Average Error: 0.5 → 0.6
Time: 12.0s
Precision: binary32
\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
↓
\[\mathsf{fma}\left(\sin \left(6.28318530718 \cdot u2\right), \sqrt{\left(1 + \mathsf{fma}\left(u1, u1, u1\right)\right) \cdot \frac{u1}{1 - {u1}^{3}}}, 0\right) \]
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))
↓
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (fma
  (sin (* 6.28318530718 u2))
  (sqrt (* (+ 1.0 (fma u1 u1 u1)) (/ u1 (- 1.0 (pow u1 3.0)))))
  0.0))
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 fmaf(sinf((6.28318530718f * u2)), sqrtf(((1.0f + fmaf(u1, u1, u1)) * (u1 / (1.0f - powf(u1, 3.0f))))), 0.0f);
}

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 fma(sin(Float32(Float32(6.28318530718) * u2)), sqrt(Float32(Float32(Float32(1.0) + fma(u1, u1, u1)) * Float32(u1 / Float32(Float32(1.0) - (u1 ^ Float32(3.0)))))), Float32(0.0))
end

\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right)
↓
\mathsf{fma}\left(\sin \left(6.28318530718 \cdot u2\right), \sqrt{\left(1 + \mathsf{fma}\left(u1, u1, u1\right)\right) \cdot \frac{u1}{1 - {u1}^{3}}}, 0\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-rr0.6

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

  3. Applied egg-rr0.6

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

  4. Applied egg-rr0.6

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(6.28318530718 \cdot u2\right), \sqrt{\left(1 + \mathsf{fma}\left(u1, u1, u1\right)\right) \cdot \frac{u1}{1 - {u1}^{3}}}, 0\right)} \]

  5. Final simplification0.6

    \[\leadsto \mathsf{fma}\left(\sin \left(6.28318530718 \cdot u2\right), \sqrt{\left(1 + \mathsf{fma}\left(u1, u1, u1\right)\right) \cdot \frac{u1}{1 - {u1}^{3}}}, 0\right) \]

Reproduce

herbie shell --seed 2022155 
(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))))