7
Error1.6
Cost6720
\[0.5 \cdot \frac{-1 + \alpha \cdot \alpha}{\pi \cdot \log \alpha} \]
Alternative 8
Error11.1
Cost6592
\[0.5 \cdot \frac{\frac{-1}{\pi}}{\log \alpha} \]
Alternative 9
Error11.1
Cost6528
\[\frac{-0.5}{\pi \cdot \log \alpha} \]
Alternative 10
Error11.1
Cost6528
\[\frac{\frac{-0.5}{\log \alpha}}{\pi} \]

Error

Reproduce

herbie shell --seed 2022308 
(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)))))