Average Error: 0.6 → 0.5
Time: 5.8s
Precision: binary32
\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
↓
\[\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{u1}{1 - u1}\right)\right)} \cdot \sin \left(\sqrt{\sqrt{{u2}^{4} \cdot 1558.5454565444493}}\right) \]
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))
↓
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (expm1 (log1p (/ u1 (- 1.0 u1)))))
  (sin (sqrt (sqrt (* (pow u2 4.0) 1558.5454565444493))))))
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(expm1f(log1pf((u1 / (1.0f - u1))))) * sinf(sqrtf(sqrtf((powf(u2, 4.0f) * 1558.5454565444493f))));
}

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(expm1(log1p(Float32(u1 / Float32(Float32(1.0) - u1))))) * sin(sqrt(sqrt(Float32((u2 ^ Float32(4.0)) * Float32(1558.5454565444493))))))
end

\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right)
↓
\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{u1}{1 - u1}\right)\right)} \cdot \sin \left(\sqrt{\sqrt{{u2}^{4} \cdot 1558.5454565444493}}\right)

Error

Bits error versus cosTheta_i

Bits error versus u1

Bits error versus u2

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

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

  3. Applied egg-rr0.5

    \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\sqrt{\color{blue}{\sqrt{{u2}^{4} \cdot 1558.5454565444493}}}\right) \]

  4. Applied egg-rr0.6

    \[\leadsto \sqrt{\color{blue}{\frac{\sqrt[3]{u1 \cdot u1}}{{\left(\sqrt[3]{1 - u1}\right)}^{2}} \cdot \sqrt[3]{\frac{u1}{1 - u1}}}} \cdot \sin \left(\sqrt{\sqrt{{u2}^{4} \cdot 1558.5454565444493}}\right) \]

  5. Applied egg-rr0.5

    \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{u1}{1 - u1}\right)\right)}} \cdot \sin \left(\sqrt{\sqrt{{u2}^{4} \cdot 1558.5454565444493}}\right) \]

  6. Final simplification0.5

    \[\leadsto \sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{u1}{1 - u1}\right)\right)} \cdot \sin \left(\sqrt{\sqrt{{u2}^{4} \cdot 1558.5454565444493}}\right) \]

Reproduce

herbie shell --seed 2022150 
(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))))