?

Average Error: 0.5 → 0.6
Time: 13.8s
Precision: binary32
Cost: 6720

?

\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]
\[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
↓
\[{\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))
↓
(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (pow (+ (/ 1.0 u1) -1.0) -0.5) (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 powf(((1.0f / u1) + -1.0f), -0.5f) * sinf((6.28318530718f * u2));
}

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 / (1.0e0 - u1))) * sin((6.28318530718e0 * u2))
end function
↓
real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = (((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0)) * sin((6.28318530718e0 * u2))
end function

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((Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)) ^ Float32(-0.5)) * sin(Float32(Float32(6.28318530718) * u2)))
end

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * sin((single(6.28318530718) * u2));
end
↓
function tmp = code(cosTheta_i, u1, u2)
	tmp = (((single(1.0) / u1) + single(-1.0)) ^ single(-0.5)) * sin((single(6.28318530718) * u2));
end

\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right)
↓
{\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \sin \left(6.28318530718 \cdot u2\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-rr0.9

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

  3. Applied egg-rr0.5

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

  4. Simplified0.5

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

    [Start]0.5

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

    div-sub [=>]0.5

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

    *-inverses [=>]0.5

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

    sub-neg [=>]0.5

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

    metadata-eval [=>]0.5

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

  5. Applied egg-rr0.6

    \[\leadsto \color{blue}{{\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \sin \left(6.28318530718 \cdot u2\right)} \]

  6. Final simplification0.6

    \[\leadsto {\left(\frac{1}{u1} + -1\right)}^{-0.5} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

Alternatives

Alternative 1
Error3.1
Cost6692
\[\begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.004939999897032976:\\ \;\;\;\;\sqrt{\frac{39.47841760436263}{\frac{1 - u1}{u2 \cdot \left(u1 \cdot u2\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
Alternative 2
Error0.5
Cost6688
\[\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}} \]
Alternative 3
Error5.0
Cost3748
\[\begin{array}{l} \mathbf{if}\;\frac{u1}{1 - u1} \leq 1.500000053056283 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot -41.341702240407926\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{39.47841760436263}{\frac{1 - u1}{u2 \cdot \left(u1 \cdot u2\right)}}}\\ \end{array} \]
Alternative 4
Error5.7
Cost3552
\[\sqrt{39.47841760436263 \cdot \left(u1 \cdot \frac{u2 \cdot u2}{1 - u1}\right)} \]
Alternative 5
Error5.7
Cost3552
\[\sqrt{39.47841760436263 \cdot \frac{u2}{\frac{1 - u1}{u1 \cdot u2}}} \]
Alternative 6
Error5.7
Cost3552
\[\sqrt{\frac{39.47841760436263}{\frac{1 - u1}{u2 \cdot \left(u1 \cdot u2\right)}}} \]
Alternative 7
Error5.8
Cost3488
\[6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]
Alternative 8
Error5.8
Cost3488
\[u2 \cdot \left(6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \]
Alternative 9
Error5.9
Cost3488
\[\frac{6.28318530718}{\frac{\sqrt{\frac{1}{u1} + -1}}{u2}} \]
Alternative 10
Error11.3
Cost3424
\[\sqrt{u2 \cdot \left(u2 \cdot \left(u1 \cdot 39.47841760436263\right)\right)} \]
Alternative 11
Error11.3
Cost3360
\[6.28318530718 \cdot \left(u2 \cdot \sqrt{u1}\right) \]
Alternative 12
Error11.3
Cost3360
\[\left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1} \]
Alternative 13
Error27.3
Cost224
\[6.28318530718 \cdot \left(u2 \cdot \left(u1 \cdot u1\right)\right) \]
Alternative 14
Error27.3
Cost224
\[u2 \cdot \left(6.28318530718 \cdot \left(u1 \cdot u1\right)\right) \]
Alternative 15
Error29.7
Cost32
\[0 \]

Error

Reproduce?

herbie shell --seed 2023083 
(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))))