Average Error: 0.7 → 0.5
Time: 13.3s
Precision: binary32
Cost: 16544
\[\left(0 < cosTheta \land cosTheta < 0.9999\right) \land \left(-1 < c \land c < 1\right)\]
\[\frac{1}{\left(1 + c\right) + \left(\frac{1}{\sqrt{\pi}} \cdot \frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta}\right) \cdot e^{\left(-cosTheta\right) \cdot cosTheta}} \]
↓
\[\frac{1}{\left(1 + c\right) + \frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta \cdot \sqrt{\pi}}}{{\left(e^{cosTheta}\right)}^{cosTheta}}} \]
(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 (sqrt PI)))
    (pow (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 * sqrtf(((float) M_PI)))) / powf(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(sqrt(Float32(Float32(Float32(1.0) - cosTheta) - cosTheta)) / Float32(cosTheta * sqrt(Float32(pi)))) / (exp(cosTheta) ^ cosTheta))))
end

function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + (((single(1.0) / sqrt(single(pi))) * (sqrt(((single(1.0) - cosTheta) - cosTheta)) / cosTheta)) * exp((-cosTheta * cosTheta))));
end
↓
function tmp = code(cosTheta, c)
	tmp = single(1.0) / ((single(1.0) + c) + ((sqrt(((single(1.0) - cosTheta) - cosTheta)) / (cosTheta * sqrt(single(pi)))) / (exp(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}{\left(1 + c\right) + \frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{cosTheta \cdot \sqrt{\pi}}}{{\left(e^{cosTheta}\right)}^{cosTheta}}}

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. Simplified0.5

    \[\leadsto \color{blue}{\frac{1}{\left(1 + c\right) + \frac{\frac{\sqrt{\left(1 - cosTheta\right) - cosTheta}}{\sqrt{\pi} \cdot cosTheta}}{{\left(e^{cosTheta}\right)}^{cosTheta}}}} \]
    Proof
    (/.f32 1 (+.f32 (+.f32 1 c) (/.f32 (/.f32 (sqrt.f32 (-.f32 (-.f32 1 cosTheta) cosTheta)) (*.f32 (sqrt.f32 (PI.f32)) cosTheta)) (pow.f32 (exp.f32 cosTheta) cosTheta)))): 0 points increase in error, 0 points decrease in error
    (/.f32 1 (+.f32 (+.f32 1 c) (/.f32 (/.f32 (Rewrite<= *-lft-identity_binary32 (*.f32 1 (sqrt.f32 (-.f32 (-.f32 1 cosTheta) cosTheta)))) (*.f32 (sqrt.f32 (PI.f32)) cosTheta)) (pow.f32 (exp.f32 cosTheta) cosTheta)))): 0 points increase in error, 0 points decrease in error
    (/.f32 1 (+.f32 (+.f32 1 c) (/.f32 (Rewrite=> times-frac_binary32 (*.f32 (/.f32 1 (sqrt.f32 (PI.f32))) (/.f32 (sqrt.f32 (-.f32 (-.f32 1 cosTheta) cosTheta)) cosTheta))) (pow.f32 (exp.f32 cosTheta) cosTheta)))): 70 points increase in error, 17 points decrease in error
    (/.f32 1 (+.f32 (+.f32 1 c) (/.f32 (*.f32 (/.f32 1 (sqrt.f32 (PI.f32))) (/.f32 (sqrt.f32 (-.f32 (-.f32 1 cosTheta) cosTheta)) cosTheta)) (Rewrite<= exp-prod_binary32 (exp.f32 (*.f32 cosTheta cosTheta)))))): 0 points increase in error, 0 points decrease in error
    (/.f32 1 (+.f32 (+.f32 1 c) (/.f32 (*.f32 (/.f32 1 (sqrt.f32 (PI.f32))) (/.f32 (sqrt.f32 (-.f32 (-.f32 1 cosTheta) cosTheta)) cosTheta)) (Rewrite<= /-rgt-identity_binary32 (/.f32 (exp.f32 (*.f32 cosTheta cosTheta)) 1))))): 0 points increase in error, 0 points decrease in error
    (/.f32 1 (+.f32 (+.f32 1 c) (Rewrite<= associate-/l*_binary32 (/.f32 (*.f32 (*.f32 (/.f32 1 (sqrt.f32 (PI.f32))) (/.f32 (sqrt.f32 (-.f32 (-.f32 1 cosTheta) cosTheta)) cosTheta)) 1) (exp.f32 (*.f32 cosTheta cosTheta)))))): 0 points increase in error, 0 points decrease in error
    (/.f32 1 (+.f32 (+.f32 1 c) (Rewrite<= associate-*r/_binary32 (*.f32 (*.f32 (/.f32 1 (sqrt.f32 (PI.f32))) (/.f32 (sqrt.f32 (-.f32 (-.f32 1 cosTheta) cosTheta)) cosTheta)) (/.f32 1 (exp.f32 (*.f32 cosTheta cosTheta))))))): 1 points increase in error, 0 points decrease in error
    (/.f32 1 (+.f32 (+.f32 1 c) (*.f32 (*.f32 (/.f32 1 (sqrt.f32 (PI.f32))) (/.f32 (sqrt.f32 (-.f32 (-.f32 1 cosTheta) cosTheta)) cosTheta)) (Rewrite<= exp-neg_binary32 (exp.f32 (neg.f32 (*.f32 cosTheta cosTheta))))))): 3 points increase in error, 4 points decrease in error
    (/.f32 1 (+.f32 (+.f32 1 c) (*.f32 (*.f32 (/.f32 1 (sqrt.f32 (PI.f32))) (/.f32 (sqrt.f32 (-.f32 (-.f32 1 cosTheta) cosTheta)) cosTheta)) (exp.f32 (Rewrite<= distribute-lft-neg-out_binary32 (*.f32 (neg.f32 cosTheta) cosTheta)))))): 0 points increase in error, 0 points decrease in error

