Average Error: 0.5 → 0.4
Time: 9.2s
Precision: binary32
\[0 \leq cosTheta \land cosTheta \leq 1 \land 0.0001 \leq \alpha \land \alpha \leq 1\]
\[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
↓
\[\begin{array}{l} t_0 := \alpha \cdot \alpha - 1\\ \frac{\frac{t_0}{\log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right)}}{1 + cosTheta \cdot \left(t_0 \cdot cosTheta\right)} \end{array} \]
\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{\frac{t_0}{\log \left({\left(\alpha \cdot \alpha\right)}^{\pi}\right)}}{1 + cosTheta \cdot \left(t_0 \cdot cosTheta\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 (log (pow (* alpha alpha) PI)))
    (+ 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 / logf(powf((alpha * alpha), ((float) M_PI)))) / (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. 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 associate-/r*_binary320.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2021204 
(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)))))