Average Error: 0.5 → 0.4
Time: 7.8s
Precision: binary32
\[\left(0 \leq cosTheta \land cosTheta \leq 1\right) \land \left(0.0001 \leq \alpha \land \alpha \leq 1\right)\]
\[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
↓
\[\begin{array}{l} t_0 := \alpha \cdot \alpha - 1\\ \frac{t_0}{\sqrt[3]{\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \left(\sqrt[3]{\pi} \cdot {\pi}^{2}\right)\right) \cdot {\log \left(\alpha \cdot \alpha\right)}^{3}} \cdot \left(1 + cosTheta \cdot \left(t_0 \cdot cosTheta\right)\right)} \end{array} \]
\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 := \alpha \cdot \alpha - 1\\
\frac{t_0}{\sqrt[3]{\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \left(\sqrt[3]{\pi} \cdot {\pi}^{2}\right)\right) \cdot {\log \left(\alpha \cdot \alpha\right)}^{3}} \cdot \left(1 + cosTheta \cdot \left(t_0 \cdot cosTheta\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 (- (* alpha alpha) 1.0)))
   (/
    t_0
    (*
     (cbrt
      (*
       (* (pow (cbrt PI) 2.0) (* (cbrt PI) (pow PI 2.0)))
       (pow (log (* alpha alpha)) 3.0)))
     (+ 1.0 (* cosTheta (* t_0 cosTheta)))))))
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;
	return t_0 / (cbrtf(((powf(cbrtf(((float) M_PI)), 2.0f) * (cbrtf(((float) M_PI)) * powf(((float) M_PI), 2.0f))) * powf(logf((alpha * alpha)), 3.0f))) * (1.0f + (cosTheta * (t_0 * cosTheta))));
}

Error

Bits error versus cosTheta

Bits error versus alpha

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

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

  3. Applied egg-rr0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2022127 
(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)))))