?

Average Error: 0.7 → 0.5
Time: 17.2s
Precision: binary32
Cost: 23104

?

\[\left(0 < cosTheta \land cosTheta < 0.9999\right) \land \left(-1 < c \land c < 1\right)\]
\[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
↓
\[\frac{1}{\left(1 + c\right) + \left(\frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}{\sqrt[3]{\pi}} \cdot \frac{1}{\sqrt[3]{\sqrt{\pi}}}\right) \cdot e^{cosTheta \cdot \left(-cosTheta\right)}} \]
(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
  (+
   (+ 1.0 c)
   (*
    (*
     (/ (/ (sqrt (- (- 1.0 cosTheta) cosTheta)) cosTheta) (cbrt PI))
     (/ 1.0 (cbrt (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 / ((1.0f + c) + ((((sqrtf(((1.0f - cosTheta) - cosTheta)) / cosTheta) / cbrtf(((float) M_PI))) * (1.0f / cbrtf(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(Float32(Float32(1.0) + c) + Float32(Float32(Float32(Float32(sqrt(Float32(Float32(Float32(1.0) - cosTheta) - cosTheta)) / cosTheta) / cbrt(Float32(pi))) * Float32(Float32(1.0) / cbrt(sqrt(Float32(pi))))) * exp(Float32(cosTheta * Float32(-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}{\left(1 + c\right) + \left(\frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}}{\sqrt[3]{\pi}} \cdot \frac{1}{\sqrt[3]{\sqrt{\pi}}}\right) \cdot e^{cosTheta \cdot \left(-cosTheta\right)}}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Applied egg-rr0.5

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

  3. Applied egg-rr0.5

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

  4. Final simplification0.5

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

Alternatives

Alternative 1
Error0.5
Cost10272
\[\frac{1}{\left(1 + c\right) + e^{cosTheta \cdot \left(-cosTheta\right)} \cdot \frac{1}{cosTheta \cdot \sqrt{\frac{\pi}{1 - \left(cosTheta + cosTheta\right)}}}} \]
Alternative 2
Error0.7
Cost10240
\[\frac{1}{c + \left(1 + \frac{\frac{1}{cosTheta} \cdot \sqrt{\frac{1 + cosTheta \cdot -2}{\pi}}}{e^{cosTheta \cdot cosTheta}}\right)} \]
Alternative 3
Error1.1
Cost6976
\[\frac{1}{\left(1 + c\right) + \sqrt{\frac{1}{\pi}} \cdot \left(cosTheta \cdot -1.5 + \left(\frac{1}{cosTheta} + -1\right)\right)} \]
Alternative 4
Error1.5
Cost6848
\[\frac{1}{\left(1 + c\right) + \sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{cosTheta} + -1\right)} \]
Alternative 5
Error2.2
Cost6464
\[cosTheta \cdot \sqrt{\pi} \]
Alternative 6
Error28.5
Cost3424
\[c \cdot \left(cosTheta \cdot \left(\pi \cdot \left(-cosTheta\right)\right)\right) \]
Alternative 7
Error28.6
Cost32
\[1 \]

Error

Reproduce?

herbie shell --seed 2023187 
(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))))))