Average Error: 0.5 → 0.6
Time: 4.5s
Precision: binary32
\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
↓
\[\sqrt{\sqrt[3]{\frac{{\left(\mathsf{fma}\left(u1, u1, u1\right)\right)}^{3}}{{\left(1 - u1 \cdot u1\right)}^{3}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
(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 (fma u1 u1 u1) 3.0) (pow (- 1.0 (* u1 u1)) 3.0))))
  (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) {
	return sqrtf(cbrtf((powf(fmaf(u1, u1, u1), 3.0f) / powf((1.0f - (u1 * u1)), 3.0f)))) * 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)
	return Float32(sqrt(cbrt(Float32((fma(u1, u1, u1) ^ Float32(3.0)) / (Float32(Float32(1.0) - Float32(u1 * u1)) ^ Float32(3.0))))) * sin(Float32(Float32(6.28318530718) * u2)))
end

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

Error

Derivation

  1. Initial program 0.5

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

  2. Applied egg-rr0.6

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

  3. Applied egg-rr0.6

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

  4. Applied egg-rr0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2022211 
(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))))