Average Error: 0.7 → 0.5
Time: 17.0s
Precision: binary32
Cost: 22848
\[\left(0 < cosTheta \land cosTheta < 0.9999\right) \land \left(-1 < c \land c < 1\right)\]
\[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
↓
\[\frac{1}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}, \frac{{\left(e^{cosTheta}\right)}^{\left(-cosTheta\right)}}{cosTheta \cdot \sqrt{\pi}}, 1 + c\right)} \]
(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
  (fma
   (sqrt (fma cosTheta -2.0 1.0))
   (/ (pow (exp cosTheta) (- cosTheta)) (* cosTheta (sqrt PI)))
   (+ 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) {
	return 1.0f / fmaf(sqrtf(fmaf(cosTheta, -2.0f, 1.0f)), (powf(expf(cosTheta), -cosTheta) / (cosTheta * sqrtf(((float) M_PI)))), (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)
	return Float32(Float32(1.0) / fma(sqrt(fma(cosTheta, Float32(-2.0), Float32(1.0))), Float32((exp(cosTheta) ^ Float32(-cosTheta)) / Float32(cosTheta * sqrt(Float32(pi)))), 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}}
↓
\frac{1}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}, \frac{{\left(e^{cosTheta}\right)}^{\left(-cosTheta\right)}}{cosTheta \cdot \sqrt{\pi}}, 1 + c\right)}

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

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

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

    *-commutative [=>]0.7

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

    associate-*r/ [=>]0.7

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

    associate-*l/ [<=]0.7

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

    associate-*r* [=>]0.7

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

    *-commutative [=>]0.7

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

    fma-def [=>]0.7

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

    sub-neg [=>]0.7

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

    sub-neg [=>]0.7

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

    associate-+l+ [=>]0.7

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

    +-commutative [=>]0.7

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

    neg-mul-1 [=>]0.7

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

    neg-mul-1 [=>]0.7

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

    distribute-rgt-out [=>]0.7

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

    fma-def [=>]0.7

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

    metadata-eval [=>]0.7

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

    associate-*r/ [=>]0.7

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

    associate-*r/ [=>]0.7

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

    associate-/l* [=>]0.7

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

    /-rgt-identity [=>]0.7

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

    associate-/l/ [=>]0.5

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

    *-commutative [=>]0.5

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

    exp-prod [=>]0.5

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

    *-commutative [=>]0.5

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

  3. Final simplification0.5

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

Alternatives

Alternative 1
Error0.4
Cost16864
\[\frac{1}{\frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{{\left(e^{cosTheta}\right)}^{cosTheta}} \cdot \left(1 - c\right) + cosTheta \cdot \left(1 - c \cdot c\right)} \cdot \left(cosTheta \cdot \left(1 - c\right)\right) \]
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
Cost10208
\[\frac{1}{c + \left(1 + \frac{\frac{\sqrt{\frac{-1 + cosTheta \cdot 2}{-\pi}}}{cosTheta}}{e^{cosTheta \cdot cosTheta}}\right)} \]
Alternative 4
Error1.2
Cost6912
\[\frac{1}{1 + \left(-1 + \left(\frac{1}{cosTheta} + cosTheta \cdot -1.5\right)\right) \cdot \sqrt{\frac{1}{\pi}}} \]
Alternative 5
Error1.0
Cost6912
\[\frac{1}{\left(1 + c\right) + \frac{-1 + \left(\frac{1}{cosTheta} + cosTheta \cdot -1.5\right)}{\sqrt{\pi}}} \]
Alternative 6
Error1.6
Cost6784
\[\frac{1}{1 + \sqrt{\frac{1}{\pi}} \cdot \left(-1 + \frac{1}{cosTheta}\right)} \]
Alternative 7
Error1.5
Cost6784
\[\frac{1}{\left(1 + c\right) + \frac{-1 + \frac{1}{cosTheta}}{\sqrt{\pi}}} \]
Alternative 8
Error2.3
Cost6464
\[cosTheta \cdot \sqrt{\pi} \]
Alternative 9
Error28.5
Cost96
\[1 - c \]
Alternative 10
Error28.5
Cost32
\[1 \]

Error

Reproduce

herbie shell --seed 2022356 
(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))))))