Average Error: 0.3 → 0.4
Time: 4.6s
Precision: binary32
\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
↓
\[\sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1.5} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(6.28318530718 \cdot u2\right)}^{3}\right)\right)} \]
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (cos (* 6.28318530718 u2))))
↓
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (cbrt
  (*
   (pow (/ u1 (- 1.0 u1)) 1.5)
   (expm1 (log1p (pow (cos (* 6.28318530718 u2)) 3.0))))))
float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * cosf((6.28318530718f * u2));
}
↓
float code(float cosTheta_i, float u1, float u2) {
	return cbrtf((powf((u1 / (1.0f - u1)), 1.5f) * expm1f(log1pf(powf(cosf((6.28318530718f * u2)), 3.0f)))));
}

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * cos(Float32(Float32(6.28318530718) * u2)))
end
↓
function code(cosTheta_i, u1, u2)
	return cbrt(Float32((Float32(u1 / Float32(Float32(1.0) - u1)) ^ Float32(1.5)) * expm1(log1p((cos(Float32(Float32(6.28318530718) * u2)) ^ Float32(3.0))))))
end

\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right)
↓
\sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1.5} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(6.28318530718 \cdot u2\right)}^{3}\right)\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 0.3

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

  2. Applied egg-rr0.3

    \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1.5} \cdot {\cos \left(6.28318530718 \cdot u2\right)}^{3}}} \]

  3. Applied egg-rr0.4

    \[\leadsto \sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1.5} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(6.28318530718 \cdot u2\right)}^{3}\right)\right)}} \]

  4. Final simplification0.4

    \[\leadsto \sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1.5} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(6.28318530718 \cdot u2\right)}^{3}\right)\right)} \]

Reproduce

herbie shell --seed 2022169 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_x"
  :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))) (cos (* 6.28318530718 u2))))