Average Error: 0.3 → 0.4
Time: 10.4s
Precision: binary32
\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
↓
\[\sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{u1}{1 - u1}\right)\right)} \cdot \cos \left(e^{\log 6.28318530718 + \log u2}\right) \]
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (cos (* 6.28318530718 u2))))
↓
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (log1p (expm1 (/ u1 (- 1.0 u1)))))
  (cos (exp (+ (log 6.28318530718) (log u2))))))
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 sqrtf(log1pf(expm1f((u1 / (1.0f - u1))))) * cosf(expf((logf(6.28318530718f) + logf(u2))));
}

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 Float32(sqrt(log1p(expm1(Float32(u1 / Float32(Float32(1.0) - u1))))) * cos(exp(Float32(log(Float32(6.28318530718)) + log(u2)))))
end

\sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(6.28318530718 \cdot u2\right)
↓
\sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{u1}{1 - u1}\right)\right)} \cdot \cos \left(e^{\log 6.28318530718 + \log u2}\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 add-exp-log_binary320.3

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

  3. Applied add-exp-log_binary320.4

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

  4. Applied prod-exp_binary320.3

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

  5. Applied log1p-expm1-u_binary320.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2022131 
(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))))