Average Error: 0.7 → 0.4
Time: 6.1s
Precision: binary32
\[\left(0 < cosTheta \land cosTheta < 0.9999\right) \land \left(-1 < c \land c < 1\right)\]
\[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
↓
\[\begin{array}{l} t_0 := \frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}\\ \frac{1}{1 + \left(c + \frac{\sqrt[3]{t_0}}{\frac{cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}}{\sqrt[3]{\sqrt{t_0}}}}\right)} \end{array} \]
(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)))
   (/
    1.0
    (+
     1.0
     (+
      c
      (/
       (cbrt t_0)
       (/ (* cosTheta (pow (exp cosTheta) cosTheta)) (cbrt (sqrt t_0)))))))))
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);
	return 1.0f / (1.0f + (c + (cbrtf(t_0) / ((cosTheta * powf(expf(cosTheta), cosTheta)) / cbrtf(sqrtf(t_0))))));
}

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))
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(c + Float32(cbrt(t_0) / Float32(Float32(cosTheta * (exp(cosTheta) ^ cosTheta)) / cbrt(sqrt(t_0)))))))
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}\\
\frac{1}{1 + \left(c + \frac{\sqrt[3]{t_0}}{\frac{cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}}{\sqrt[3]{\sqrt{t_0}}}}\right)}
\end{array}

Error

Bits error versus cosTheta

Bits error versus c

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

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

  4. Applied egg-rr0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2022153 
(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))))))