Average Error: 0.5 → 0.5
Time: 6.7s
Precision: binary32
\[\left(0 \leq cosTheta \land cosTheta \leq 1\right) \land \left(0.0001 \leq \alpha \land \alpha \leq 1\right)\]
\[\frac{\alpha \cdot \alpha - 1}{\left(\pi \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
↓
\[\frac{-\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\mathsf{fma}\left(cosTheta, \mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot cosTheta, 1\right) \cdot \left(\pi \cdot \left(\log \alpha \cdot -2\right)\right)} \]
(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))
  (*
   (fma cosTheta (* (fma alpha alpha -1.0) cosTheta) 1.0)
   (* PI (* (log alpha) -2.0)))))
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) / (fmaf(cosTheta, (fmaf(alpha, alpha, -1.0f) * cosTheta), 1.0f) * (((float) M_PI) * (logf(alpha) * -2.0f)));
}

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(fma(cosTheta, Float32(fma(alpha, alpha, Float32(-1.0)) * cosTheta), Float32(1.0)) * Float32(Float32(pi) * Float32(log(alpha) * Float32(-2.0)))))
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)}{\mathsf{fma}\left(cosTheta, \mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot cosTheta, 1\right) \cdot \left(\pi \cdot \left(\log \alpha \cdot -2\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-rr0.5

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

  3. Applied egg-rr0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2022151 
(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)))))