Average Error: 0.7 → 0.4
Time: 13.8s
Precision: binary32
\[\left(0 < cosTheta \land cosTheta < 0.9999\right) \land \left(-1 < c \land c < 1\right)\]
\[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
↓
\[\begin{array}{l} t_0 := \left(1 + c \cdot c\right) - c\\ t_1 := \sqrt[3]{\pi} \cdot \left(\left(cosTheta \cdot e^{cosTheta \cdot cosTheta}\right) \cdot \sqrt{\sqrt[3]{\pi}}\right)\\ \frac{t_0 \cdot t_1}{\mathsf{fma}\left(1 + {c}^{3}, t_1, t_0 \cdot \sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}\right)} \end{array} \]
\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 := \left(1 + c \cdot c\right) - c\\
t_1 := \sqrt[3]{\pi} \cdot \left(\left(cosTheta \cdot e^{cosTheta \cdot cosTheta}\right) \cdot \sqrt{\sqrt[3]{\pi}}\right)\\
\frac{t_0 \cdot t_1}{\mathsf{fma}\left(1 + {c}^{3}, t_1, t_0 \cdot \sqrt{\mathsf{fma}\left(cosTheta, -2, 1\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 (- (+ 1.0 (* c c)) c))
        (t_1
         (*
          (cbrt PI)
          (* (* cosTheta (exp (* cosTheta cosTheta))) (sqrt (cbrt PI))))))
   (/
    (* t_0 t_1)
    (fma (+ 1.0 (pow c 3.0)) t_1 (* t_0 (sqrt (fma cosTheta -2.0 1.0)))))))
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 = (1.0f + (c * c)) - c;
	float t_1 = cbrtf((float) M_PI) * ((cosTheta * expf(cosTheta * cosTheta)) * sqrtf(cbrtf((float) M_PI)));
	return (t_0 * t_1) / fmaf((1.0f + powf(c, 3.0f)), t_1, (t_0 * sqrtf(fmaf(cosTheta, -2.0f, 1.0f))));
}

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. Simplified0.5

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

  7. Applied flip3-+_binary320.5

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

  8. Applied frac-add_binary320.5

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

  9. Applied associate-/r/_binary320.4

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

  10. Applied *-un-lft-identity_binary320.4

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

  11. Applied *-un-lft-identity_binary320.4

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

  12. Applied times-frac_binary320.4

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

  13. Applied associate-*l*_binary320.4

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

  14. Simplified0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2022081 
(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))))))