Average Error: 0.7 → 0.5
Time: 39.7s
Precision: binary32
\[0 < cosTheta \land cosTheta < 0.9999 \land -1 < c \land c < 1\]
\[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}}\]
↓
\[\frac{1}{\sqrt[3]{\pi} \cdot \left(1 - c \cdot c\right) + \frac{\sqrt{1 - \left(cosTheta + cosTheta\right)}}{\left(cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}\right) \cdot \sqrt{\sqrt[3]{\pi}}} \cdot \left(1 - c\right)} \cdot \left(\left(1 - c\right) \cdot \left|\sqrt[3]{\pi}\right|\right)\]
\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}{\sqrt[3]{\pi} \cdot \left(1 - c \cdot c\right) + \frac{\sqrt{1 - \left(cosTheta + cosTheta\right)}}{\left(cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}\right) \cdot \sqrt{\sqrt[3]{\pi}}} \cdot \left(1 - c\right)} \cdot \left(\left(1 - c\right) \cdot \left|\sqrt[3]{\pi}\right|\right)
(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
   (+
    (* (cbrt PI) (- 1.0 (* c c)))
    (*
     (/
      (sqrt (- 1.0 (+ cosTheta cosTheta)))
      (* (* cosTheta (pow (exp cosTheta) cosTheta)) (sqrt (cbrt PI))))
     (- 1.0 c))))
  (* (- 1.0 c) (fabs (cbrt PI)))))
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 / ((cbrtf((float) M_PI) * (1.0f - (c * c))) + ((sqrtf(1.0f - (cosTheta + cosTheta)) / ((cosTheta * powf(expf(cosTheta), cosTheta)) * sqrtf(cbrtf((float) M_PI)))) * (1.0f - c)))) * ((1.0f - c) * fabsf(cbrtf((float) M_PI)));
}

Error

Bits error versus cosTheta

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

    \[\leadsto \color{blue}{\frac{1}{\left(1 + c\right) + \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{\sqrt{\pi} \cdot \left(cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}\right)}}}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt_binary320.5

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

  5. Applied associate-/l*_binary320.5

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

  6. Simplified0.6

    \[\leadsto \frac{1}{\left(1 + c\right) + \frac{\sqrt[3]{\sqrt{\left(1 - cosTheta\right) - cosTheta}} \cdot \sqrt[3]{\sqrt{\left(1 - cosTheta\right) - cosTheta}}}{\color{blue}{\frac{\sqrt{\pi}}{\frac{\sqrt[3]{\sqrt{1 - \left(cosTheta + cosTheta\right)}}}{cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}}}}}}\]
  7. Using strategy rm

  8. Applied div-inv_binary320.6

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

  9. Applied add-cube-cbrt_binary320.6

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

  10. Applied sqrt-prod_binary320.6

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

  11. Applied times-frac_binary320.6

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

  12. Applied times-frac_binary320.6

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

  13. Simplified0.6

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

  14. Simplified0.6

    \[\leadsto \frac{1}{\left(1 + c\right) + \left(\frac{\sqrt[3]{\sqrt{1 - \left(cosTheta + cosTheta\right)}}}{\left|\sqrt[3]{\pi}\right|} \cdot \sqrt[3]{\sqrt{1 - \left(cosTheta + cosTheta\right)}}\right) \cdot \color{blue}{\frac{\sqrt[3]{\sqrt{1 - \left(cosTheta + cosTheta\right)}}}{\sqrt{\sqrt[3]{\pi}} \cdot \left(cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}\right)}}}\]
  15. Using strategy rm

  16. Applied associate-*l/_binary320.6

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

  17. Applied associate-*l/_binary320.6

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

  18. Applied flip-+_binary320.6

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

  19. Applied frac-add_binary320.6

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

  20. Applied associate-/r/_binary320.5

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

  21. Simplified0.5

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

  22. Final simplification0.5

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

Reproduce

herbie shell --seed 2021175 
(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))))))