Average Error: 0.5 → 0.5
Time: 6.0s
Precision: binary32
\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
↓
\[\sqrt{\frac{u1 + u1 \cdot \mathsf{fma}\left(u1, u1, u1\right)}{1 - {u1}^{3}}} \cdot \sin \left(\sqrt{\left(u2 \cdot u2\right) \cdot 39.47841760436263}\right) \]
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))
↓
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ (+ u1 (* u1 (fma u1 u1 u1))) (- 1.0 (pow u1 3.0))))
  (sin (sqrt (* (* u2 u2) 39.47841760436263)))))
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(((u1 + (u1 * fmaf(u1, u1, u1))) / (1.0f - powf(u1, 3.0f)))) * sinf(sqrtf(((u2 * u2) * 39.47841760436263f)));
}

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(Float32(Float32(u1 + Float32(u1 * fma(u1, u1, u1))) / Float32(Float32(1.0) - (u1 ^ Float32(3.0))))) * sin(sqrt(Float32(Float32(u2 * u2) * Float32(39.47841760436263)))))
end

\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right)
↓
\sqrt{\frac{u1 + u1 \cdot \mathsf{fma}\left(u1, u1, u1\right)}{1 - {u1}^{3}}} \cdot \sin \left(\sqrt{\left(u2 \cdot u2\right) \cdot 39.47841760436263}\right)

Error

Bits error versus cosTheta_i

Bits error versus u1

Bits error versus u2

Derivation

  1. Initial program 0.5

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

  2. Applied egg-rr0.5

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

  3. Applied egg-rr0.5

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

  4. Applied egg-rr0.5

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

  5. Applied egg-rr0.5

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

  6. Final simplification0.5

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

Reproduce

herbie shell --seed 2022151 
(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))))