Average Error: 0.5 → 0.4
Time: 8.2s
Precision: binary32
\[0 \leq cosTheta \land cosTheta \leq 1 \land 0.0001 \leq \alpha \land \alpha \leq 1\]
\[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
↓
\[\frac{\alpha \cdot \alpha - 1}{\log \left({\alpha}^{\left(2 \cdot \pi\right)}\right) \cdot \left(1 + cosTheta \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta - cosTheta\right)\right)} \]
\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)}
↓
\frac{\alpha \cdot \alpha - 1}{\log \left({\alpha}^{\left(2 \cdot \pi\right)}\right) \cdot \left(1 + cosTheta \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta - cosTheta\right)\right)}

(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
 (/
  (- (* alpha alpha) 1.0)
  (*
   (log (pow alpha (* 2.0 PI)))
   (+ 1.0 (* cosTheta (- (* (* alpha alpha) cosTheta) 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) {
	return ((alpha * alpha) - 1.0f) / (logf(powf(alpha, (2.0f * ((float) M_PI)))) * (1.0f + (cosTheta * (((alpha * alpha) * cosTheta) - 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. Taylor expanded around 0 0.5

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

  3. Simplified0.5

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

  5. Applied add-log-exp_binary320.5

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

  6. Simplified0.4

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

  8. Applied pow2_binary320.4

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

  9. Applied pow-pow_binary320.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2021190 
(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)))))