Average Error: 0.7 → 0.4
Time: 16.6s
Precision: binary32
\[\left(0 < cosTheta \land cosTheta < 0.9999\right) \land \left(-1 < c \land c < 1\right)\]
\[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
↓
\[\begin{array}{l} t_0 := \left(cosTheta \cdot e^{cosTheta \cdot cosTheta}\right) \cdot \sqrt{\sqrt[3]{\pi}}\\ \frac{\left(1 - c\right) \cdot t_0}{\mathsf{fma}\left(1 - c, \frac{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}}{\sqrt[3]{\pi}}, t_0 \cdot \left(1 - c \cdot c\right)\right)} \end{array} \]
(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 (* (* cosTheta (exp (* cosTheta cosTheta))) (sqrt (cbrt PI)))))
   (/
    (* (- 1.0 c) t_0)
    (fma
     (- 1.0 c)
     (/ (sqrt (fma cosTheta -2.0 1.0)) (cbrt PI))
     (* t_0 (- 1.0 (* c 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 = (cosTheta * expf((cosTheta * cosTheta))) * sqrtf(cbrtf(((float) M_PI)));
	return ((1.0f - c) * t_0) / fmaf((1.0f - c), (sqrtf(fmaf(cosTheta, -2.0f, 1.0f)) / cbrtf(((float) M_PI))), (t_0 * (1.0f - (c * c))));
}

function code(cosTheta, c)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + c) + Float32(Float32(Float32(Float32(1.0) / sqrt(Float32(pi))) * Float32(sqrt(Float32(Float32(Float32(1.0) - cosTheta) - cosTheta)) / cosTheta)) * exp(Float32(Float32(-cosTheta) * cosTheta)))))
end
↓
function code(cosTheta, c)
	t_0 = Float32(Float32(cosTheta * exp(Float32(cosTheta * cosTheta))) * sqrt(cbrt(Float32(pi))))
	return Float32(Float32(Float32(Float32(1.0) - c) * t_0) / fma(Float32(Float32(1.0) - c), Float32(sqrt(fma(cosTheta, Float32(-2.0), Float32(1.0))) / cbrt(Float32(pi))), Float32(t_0 * Float32(Float32(1.0) - Float32(c * c)))))
end

\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(cosTheta \cdot e^{cosTheta \cdot cosTheta}\right) \cdot \sqrt{\sqrt[3]{\pi}}\\
\frac{\left(1 - c\right) \cdot t_0}{\mathsf{fma}\left(1 - c, \frac{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}}{\sqrt[3]{\pi}}, t_0 \cdot \left(1 - c \cdot c\right)\right)}
\end{array}

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*_binary320.4

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

  7. Simplified0.4

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

  8. Applied flip-+_binary320.4

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

  9. Applied frac-add_binary320.5

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

  10. Applied associate-/r/_binary320.4

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

  11. Simplified0.4

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

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

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

  13. Applied associate-*l*_binary320.4

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

  14. Simplified0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2022131 
(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))))))