Average Error: 0.7 → 0.4
Time: 5.3s
Precision: binary32
\[\left(0 < cosTheta \land cosTheta < 0.9999\right) \land \left(-1 < c \land c < 1\right)\]
\[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
↓
\[\begin{array}{l} t_0 := \frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}\\ t_1 := \sqrt{t_0}\\ t_2 := cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}\\ {\left(1 + \left(\frac{t_1}{t_2} + c\right)\right)}^{-0.5} \cdot {\left(1 + \left(c + \frac{\sqrt[3]{t_1} \cdot \sqrt[3]{t_0}}{t_2}\right)\right)}^{-0.5} \end{array} \]
(FPCore (cosTheta c)
 :precision binary32
 (/
  1.0
  (+
   (+ 1.0 c)
   (*
    (* (/ 1.0 (sqrt PI)) (/ (sqrt (- (- 1.0 cosTheta) cosTheta)) cosTheta))
    (exp (* (- cosTheta) cosTheta))))))
↓
(FPCore (cosTheta c)
 :precision binary32
 (let* ((t_0 (/ (fma cosTheta -2.0 1.0) PI))
        (t_1 (sqrt t_0))
        (t_2 (* cosTheta (pow (exp cosTheta) cosTheta))))
   (*
    (pow (+ 1.0 (+ (/ t_1 t_2) c)) -0.5)
    (pow (+ 1.0 (+ c (/ (* (cbrt t_1) (cbrt t_0)) t_2))) -0.5))))
float code(float cosTheta, float c) {
	return 1.0f / ((1.0f + c) + (((1.0f / sqrtf(((float) M_PI))) * (sqrtf(((1.0f - cosTheta) - cosTheta)) / cosTheta)) * expf((-cosTheta * cosTheta))));
}
↓
float code(float cosTheta, float c) {
	float t_0 = fmaf(cosTheta, -2.0f, 1.0f) / ((float) M_PI);
	float t_1 = sqrtf(t_0);
	float t_2 = cosTheta * powf(expf(cosTheta), cosTheta);
	return powf((1.0f + ((t_1 / t_2) + c)), -0.5f) * powf((1.0f + (c + ((cbrtf(t_1) * cbrtf(t_0)) / t_2))), -0.5f);
}

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32(Float32(Float32(Float32(1.0) / sqrt(Float32(pi))) * Float32(sqrt(Float32(Float32(Float32(1.0) - cosTheta) - cosTheta)) / cosTheta)) * exp(Float32(Float32(-cosTheta) * cosTheta)))))
end
↓
function code(cosTheta, c)
	t_0 = Float32(fma(cosTheta, Float32(-2.0), Float32(1.0)) / Float32(pi))
	t_1 = sqrt(t_0)
	t_2 = Float32(cosTheta * (exp(cosTheta) ^ cosTheta))
	return Float32((Float32(Float32(1.0) + Float32(Float32(t_1 / t_2) + c)) ^ Float32(-0.5)) * (Float32(Float32(1.0) + Float32(c + Float32(Float32(cbrt(t_1) * cbrt(t_0)) / t_2))) ^ Float32(-0.5)))
end

\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}}
↓
\begin{array}{l}
t_0 := \frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}\\
t_1 := \sqrt{t_0}\\
t_2 := cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}\\
{\left(1 + \left(\frac{t_1}{t_2} + c\right)\right)}^{-0.5} \cdot {\left(1 + \left(c + \frac{\sqrt[3]{t_1} \cdot \sqrt[3]{t_0}}{t_2}\right)\right)}^{-0.5}
\end{array}

Error

Derivation

  1. Initial program 0.7

    \[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]

  2. Simplified0.5

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

  3. Applied egg-rr0.7

    \[\leadsto \color{blue}{{\left(1 + \left(\frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}} + c\right)\right)}^{-0.5} \cdot {\left(1 + \left(\frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}} + c\right)\right)}^{-0.5}} \]

  4. Applied egg-rr0.4

    \[\leadsto {\left(1 + \left(\frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}} + c\right)\right)}^{-0.5} \cdot {\left(1 + \left(\frac{\color{blue}{\sqrt[3]{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}} \cdot \sqrt[3]{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}}{cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}} + c\right)\right)}^{-0.5} \]

  5. Final simplification0.4

    \[\leadsto {\left(1 + \left(\frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}} + c\right)\right)}^{-0.5} \cdot {\left(1 + \left(c + \frac{\sqrt[3]{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}} \cdot \sqrt[3]{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}}\right)\right)}^{-0.5} \]

Reproduce

herbie shell --seed 2022192 
(FPCore (cosTheta c)
  :name "Beckmann Sample, normalization factor"
  :precision binary32
  :pre (and (and (< 0.0 cosTheta) (< cosTheta 0.9999)) (and (< -1.0 c) (< c 1.0)))
  (/ 1.0 (+ (+ 1.0 c) (* (* (/ 1.0 (sqrt PI)) (/ (sqrt (- (- 1.0 cosTheta) cosTheta)) cosTheta)) (exp (* (- cosTheta) cosTheta))))))