Average Error: 0.5 → 0.4
Time: 7.9s
Precision: binary32
\[0 \leq cosTheta \land cosTheta \leq 1 \land 0.0001 \leq \alpha \land \alpha \leq 1\]
\[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{cosTheta \cdot \left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right)\right)}\\ \frac{\alpha \cdot \alpha - 1}{\log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right) \cdot \left(1 + t_0 \cdot \left(t_0 \cdot t_0\right)\right)} \end{array} \]
\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 := \sqrt[3]{cosTheta \cdot \left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right)\right)}\\
\frac{\alpha \cdot \alpha - 1}{\log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right) \cdot \left(1 + t_0 \cdot \left(t_0 \cdot t_0\right)\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 (cbrt (* cosTheta (* cosTheta (fma alpha alpha -1.0))))))
   (/
    (- (* alpha alpha) 1.0)
    (* (log (pow (* alpha alpha) PI)) (+ 1.0 (* t_0 (* t_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 = cbrtf(cosTheta * (cosTheta * fmaf(alpha, alpha, -1.0f)));
	return ((alpha * alpha) - 1.0f) / (logf(powf((alpha * alpha), ((float) M_PI))) * (1.0f + (t_0 * (t_0 * t_0))));
}

Error

Bits error versus cosTheta

Bits error versus alpha

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. Using strategy rm

  3. Applied add-log-exp_binary320.5

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

  4. Simplified0.4

    \[\leadsto \frac{\alpha \cdot \alpha - 1}{\log \color{blue}{\left({\left(\alpha \cdot \alpha\right)}^{\pi}\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
  5. Using strategy rm

  6. Applied add-cube-cbrt_binary320.4

    \[\leadsto \frac{\alpha \cdot \alpha - 1}{\log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right) \cdot \left(1 + \color{blue}{\left(\sqrt[3]{\left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta} \cdot \sqrt[3]{\left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta}\right) \cdot \sqrt[3]{\left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta}}\right)} \]

  7. Simplified0.4

    \[\leadsto \frac{\alpha \cdot \alpha - 1}{\log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right) \cdot \left(1 + \color{blue}{\left(\sqrt[3]{cosTheta \cdot \left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right)\right)} \cdot \sqrt[3]{cosTheta \cdot \left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right)\right)}\right)} \cdot \sqrt[3]{\left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta}\right)} \]

  8. Simplified0.4

    \[\leadsto \frac{\alpha \cdot \alpha - 1}{\log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right) \cdot \left(1 + \left(\sqrt[3]{cosTheta \cdot \left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right)\right)} \cdot \sqrt[3]{cosTheta \cdot \left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right)\right)}\right) \cdot \color{blue}{\sqrt[3]{cosTheta \cdot \left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right)\right)}}\right)} \]

  9. Final simplification0.4

    \[\leadsto \frac{\alpha \cdot \alpha - 1}{\log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right) \cdot \left(1 + \sqrt[3]{cosTheta \cdot \left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right)\right)} \cdot \left(\sqrt[3]{cosTheta \cdot \left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right)\right)} \cdot \sqrt[3]{cosTheta \cdot \left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right)\right)}\right)\right)} \]

Reproduce

herbie shell --seed 2021211 
(FPCore (cosTheta alpha)
  :name "GTR1 distribution"
  :precision binary32
  :pre (and (<= 0.0 cosTheta 1.0) (<= 0.0001 alpha 1.0))
  (/ (- (* alpha alpha) 1.0) (* (* PI (log (* alpha alpha))) (+ 1.0 (* (* (- (* alpha alpha) 1.0) cosTheta) cosTheta)))))