  3. Final simplification0.5

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

Alternatives

Alternative 1
Error0.6
Cost13344
\[\frac{1}{1 + \mathsf{fma}\left(\frac{\sqrt{\frac{1 + cosTheta \cdot -2}{\pi}}}{cosTheta}, e^{cosTheta \cdot \left(-cosTheta\right)}, c\right)} \]
Alternative 2
Error0.6
Cost13312
\[\frac{1}{1 + \left(c + \frac{\sqrt{\frac{\mathsf{fma}\left(cosTheta, -2, 1\right)}{\pi}}}{cosTheta \cdot e^{cosTheta \cdot cosTheta}}\right)} \]
Alternative 3
Error0.8
Cost10144
\[\frac{1}{1 + \sqrt{\frac{1 + cosTheta \cdot -2}{\pi}} \cdot \frac{e^{cosTheta \cdot \left(-cosTheta\right)}}{cosTheta}} \]
Alternative 4
Error1.2
Cost6944
\[\frac{1}{1 + \left(c + {\pi}^{-0.5} \cdot \left(cosTheta \cdot -1.5 + \left(-1 + \frac{1}{cosTheta}\right)\right)\right)} \]
Alternative 5
Error1.2
Cost6912
\[\frac{1}{1 + \sqrt{\frac{1}{\pi}} \cdot \left(cosTheta \cdot -1.5 + \left(-1 + \frac{1}{cosTheta}\right)\right)} \]
Alternative 6
Error1.4
Cost6784
\[\frac{1}{1 + \left(c + \frac{1 - cosTheta}{cosTheta \cdot \sqrt{\pi}}\right)} \]
Alternative 7
Error1.5
Cost6720
\[\frac{1}{1 + \frac{-1 + \frac{1}{cosTheta}}{\sqrt{\pi}}} \]
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 2022325 
(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))))))