?

Average Error: 0.7 → 0.4
Time: 16.2s
Precision: binary32
Cost: 23360

?

\[\left(0 < cosTheta \land cosTheta < 0.9999\right) \land \left(-1 < c \land c < 1\right)\]
\[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
↓
\[\begin{array}{l} t_0 := cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}\\ \frac{1}{\left(1 - c \cdot c\right) \cdot t_0 + \left(1 - c\right) \cdot \sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}} \cdot \left(t_0 \cdot \left(1 - c\right)\right) \end{array} \]
(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 (pow (exp cosTheta) cosTheta))))
   (*
    (/
     1.0
     (+
      (* (- 1.0 (* c c)) t_0)
      (* (- 1.0 c) (sqrt (/ (fma cosTheta -2.0 1.0) PI)))))
    (* t_0 (- 1.0 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 * powf(expf(cosTheta), cosTheta);
	return (1.0f / (((1.0f - (c * c)) * t_0) + ((1.0f - c) * sqrtf((fmaf(cosTheta, -2.0f, 1.0f) / ((float) M_PI)))))) * (t_0 * (1.0f - 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(cosTheta * (exp(cosTheta) ^ cosTheta))
	return Float32(Float32(Float32(1.0) / Float32(Float32(Float32(Float32(1.0) - Float32(c * c)) * t_0) + Float32(Float32(Float32(1.0) - c) * sqrt(Float32(fma(cosTheta, Float32(-2.0), Float32(1.0)) / Float32(pi)))))) * Float32(t_0 * Float32(Float32(1.0) - 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 := cosTheta \cdot {\left(e^{cosTheta}\right)}^{cosTheta}\\
\frac{1}{\left(1 - c \cdot c\right) \cdot t_0 + \left(1 - c\right) \cdot \sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}} \cdot \left(t_0 \cdot \left(1 - c\right)\right)
\end{array}

Error?

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

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

    [Start]0.7

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

    +-commutative [=>]0.7

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

    associate-+l+ [=>]0.7

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

    distribute-lft-neg-out [=>]0.7

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

    exp-neg [=>]0.7

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

    associate-*r/ [=>]0.7

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

    associate-/l* [=>]0.7

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

    associate-*l/ [=>]0.5

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

    *-lft-identity [=>]0.5

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

  3. Applied egg-rr0.8

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

  4. Simplified0.7

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

    [Start]0.8

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

    pow-sqr [=>]0.7

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

    associate-+r+ [=>]0.7

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

    +-commutative [<=]0.7

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

    *-commutative [=>]0.7

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

    metadata-eval [=>]0.7

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

  5. Applied egg-rr0.4

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

  6. Final simplification0.4

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

Alternatives

Alternative 1
Error0.5
Cost16640
\[\frac{1}{c + \left(1 + \frac{\frac{\sqrt{1 + cosTheta \cdot -2}}{\sqrt{cosTheta \cdot \pi} \cdot \sqrt{cosTheta}}}{e^{cosTheta \cdot cosTheta}}\right)} \]
Alternative 2
Error0.5
Cost13376
\[\frac{1}{c + \left(1 + \frac{\frac{\sqrt{1 + cosTheta \cdot -2}}{cosTheta \cdot \sqrt{\pi}}}{e^{cosTheta \cdot cosTheta}}\right)} \]
Alternative 3
Error0.6
Cost13312
\[\frac{1}{c + \left(1 + \frac{\frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta}}{e^{cosTheta \cdot cosTheta}}\right)} \]
Alternative 4
Error0.7
Cost13248
\[\frac{1}{1 + \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta \cdot e^{cosTheta \cdot cosTheta}}} \]
Alternative 5
Error1.0
Cost10048
\[\frac{1}{\frac{-1 + \mathsf{fma}\left(cosTheta, -1.5, \frac{1}{cosTheta}\right)}{\sqrt{\pi}} + \left(1 + c\right)} \]
Alternative 6
Error1.2
Cost6912
\[\frac{1}{1 + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{cosTheta} + \left(-1 + cosTheta \cdot -1.5\right)\right)} \]
Alternative 7
Error1.6
Cost6784
\[\frac{1}{1 + \sqrt{\frac{1}{\pi}} \cdot \left(-1 + \frac{1}{cosTheta}\right)} \]
Alternative 8
Error1.4
Cost6784
\[\frac{1}{\left(1 + c\right) + \frac{-1 + \frac{1}{cosTheta}}{\sqrt{\pi}}} \]
Alternative 9
Error2.2
Cost6464
\[cosTheta \cdot \sqrt{\pi} \]
Alternative 10
Error28.6
Cost96
\[1 - c \]
Alternative 11
Error28.6
Cost32
\[1 \]

Error

Reproduce?

herbie shell --seed 2023025 
(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))))))