Average Error: 0.5 → 0.6
Time: 10.9s
Precision: binary32
Cost: 16704
\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
↓
\[\begin{array}{l} t_0 := \frac{u1}{1 - u1}\\ \sqrt{\left({t_0}^{0.16666666666666666} \cdot \sqrt{t_0}\right) \cdot \sqrt[3]{t_0}} \cdot \sin \left(6.28318530718 \cdot u2\right) \end{array} \]
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))
↓
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (/ u1 (- 1.0 u1))))
   (*
    (sqrt (* (* (pow t_0 0.16666666666666666) (sqrt t_0)) (cbrt t_0)))
    (sin (* 6.28318530718 u2)))))
float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * sinf((6.28318530718f * u2));
}
↓
float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u1 / (1.0f - u1);
	return sqrtf(((powf(t_0, 0.16666666666666666f) * sqrtf(t_0)) * cbrtf(t_0))) * sinf((6.28318530718f * u2));
}

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * sin(Float32(Float32(6.28318530718) * u2)))
end
↓
function code(cosTheta_i, u1, u2)
	t_0 = Float32(u1 / Float32(Float32(1.0) - u1))
	return Float32(sqrt(Float32(Float32((t_0 ^ Float32(0.16666666666666666)) * sqrt(t_0)) * cbrt(t_0))) * sin(Float32(Float32(6.28318530718) * u2)))
end

\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right)
↓
\begin{array}{l}
t_0 := \frac{u1}{1 - u1}\\
\sqrt{\left({t_0}^{0.16666666666666666} \cdot \sqrt{t_0}\right) \cdot \sqrt[3]{t_0}} \cdot \sin \left(6.28318530718 \cdot u2\right)
\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.5

    \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

  2. Applied egg-rr0.7

    \[\leadsto \sqrt{\color{blue}{{\left(\sqrt[3]{\frac{u1}{1 - u1}}\right)}^{3}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

  3. Applied egg-rr0.6

    \[\leadsto \sqrt{\color{blue}{\left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt[3]{\frac{u1}{1 - u1}}\right) \cdot {\left(\frac{u1}{1 - u1}\right)}^{0.16666666666666666}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

  4. Simplified0.6

    \[\leadsto \sqrt{\color{blue}{\left({\left(\frac{u1}{1 - u1}\right)}^{0.16666666666666666} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \sqrt[3]{\frac{u1}{1 - u1}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
    Proof
    (*.f32 (*.f32 (pow.f32 (/.f32 u1 (-.f32 1 u1)) 1/6) (sqrt.f32 (/.f32 u1 (-.f32 1 u1)))) (cbrt.f32 (/.f32 u1 (-.f32 1 u1)))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-*r*_binary32 (*.f32 (pow.f32 (/.f32 u1 (-.f32 1 u1)) 1/6) (*.f32 (sqrt.f32 (/.f32 u1 (-.f32 1 u1))) (cbrt.f32 (/.f32 u1 (-.f32 1 u1)))))): 37 points increase in error, 52 points decrease in error
    (Rewrite<= *-commutative_binary32 (*.f32 (*.f32 (sqrt.f32 (/.f32 u1 (-.f32 1 u1))) (cbrt.f32 (/.f32 u1 (-.f32 1 u1)))) (pow.f32 (/.f32 u1 (-.f32 1 u1)) 1/6))): 0 points increase in error, 0 points decrease in error

  5. Final simplification0.6

    \[\leadsto \sqrt{\left({\left(\frac{u1}{1 - u1}\right)}^{0.16666666666666666} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \sqrt[3]{\frac{u1}{1 - u1}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

Alternatives

Alternative 1
Error0.5
Cost6720
\[\sin \left(6.28318530718 \cdot u2\right) \cdot {\left(-1 + \frac{1}{u1}\right)}^{-0.5} \]
Alternative 2
Error3.2
Cost6692
\[\begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.010999999940395355:\\ \;\;\;\;\sqrt{\frac{u1 \cdot \left(u2 \cdot \left(u2 \cdot 39.47841760436263\right)\right)}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
Alternative 3
Error0.5
Cost6688
\[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
Alternative 4
Error5.9
Cost3552
\[\sqrt{u1 \cdot \frac{39.47841760436263 \cdot \left(u2 \cdot u2\right)}{1 - u1}} \]
Alternative 5
Error5.9
Cost3552
\[\sqrt{\frac{u1 \cdot \left(u2 \cdot \left(u2 \cdot 39.47841760436263\right)\right)}{1 - u1}} \]
Alternative 6
Error6.0
Cost3488
\[6.28318530718 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right) \]
Alternative 7
Error6.0
Cost3488
\[\sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 \cdot u2\right) \]
Alternative 8
Error11.4
Cost3424
\[\sqrt{u2 \cdot \left(39.47841760436263 \cdot \left(u1 \cdot u2\right)\right)} \]
Alternative 9
Error11.4
Cost3424
\[\sqrt{39.47841760436263 \cdot \left(u1 \cdot \left(u2 \cdot u2\right)\right)} \]
Alternative 10
Error11.4
Cost3360
\[6.28318530718 \cdot \left(u2 \cdot \sqrt{u1}\right) \]
Alternative 11
Error11.4
Cost3360
\[u2 \cdot \left(6.28318530718 \cdot \sqrt{u1}\right) \]
Alternative 12
Error11.4
Cost3360
\[\left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1} \]
Alternative 13
Error25.8
Cost160
\[6.28318530718 \cdot \left(u1 \cdot u2\right) \]

Error

Reproduce

herbie shell --seed 2022338 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_y"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))