Average Error: 0.5 → 0.4
Time: 10.4s
Precision: binary32
\[\left(0 \leq cosTheta \land cosTheta \leq 1\right) \land \left(0.0001 \leq \alpha \land \alpha \leq 1\right)\]
\[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
↓
\[\begin{array}{l} t_0 := \alpha \cdot \alpha - 1\\ t_1 := {\left({\left(\sqrt{\pi}\right)}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)}\\ \sqrt[3]{\left(t_0 \cdot \left(t_0 \cdot t_0\right)\right) \cdot \frac{1}{\left(t_1 \cdot \left(t_1 \cdot \left(8 \cdot {\log \alpha}^{3}\right)\right)\right) \cdot {\left(\mathsf{fma}\left(cosTheta, cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), 1\right)\right)}^{3}}} \end{array} \]
\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 := \alpha \cdot \alpha - 1\\
t_1 := {\left({\left(\sqrt{\pi}\right)}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)}\\
\sqrt[3]{\left(t_0 \cdot \left(t_0 \cdot t_0\right)\right) \cdot \frac{1}{\left(t_1 \cdot \left(t_1 \cdot \left(8 \cdot {\log \alpha}^{3}\right)\right)\right) \cdot {\left(\mathsf{fma}\left(cosTheta, cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), 1\right)\right)}^{3}}}
\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 (- (* alpha alpha) 1.0))
        (t_1 (pow (pow (sqrt PI) (sqrt 3.0)) (sqrt 3.0))))
   (cbrt
    (*
     (* t_0 (* t_0 t_0))
     (/
      1.0
      (*
       (* t_1 (* t_1 (* 8.0 (pow (log alpha) 3.0))))
       (pow (fma cosTheta (* cosTheta (fma alpha alpha -1.0)) 1.0) 3.0)))))))
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 = (alpha * alpha) - 1.0f;
	float t_1 = powf(powf(sqrtf((float) M_PI), sqrtf(3.0f)), sqrtf(3.0f));
	return cbrtf((t_0 * (t_0 * t_0)) * (1.0f / ((t_1 * (t_1 * (8.0f * powf(logf(alpha), 3.0f)))) * powf(fmaf(cosTheta, (cosTheta * fmaf(alpha, alpha, -1.0f)), 1.0f), 3.0f))));
}

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. Applied add-cbrt-cube_binary320.5

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

  3. Applied add-cbrt-cube_binary320.5

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

  4. Applied add-cbrt-cube_binary320.5

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

  5. Applied cbrt-unprod_binary320.5

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

  6. Applied cbrt-unprod_binary320.5

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

  7. Applied add-cbrt-cube_binary320.5

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

  8. Applied cbrt-undiv_binary320.4

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

  9. Applied div-inv_binary320.5

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

  10. Simplified0.5

    \[\leadsto \sqrt[3]{\left(\left(\left(\alpha \cdot \alpha - 1\right) \cdot \left(\alpha \cdot \alpha - 1\right)\right) \cdot \left(\alpha \cdot \alpha - 1\right)\right) \cdot \color{blue}{\frac{1}{\left({\pi}^{3} \cdot \left(8 \cdot {\log \alpha}^{3}\right)\right) \cdot {\left(\mathsf{fma}\left(cosTheta, cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), 1\right)\right)}^{3}}}} \]

  11. Applied add-sqr-sqrt_binary320.5

    \[\leadsto \sqrt[3]{\left(\left(\left(\alpha \cdot \alpha - 1\right) \cdot \left(\alpha \cdot \alpha - 1\right)\right) \cdot \left(\alpha \cdot \alpha - 1\right)\right) \cdot \frac{1}{\left({\pi}^{\color{blue}{\left(\sqrt{3} \cdot \sqrt{3}\right)}} \cdot \left(8 \cdot {\log \alpha}^{3}\right)\right) \cdot {\left(\mathsf{fma}\left(cosTheta, cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), 1\right)\right)}^{3}}} \]

