Average Error: 0.7 → 0.4
Time: 22.3s
Precision: binary32
\[\left(0 < cosTheta \land cosTheta < 0.9999\right) \land \left(-1 < c \land c < 1\right)\]
\[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
↓
\[\frac{1}{\left(1 + c\right) + \sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)} \cdot \left(\frac{e^{-cosTheta \cdot cosTheta}}{\left|\sqrt[3]{\pi}\right|} \cdot \frac{{\left(\frac{1}{\pi}\right)}^{0.16666666666666666}}{cosTheta}\right)} \]
\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) + \sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)} \cdot \left(\frac{e^{-cosTheta \cdot cosTheta}}{\left|\sqrt[3]{\pi}\right|} \cdot \frac{{\left(\frac{1}{\pi}\right)}^{0.16666666666666666}}{cosTheta}\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
  (+
   (+ 1.0 c)
   (*
    (sqrt (fma cosTheta -2.0 1.0))
    (*
     (/ (exp (- (* cosTheta cosTheta))) (fabs (cbrt PI)))
     (/ (pow (/ 1.0 PI) 0.16666666666666666) 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) + (sqrtf(fmaf(cosTheta, -2.0f, 1.0f)) * ((expf(-(cosTheta * cosTheta)) / fabsf(cbrtf((float) M_PI))) * (powf((1.0f / ((float) M_PI)), 0.16666666666666666f) / 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 div-inv_binary320.5

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

  4. Simplified0.5

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

  5. Applied add-cube-cbrt_binary320.5

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

  6. Applied sqrt-prod_binary320.5

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

  7. Applied div-inv_binary320.5

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

  8. Applied times-frac_binary320.5

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

  9. Simplified0.5

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

  10. Taylor expanded in cosTheta around 0 0.5

    \[\leadsto \frac{1}{\left(1 + c\right) + \sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)} \cdot \left(\frac{e^{-cosTheta \cdot cosTheta}}{\left|\sqrt[3]{\pi}\right|} \cdot \color{blue}{\left({\left(\frac{1}{\pi}\right)}^{0.16666666666666666} \cdot \frac{1}{cosTheta}\right)}\right)} \]

  11. Simplified0.4

    \[\leadsto \frac{1}{\left(1 + c\right) + \sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)} \cdot \left(\frac{e^{-cosTheta \cdot cosTheta}}{\left|\sqrt[3]{\pi}\right|} \cdot \color{blue}{\frac{{\left(\frac{1}{\pi}\right)}^{0.16666666666666666}}{cosTheta}}\right)} \]

  12. Final simplification0.4

    \[\leadsto \frac{1}{\left(1 + c\right) + \sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)} \cdot \left(\frac{e^{-cosTheta \cdot cosTheta}}{\left|\sqrt[3]{\pi}\right|} \cdot \frac{{\left(\frac{1}{\pi}\right)}^{0.16666666666666666}}{cosTheta}\right)} \]

Reproduce

herbie shell --seed 2022004 
(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))))))