Average Error: 0.6 → 0.5
Time: 4.8s
Precision: binary32
\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
↓
\[\sqrt{\sqrt[3]{\frac{{u1}^{3}}{\frac{{\left(1 - u1 \cdot u1\right)}^{3}}{1 + \left(3 \cdot {u1}^{2} + \left({u1}^{3} + u1 \cdot 3\right)\right)}}}} \cdot \sin \left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right) \]
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))
↓
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt
   (cbrt
    (/
     (pow u1 3.0)
     (/
      (pow (- 1.0 (* u1 u1)) 3.0)
      (+ 1.0 (+ (* 3.0 (pow u1 2.0)) (+ (pow u1 3.0) (* u1 3.0))))))))
  (sin (sqrt (* 39.47841760436263 (* u2 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) {
	return sqrtf(cbrtf((powf(u1, 3.0f) / (powf((1.0f - (u1 * u1)), 3.0f) / (1.0f + ((3.0f * powf(u1, 2.0f)) + (powf(u1, 3.0f) + (u1 * 3.0f)))))))) * sinf(sqrtf((39.47841760436263f * (u2 * 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)
	return Float32(sqrt(cbrt(Float32((u1 ^ Float32(3.0)) / Float32((Float32(Float32(1.0) - Float32(u1 * u1)) ^ Float32(3.0)) / Float32(Float32(1.0) + Float32(Float32(Float32(3.0) * (u1 ^ Float32(2.0))) + Float32((u1 ^ Float32(3.0)) + Float32(u1 * Float32(3.0))))))))) * sin(sqrt(Float32(Float32(39.47841760436263) * Float32(u2 * u2)))))
end

\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right)
↓
\sqrt{\sqrt[3]{\frac{{u1}^{3}}{\frac{{\left(1 - u1 \cdot u1\right)}^{3}}{1 + \left(3 \cdot {u1}^{2} + \left({u1}^{3} + u1 \cdot 3\right)\right)}}}} \cdot \sin \left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right)

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.6

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

  2. Applied egg-rr0.5

    \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{\left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right)} \]

  3. Applied egg-rr0.5

    \[\leadsto \sqrt{\color{blue}{\sqrt[3]{\frac{{u1}^{3}}{{\left(1 - u1\right)}^{3}}}}} \cdot \sin \left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right) \]

  4. Applied egg-rr0.5

    \[\leadsto \sqrt{\sqrt[3]{\frac{{u1}^{3}}{\color{blue}{\frac{{\left(1 - u1 \cdot u1\right)}^{3}}{{\left(1 + u1\right)}^{3}}}}}} \cdot \sin \left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right) \]

  5. Taylor expanded in u1 around inf 0.5

    \[\leadsto \sqrt{\sqrt[3]{\frac{{u1}^{3}}{\frac{{\left(1 - u1 \cdot u1\right)}^{3}}{\color{blue}{1 + \left(3 \cdot {u1}^{2} + \left(3 \cdot u1 + {u1}^{3}\right)\right)}}}}} \cdot \sin \left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right) \]

  6. Final simplification0.5

    \[\leadsto \sqrt{\sqrt[3]{\frac{{u1}^{3}}{\frac{{\left(1 - u1 \cdot u1\right)}^{3}}{1 + \left(3 \cdot {u1}^{2} + \left({u1}^{3} + u1 \cdot 3\right)\right)}}}} \cdot \sin \left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right) \]

Reproduce

herbie shell --seed 2022210 
(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))))