  12. Applied pow-unpow_binary320.4

    \[\leadsto \sqrt[3]{\left(\left(\left(\alpha \cdot \alpha - 1\right) \cdot \left(\alpha \cdot \alpha - 1\right)\right) \cdot \left(\alpha \cdot \alpha - 1\right)\right) \cdot \frac{1}{\left(\color{blue}{{\left({\pi}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)}} \cdot \left(8 \cdot {\log \alpha}^{3}\right)\right) \cdot {\left(\mathsf{fma}\left(cosTheta, cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), 1\right)\right)}^{3}}} \]

  13. Applied add-sqr-sqrt_binary320.4

    \[\leadsto \sqrt[3]{\left(\left(\left(\alpha \cdot \alpha - 1\right) \cdot \left(\alpha \cdot \alpha - 1\right)\right) \cdot \left(\alpha \cdot \alpha - 1\right)\right) \cdot \frac{1}{\left({\left({\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right)}}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)} \cdot \left(8 \cdot {\log \alpha}^{3}\right)\right) \cdot {\left(\mathsf{fma}\left(cosTheta, cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), 1\right)\right)}^{3}}} \]

  14. Applied unpow-prod-down_binary320.4

    \[\leadsto \sqrt[3]{\left(\left(\left(\alpha \cdot \alpha - 1\right) \cdot \left(\alpha \cdot \alpha - 1\right)\right) \cdot \left(\alpha \cdot \alpha - 1\right)\right) \cdot \frac{1}{\left({\color{blue}{\left({\left(\sqrt{\pi}\right)}^{\left(\sqrt{3}\right)} \cdot {\left(\sqrt{\pi}\right)}^{\left(\sqrt{3}\right)}\right)}}^{\left(\sqrt{3}\right)} \cdot \left(8 \cdot {\log \alpha}^{3}\right)\right) \cdot {\left(\mathsf{fma}\left(cosTheta, cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), 1\right)\right)}^{3}}} \]

  15. Applied unpow-prod-down_binary320.4

    \[\leadsto \sqrt[3]{\left(\left(\left(\alpha \cdot \alpha - 1\right) \cdot \left(\alpha \cdot \alpha - 1\right)\right) \cdot \left(\alpha \cdot \alpha - 1\right)\right) \cdot \frac{1}{\left(\color{blue}{\left({\left({\left(\sqrt{\pi}\right)}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)} \cdot {\left({\left(\sqrt{\pi}\right)}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)}\right)} \cdot \left(8 \cdot {\log \alpha}^{3}\right)\right) \cdot {\left(\mathsf{fma}\left(cosTheta, cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), 1\right)\right)}^{3}}} \]

  16. Applied associate-*l*_binary320.4

    \[\leadsto \sqrt[3]{\left(\left(\left(\alpha \cdot \alpha - 1\right) \cdot \left(\alpha \cdot \alpha - 1\right)\right) \cdot \left(\alpha \cdot \alpha - 1\right)\right) \cdot \frac{1}{\color{blue}{\left({\left({\left(\sqrt{\pi}\right)}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)} \cdot \left({\left({\left(\sqrt{\pi}\right)}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)} \cdot \left(8 \cdot {\log \alpha}^{3}\right)\right)\right)} \cdot {\left(\mathsf{fma}\left(cosTheta, cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), 1\right)\right)}^{3}}} \]

  17. Final simplification0.4

    \[\leadsto \sqrt[3]{\left(\left(\alpha \cdot \alpha - 1\right) \cdot \left(\left(\alpha \cdot \alpha - 1\right) \cdot \left(\alpha \cdot \alpha - 1\right)\right)\right) \cdot \frac{1}{\left({\left({\left(\sqrt{\pi}\right)}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)} \cdot \left({\left({\left(\sqrt{\pi}\right)}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)} \cdot \left(8 \cdot {\log \alpha}^{3}\right)\right)\right) \cdot {\left(\mathsf{fma}\left(cosTheta, cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), 1\right)\right)}^{3}}} \]

Reproduce

herbie shell --seed 2022082 
(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)))))