Average Error: 0.5 → 0.4
Time: 21.0s
Precision: binary32
\[\left(0 \leq cosTheta \land cosTheta \leq 1\right) \land \left(0.0001 \leq \alpha \land \alpha \leq 1\right)\]
\[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
↓
\[\sqrt[3]{\frac{{\left(\mathsf{fma}\left(\alpha, \alpha, -1\right)\right)}^{3}}{\left({\left({\pi}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)} \cdot {\left(2 \cdot \log \alpha\right)}^{3}\right) \cdot {\left(\mathsf{fma}\left(cosTheta, \mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot cosTheta, 1\right)\right)}^{3}}} \]
\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)}
↓
\sqrt[3]{\frac{{\left(\mathsf{fma}\left(\alpha, \alpha, -1\right)\right)}^{3}}{\left({\left({\pi}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)} \cdot {\left(2 \cdot \log \alpha\right)}^{3}\right) \cdot {\left(\mathsf{fma}\left(cosTheta, \mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot cosTheta, 1\right)\right)}^{3}}}

(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
 (cbrt
  (/
   (pow (fma alpha alpha -1.0) 3.0)
   (*
    (* (pow (pow PI (sqrt 3.0)) (sqrt 3.0)) (pow (* 2.0 (log alpha)) 3.0))
    (pow (fma cosTheta (* (fma alpha alpha -1.0) cosTheta) 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) {
	return cbrtf(powf(fmaf(alpha, alpha, -1.0f), 3.0f) / ((powf(powf(((float) M_PI), sqrtf(3.0f)), sqrtf(3.0f)) * powf((2.0f * logf(alpha)), 3.0f)) * powf(fmaf(cosTheta, (fmaf(alpha, alpha, -1.0f) * cosTheta), 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. Simplified0.4

    \[\leadsto \sqrt[3]{\color{blue}{\frac{{\left(\mathsf{fma}\left(\alpha, \alpha, -1\right)\right)}^{3}}{\left({\pi}^{3} \cdot {\left(2 \cdot \log \alpha\right)}^{3}\right) \cdot {\left(\mathsf{fma}\left(cosTheta, cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), 1\right)\right)}^{3}}}} \]

  10. Applied add-sqr-sqrt_binary320.4

    \[\leadsto \sqrt[3]{\frac{{\left(\mathsf{fma}\left(\alpha, \alpha, -1\right)\right)}^{3}}{\left({\pi}^{\color{blue}{\left(\sqrt{3} \cdot \sqrt{3}\right)}} \cdot {\left(2 \cdot \log \alpha\right)}^{3}\right) \cdot {\left(\mathsf{fma}\left(cosTheta, cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), 1\right)\right)}^{3}}} \]

  11. Applied pow-unpow_binary320.4

    \[\leadsto \sqrt[3]{\frac{{\left(\mathsf{fma}\left(\alpha, \alpha, -1\right)\right)}^{3}}{\left(\color{blue}{{\left({\pi}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)}} \cdot {\left(2 \cdot \log \alpha\right)}^{3}\right) \cdot {\left(\mathsf{fma}\left(cosTheta, cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), 1\right)\right)}^{3}}} \]

  12. Final simplification0.4

    \[\leadsto \sqrt[3]{\frac{{\left(\mathsf{fma}\left(\alpha, \alpha, -1\right)\right)}^{3}}{\left({\left({\pi}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)} \cdot {\left(2 \cdot \log \alpha\right)}^{3}\right) \cdot {\left(\mathsf{fma}\left(cosTheta, \mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot cosTheta, 1\right)\right)}^{3}}} \]

Reproduce

herbie shell --seed 2022055 
(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)))))