Average Error: 0.6 → 0.5
Time: 4.5s
Precision: binary32
\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
↓
\[\sqrt[3]{{\sin \left(6.28318530718 \cdot u2\right)}^{3} \cdot \sqrt{\frac{{u1}^{3}}{{\left(1 - u1\right)}^{3}}}} \]
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))
↓
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (cbrt
  (*
   (pow (sin (* 6.28318530718 u2)) 3.0)
   (sqrt (/ (pow u1 3.0) (pow (- 1.0 u1) 3.0))))))
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) {
	return cbrtf((powf(sinf((6.28318530718f * u2)), 3.0f) * sqrtf((powf(u1, 3.0f) / powf((1.0f - u1), 3.0f)))));
}

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)
	return cbrt(Float32((sin(Float32(Float32(6.28318530718) * u2)) ^ Float32(3.0)) * sqrt(Float32((u1 ^ Float32(3.0)) / (Float32(Float32(1.0) - u1) ^ Float32(3.0))))))
end

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

Error

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Your Program's Arguments

Results

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Derivation

  1. Initial program 0.6

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

  2. Applied egg-rr0.5

    \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1.5} \cdot {\sin \left(6.28318530718 \cdot u2\right)}^{3}}} \]

  3. Taylor expanded in u2 around inf 0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2022204 
(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))))