Average Error: 0.5 → 0.6
Time: 12.2s
Precision: binary32
Cost: 13088
\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
↓
\[\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(6.28318530718 \cdot u2\right)\right)\right) \]
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))
↓
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (expm1 (log1p (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((u1 / (1.0f - u1))) * expm1f(log1pf(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(Float32(u1 / Float32(Float32(1.0) - u1))) * expm1(log1p(sin(Float32(Float32(6.28318530718) * u2)))))
end

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

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.5

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

  2. Applied egg-rr0.6

    \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(6.28318530718 \cdot u2\right)\right)\right)} \]

  3. Final simplification0.6

    \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(6.28318530718 \cdot u2\right)\right)\right) \]

Alternatives

Alternative 1
Error1.9
Cost6820
\[\begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.002050000010058284:\\ \;\;\;\;\sqrt{\frac{u1}{\frac{1 - u1}{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 + u1 \cdot u1}\\ \end{array} \]
Alternative 2
Error3.1
Cost6692
\[\begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.012000000104308128:\\ \;\;\;\;\sqrt{\frac{u1}{\frac{1 - u1}{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
Alternative 3
Error0.5
Cost6688
\[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
Alternative 4
Error5.9
Cost3552
\[\sqrt{39.47841760436263 \cdot \left(u2 \cdot \frac{u1 \cdot u2}{1 - u1}\right)} \]
Alternative 5
Error5.9
Cost3552
\[\sqrt{\frac{u1}{\frac{1 - u1}{u2 \cdot \left(u2 \cdot 39.47841760436263\right)}}} \]
Alternative 6
Error6.0
Cost3488
\[6.28318530718 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right) \]
Alternative 7
Error11.3
Cost3424
\[\sqrt{u2 \cdot \left(39.47841760436263 \cdot \left(u1 \cdot u2\right)\right)} \]
Alternative 8
Error11.3
Cost3424
\[\sqrt{39.47841760436263 \cdot \left(u1 \cdot \left(u2 \cdot u2\right)\right)} \]
Alternative 9
Error11.3
Cost3360
\[6.28318530718 \cdot \left(u2 \cdot \sqrt{u1}\right) \]
Alternative 10
Error11.3
Cost3360
\[\left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1} \]
Alternative 11
Error32.0
Cost3296
\[u2 \cdot \sqrt{-39.47841760436263} \]

Error

Reproduce

herbie shell --seed 2022325 
(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))))