?

Average Error: 0.7 → 0.5
Time: 16.3s
Precision: binary32
Cost: 16512

?

\[\left(0 < cosTheta \land cosTheta < 0.9999\right) \land \left(-1 < c \land c < 1\right)\]
\[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
↓
\[\frac{1}{c + \left(1 + \frac{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}}{cosTheta \cdot \left(\sqrt{\pi} \cdot e^{cosTheta \cdot cosTheta}\right)}\right)} \]
(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
    (/
     (sqrt (fma cosTheta -2.0 1.0))
     (* cosTheta (* (sqrt 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 + (sqrtf(fmaf(cosTheta, -2.0f, 1.0f)) / (cosTheta * (sqrtf(((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(sqrt(fma(cosTheta, Float32(-2.0), Float32(1.0))) / Float32(cosTheta * Float32(sqrt(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{\sqrt{\mathsf{fma}\left(cosTheta, -2, 1\right)}}{cosTheta \cdot \left(\sqrt{\pi} \cdot e^{cosTheta \cdot cosTheta}\right)}\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

    \[\leadsto \color{blue}{\frac{1}{c + \left(1 + \frac{\frac{\sqrt{1 + cosTheta \cdot -2}}{\sqrt{\pi} \cdot cosTheta}}{e^{cosTheta \cdot cosTheta}}\right)}} \]
    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-rr0.5

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

  4. Simplified0.5

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

    [Start]0.5

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

    associate-*r/ [=>]0.5

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

    *-rgt-identity [=>]0.5

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

    *-commutative [<=]0.5

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

    associate-*l* [=>]0.5

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

    exp-prod [<=]0.5

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

  5. Final simplification0.5

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

Alternatives

Alternative 1
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 2
Error0.6
Cost13312
\[\frac{1}{c + \left(1 + \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta \cdot e^{cosTheta \cdot cosTheta}}\right)} \]
Alternative 3
Error0.8
Cost10144
\[\frac{1}{1 + \frac{\sqrt{\frac{-1 + cosTheta \cdot 2}{-\pi}}}{cosTheta \cdot e^{cosTheta \cdot cosTheta}}} \]
Alternative 4
Error1.4
Cost10112
\[\frac{1}{c + \left(1 + \frac{1 - cosTheta}{cosTheta \cdot \left(\sqrt{\pi} \cdot e^{cosTheta \cdot cosTheta}\right)}\right)} \]
Alternative 5
Error1.5
Cost6848
\[\frac{1}{-1 + \left(2 + \left(c + \frac{-1 + \frac{1}{cosTheta}}{\sqrt{\pi}}\right)\right)} \]
Alternative 6
Error1.7
Cost6784
\[\frac{1}{1 + \left(-1 + \frac{1}{cosTheta}\right) \cdot \sqrt{\frac{1}{\pi}}} \]
Alternative 7
Error1.5
Cost6784
\[\frac{1}{\frac{-1 + \frac{1}{cosTheta}}{\sqrt{\pi}} + \left(1 + c\right)} \]
Alternative 8
Error2.3
Cost6464
\[cosTheta \cdot \sqrt{\pi} \]
Alternative 9
Error28.5
Cost3424
\[c \cdot \left(\pi \cdot \left(cosTheta \cdot \left(-cosTheta\right)\right)\right) \]
Alternative 10
Error28.5
Cost32
\[1 \]

Error

Reproduce?

herbie shell --seed 2023018 
(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))))))