?

Average Accuracy: 97.9% → 98.5%
Time: 17.6s
Precision: binary32
Cost: 16608

?

\[\left(0 < cosTheta \land cosTheta < 0.9999\right) \land \left(-1 < c \land c < 1\right)\]
\[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
↓
\[\frac{1}{c + \left(1 + \frac{{cosTheta}^{-0.5} \cdot \sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{cosTheta \cdot \pi}}}{e^{cosTheta \cdot cosTheta}}\right)} \]
(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
  (+
   c
   (+
    1.0
    (/
     (* (pow cosTheta -0.5) (sqrt (/ (fma cosTheta -2.0 1.0) (* cosTheta PI))))
     (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 / (c + (1.0f + ((powf(cosTheta, -0.5f) * sqrtf((fmaf(cosTheta, -2.0f, 1.0f) / (cosTheta * ((float) M_PI))))) / expf((cosTheta * cosTheta)))));
}

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)
	return Float32(Float32(1.0) / Float32(c + Float32(Float32(1.0) + Float32(Float32((cosTheta ^ Float32(-0.5)) * sqrt(Float32(fma(cosTheta, Float32(-2.0), Float32(1.0)) / Float32(cosTheta * Float32(pi))))) / exp(Float32(cosTheta * cosTheta))))))
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}}
↓
\frac{1}{c + \left(1 + \frac{{cosTheta}^{-0.5} \cdot \sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{cosTheta \cdot \pi}}}{e^{cosTheta \cdot cosTheta}}\right)}

Error?

Derivation?

  1. Initial program 97.9%

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

  2. Simplified98.5%

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

    [Start]97.9

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

    +-commutative [=>]97.9

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

    associate-+l+ [=>]97.9

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

    distribute-lft-neg-out [=>]97.9

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

    exp-neg [=>]97.9

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

    associate-*r/ [=>]97.9

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

    associate-/l* [=>]97.9

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

    associate-*l/ [=>]98.4

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

    *-lft-identity [=>]98.4

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

  3. Applied egg-rr61.1%

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

    [Start]98.5

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

    add-sqr-sqrt [=>]97.8

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

    sqrt-unprod [=>]61.5

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

    swap-sqr [=>]61.1

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

    add-sqr-sqrt [<=]61.1

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

  4. Simplified61.7%

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

    [Start]61.1

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

    *-commutative [=>]61.1

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

    associate-*l* [=>]61.7

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

  5. Applied egg-rr98.5%

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

    [Start]61.7

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

    sqrt-prod [=>]98.5

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

    *-commutative [=>]98.5

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

  6. Applied egg-rr98.3%

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

    [Start]98.5

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

    associate-/r* [=>]98.3

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

    div-inv [=>]98.2

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

    sqrt-undiv [=>]98.3

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

    +-commutative [=>]98.3

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

    fma-def [=>]98.3

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

  7. Simplified98.5%

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

    [Start]98.3

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

    associate-*r/ [=>]98.5

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

    *-rgt-identity [=>]98.5

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

  8. Applied egg-rr98.5%

    \[\leadsto \frac{1}{c + \left(1 + \frac{\color{blue}{{cosTheta}^{-0.5} \cdot \sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{cosTheta \cdot \pi}}}}{e^{cosTheta \cdot cosTheta}}\right)} \]
    Proof

    [Start]98.5

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

    div-inv [=>]98.3

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

    *-commutative [=>]98.3

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

    pow1/2 [=>]98.3

    \[ \frac{1}{c + \left(1 + \frac{\frac{1}{\color{blue}{{cosTheta}^{0.5}}} \cdot \sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{cosTheta \cdot \pi}}}{e^{cosTheta \cdot cosTheta}}\right)} \]

    pow-flip [=>]98.5

    \[ \frac{1}{c + \left(1 + \frac{\color{blue}{{cosTheta}^{\left(-0.5\right)}} \cdot \sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{cosTheta \cdot \pi}}}{e^{cosTheta \cdot cosTheta}}\right)} \]

    metadata-eval [=>]98.5

    \[ \frac{1}{c + \left(1 + \frac{{cosTheta}^{\color{blue}{-0.5}} \cdot \sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{cosTheta \cdot \pi}}}{e^{cosTheta \cdot cosTheta}}\right)} \]

  9. Final simplification98.5%

    \[\leadsto \frac{1}{c + \left(1 + \frac{{cosTheta}^{-0.5} \cdot \sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{cosTheta \cdot \pi}}}{e^{cosTheta \cdot cosTheta}}\right)} \]

Alternatives

Alternative 1
Accuracy98.5%
Cost13376
\[\frac{1}{c + \left(1 + \frac{\frac{\sqrt{1 + cosTheta \cdot -2}}{cosTheta \cdot \sqrt{\pi}}}{e^{cosTheta \cdot cosTheta}}\right)} \]
Alternative 2
Accuracy97.4%
Cost10432
\[\frac{1}{c + \left(1 + \frac{\frac{\left(1 - cosTheta\right) + cosTheta \cdot \left(cosTheta \cdot \left(-0.5 + cosTheta \cdot -0.5\right)\right)}{cosTheta \cdot \sqrt{\pi}}}{e^{cosTheta \cdot cosTheta}}\right)} \]
Alternative 3
Accuracy96.2%
Cost10304
\[\frac{1}{c + \left(1 + \frac{\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{cosTheta} + \left(cosTheta \cdot -0.5 + -1\right)\right)}{e^{cosTheta \cdot cosTheta}}\right)} \]
Alternative 4
Accuracy96.9%
Cost10304
\[\frac{1}{c + \left(1 + \frac{\frac{1 + \left(cosTheta \cdot \left(cosTheta \cdot -0.5\right) - cosTheta\right)}{cosTheta \cdot \sqrt{\pi}}}{e^{cosTheta \cdot cosTheta}}\right)} \]
Alternative 5
Accuracy95.9%
Cost10112
\[\frac{1}{c + \left(1 + \frac{\frac{1 - cosTheta}{cosTheta \cdot \sqrt{\pi}}}{e^{cosTheta \cdot cosTheta}}\right)} \]
Alternative 6
Accuracy95.3%
Cost6912
\[\frac{1}{\sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{1}{cosTheta} + -1\right) - cosTheta\right) + \left(1 + c\right)} \]
Alternative 7
Accuracy94.9%
Cost6848
\[\frac{1}{\left(1 + c\right) + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{cosTheta} + -1\right)} \]
Alternative 8
Accuracy92.9%
Cost6464
\[cosTheta \cdot \sqrt{\pi} \]
Alternative 9
Accuracy10.8%
Cost96
\[1 - c \]
Alternative 10
Accuracy10.8%
Cost32
\[1 \]

Error

Reproduce?

herbie shell --seed 2023137 
(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))))))