Average Error: 0.7 → 0.4
Time: 28.2s
Precision: binary32
\[0 < cosTheta \land cosTheta < 0.9999 \land -1 < c \land c < 1\]
\[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
↓
\[\begin{array}{l} t_0 := \sqrt{\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}} \cdot \left(\sqrt{\sqrt[3]{\pi}} \cdot \left(cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}\right)\right)\\ \frac{1}{\left(1 - c \cdot c\right) \cdot t_0 + \left(1 - c\right) \cdot \sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}} \cdot \left(t_0 \cdot \left(1 - c\right)\right) \end{array} \]
\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 := \sqrt{\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}} \cdot \left(\sqrt{\sqrt[3]{\pi}} \cdot \left(cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}\right)\right)\\
\frac{1}{\left(1 - c \cdot c\right) \cdot t_0 + \left(1 - c\right) \cdot \sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}} \cdot \left(t_0 \cdot \left(1 - c\right)\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
         (*
          (sqrt (* (cbrt PI) (cbrt PI)))
          (* (sqrt (cbrt PI)) (* cosTheta (pow (exp cosTheta) cosTheta))))))
   (*
    (/
     1.0
     (+ (* (- 1.0 (* c c)) t_0) (* (- 1.0 c) (sqrt (fma cosTheta -2.0 1.0)))))
    (* t_0 (- 1.0 c)))))
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 = sqrtf(cbrtf((float) M_PI) * cbrtf((float) M_PI)) * (sqrtf(cbrtf((float) M_PI)) * (cosTheta * powf(expf(cosTheta), cosTheta)));
	return (1.0f / (((1.0f - (c * c)) * t_0) + ((1.0f - c) * sqrtf(fmaf(cosTheta, -2.0f, 1.0f))))) * (t_0 * (1.0f - c));
}

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

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

  3. Applied add-cube-cbrt_binary320.5

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

  4. Applied sqrt-prod_binary320.5

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

  5. Applied associate-*l*_binary320.5

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

  6. Applied flip-+_binary320.5

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

  7. Applied frac-add_binary320.5

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

  8. Applied associate-/r/_binary320.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2021215 
(FPCore (cosTheta c)
  :name "Beckmann Sample, normalization factor"
  :precision binary32
  :pre (and (< 0.0 cosTheta 0.9999) (< -1.0 c 1.0))
  (/ 1.0 (+ (+ 1.0 c) (* (* (/ 1.0 (sqrt PI)) (/ (sqrt (- (- 1.0 cosTheta) cosTheta)) cosTheta)) (exp (* (- cosTheta) cosTheta))))))