Average Error: 0.5 → 0.4
Time: 5.3s
Precision: binary32
\[\left(0 \leq cosTheta \land cosTheta \leq 1\right) \land \left(0.0001 \leq \alpha \land \alpha \leq 1\right)\]
\[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
↓
\[\frac{-\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \left({\alpha}^{\left(2 \cdot \pi\right)}\right) \cdot \left(-\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)\right)} \]
(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
 (/
  (- (fma alpha alpha -1.0))
  (*
   (log (pow alpha (* 2.0 PI)))
   (- (fma (fma alpha alpha -1.0) (* cosTheta cosTheta) 1.0)))))
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 -fmaf(alpha, alpha, -1.0f) / (logf(powf(alpha, (2.0f * ((float) M_PI)))) * -fmaf(fmaf(alpha, alpha, -1.0f), (cosTheta * cosTheta), 1.0f));
}

function code(cosTheta, alpha)
	return Float32(Float32(Float32(alpha * alpha) - Float32(1.0)) / Float32(Float32(Float32(pi) * log(Float32(alpha * alpha))) * Float32(Float32(1.0) + Float32(Float32(Float32(Float32(alpha * alpha) - Float32(1.0)) * cosTheta) * cosTheta))))
end
↓
function code(cosTheta, alpha)
	return Float32(Float32(-fma(alpha, alpha, Float32(-1.0))) / Float32(log((alpha ^ Float32(Float32(2.0) * Float32(pi)))) * Float32(-fma(fma(alpha, alpha, Float32(-1.0)), Float32(cosTheta * cosTheta), Float32(1.0)))))
end

\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{-\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\log \left({\alpha}^{\left(2 \cdot \pi\right)}\right) \cdot \left(-\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)\right)}

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 egg-rr0.4

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

  3. Taylor expanded in alpha around 0 0.5

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

  4. Simplified0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2022166 
(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)))))