Average Error: 0.7 → 0.4
Time: 13.7s
Precision: binary32
\[\left(0 < cosTheta \land cosTheta < 0.9999\right) \land \left(-1 < c \land c < 1\right)\]
\[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
↓
\[\frac{1}{\left(1 + c\right) + \frac{1}{\sqrt{\sqrt[3]{\pi}}} \cdot \frac{\frac{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}}{\sqrt[3]{\pi}}}{cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}}} \]
\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}}
↓
\frac{1}{\left(1 + c\right) + \frac{1}{\sqrt{\sqrt[3]{\pi}}} \cdot \frac{\frac{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}}{\sqrt[3]{\pi}}}{cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}}}

(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
 (/
  1.0
  (+
   (+ 1.0 c)
   (*
    (/ 1.0 (sqrt (cbrt PI)))
    (/
     (/ (sqrt (fma cosTheta -2.0 1.0)) (cbrt PI))
     (* cosTheta (pow (exp cosTheta) cosTheta)))))))
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) {
	return 1.0f / ((1.0f + c) + ((1.0f / sqrtf(cbrtf((float) M_PI))) * ((sqrtf(fmaf(cosTheta, -2.0f, 1.0f)) / cbrtf((float) M_PI)) / (cosTheta * powf(expf(cosTheta), cosTheta)))));
}

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 associate-/r*_binary320.5

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

  7. Simplified0.5

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

  8. Applied *-un-lft-identity_binary320.5

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

  9. Applied cbrt-prod_binary320.5

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

  10. Applied *-un-lft-identity_binary320.5

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

  11. Applied sqrt-prod_binary320.5

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

  12. Applied times-frac_binary320.5

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

  13. Applied times-frac_binary320.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2021280 
(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))))))