Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.2%

Time: 9.5s

Alternatives: 10

Speedup: 1.0×

Specification

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * sinf((6.28318530718f * u2));
}

Fortran

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

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * sin(Float32(Float32(6.28318530718) * u2)))
end

MATLAB

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

Initial Program: 98.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * sinf((6.28318530718f * u2));
}

Fortran

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

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * sin(Float32(Float32(6.28318530718) * u2)))
end

MATLAB

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

Alternative 1: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{u1}{\frac{1 - u1 \cdot u1}{1 + u1}}} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sin (* u2 6.28318530718)) (sqrt (/ u1 (/ (- 1.0 (* u1 u1)) (+ 1.0 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf((u2 * 6.28318530718f)) * sqrtf((u1 / ((1.0f - (u1 * u1)) / (1.0f + u1))));
}

Fortran

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 = sin((u2 * 6.28318530718e0)) * sqrt((u1 / ((1.0e0 - (u1 * u1)) / (1.0e0 + u1))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sin(Float32(u2 * Float32(6.28318530718))) * sqrt(Float32(u1 / Float32(Float32(Float32(1.0) - Float32(u1 * u1)) / Float32(Float32(1.0) + u1)))))
end

MATLAB

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

Derivation

1. Initial program 98.1%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{1 - u1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   2. sub-neg N/A

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{1 + \left(\mathsf{neg}\left(u1\right)\right)}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   3. +-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) + 1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   4. flip-+ N/A

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{\frac{\left(\mathsf{neg}\left(u1\right)\right) \cdot \left(\mathsf{neg}\left(u1\right)\right) - 1 \cdot 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   5. sqr-neg N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{\color{blue}{u1 \cdot u1} - 1 \cdot 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   6. lower-/.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{\frac{u1 \cdot u1 - 1 \cdot 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   7. metadata-eval N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - \color{blue}{1}}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   8. lower--.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{\color{blue}{u1 \cdot u1 - 1}}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   9. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{\color{blue}{u1 \cdot u1} - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   10. lower--.f32 N/A

       \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   11. lower-neg.f32 98.1

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

4. Applied rewrites 98.1%

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

5. Final simplification 98.1%

   \[\leadsto \sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{u1}{\frac{1 - u1 \cdot u1}{1 + u1}}} \]
6. Add Preprocessing

Alternative 2: 98.2% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (* (- -1.0 (/ -1.0 u1)) u1))) (sin (* u2 6.28318530718))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / ((-1.0f - (-1.0f / u1)) * u1))) * sinf((u2 * 6.28318530718f));
}

Fortran

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) - ((-1.0e0) / u1)) * u1))) * sin((u2 * 6.28318530718e0))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(Float32(-1.0) - Float32(Float32(-1.0) / u1)) * u1))) * sin(Float32(u2 * Float32(6.28318530718))))
end

MATLAB

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

Derivation

1. Initial program 98.1%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{1 - u1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   2. sub-neg N/A

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{1 + \left(\mathsf{neg}\left(u1\right)\right)}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   3. +-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) + 1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   4. flip-+ N/A

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{\frac{\left(\mathsf{neg}\left(u1\right)\right) \cdot \left(\mathsf{neg}\left(u1\right)\right) - 1 \cdot 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   5. sqr-neg N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{\color{blue}{u1 \cdot u1} - 1 \cdot 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   6. lower-/.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{\color{blue}{\frac{u1 \cdot u1 - 1 \cdot 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   7. metadata-eval N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - \color{blue}{1}}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   8. lower--.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{\color{blue}{u1 \cdot u1 - 1}}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   9. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{\color{blue}{u1 \cdot u1} - 1}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   10. lower--.f32 N/A

       \[\leadsto \sqrt{\frac{u1}{\frac{u1 \cdot u1 - 1}{\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) - 1}}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   11. lower-neg.f32 98.1

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

4. Applied rewrites 98.1%

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

5. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{\frac{u1}{\frac{\color{blue}{-1}}{\left(-u1\right) - 1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]
6. Step-by-step derivation

7. Applied rewrites 88.6%

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

8. Taylor expanded in u1 around inf

   \[\leadsto \sqrt{\frac{u1}{\frac{\color{blue}{{u1}^{2} \cdot \left(1 - \frac{1}{{u1}^{2}}\right)}}{\left(-u1\right) - 1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{\color{blue}{\left(1 - \frac{1}{{u1}^{2}}\right) \cdot {u1}^{2}}}{\left(-u1\right) - 1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   2. unpow2 N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{\left(1 - \frac{1}{{u1}^{2}}\right) \cdot \color{blue}{\left(u1 \cdot u1\right)}}{\left(-u1\right) - 1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   3. associate-*r* N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{\color{blue}{\left(\left(1 - \frac{1}{{u1}^{2}}\right) \cdot u1\right) \cdot u1}}{\left(-u1\right) - 1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   4. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{\color{blue}{\left(\left(1 - \frac{1}{{u1}^{2}}\right) \cdot u1\right) \cdot u1}}{\left(-u1\right) - 1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{\color{blue}{\left(\left(1 - \frac{1}{{u1}^{2}}\right) \cdot u1\right)} \cdot u1}{\left(-u1\right) - 1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   6. lower--.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{\left(\color{blue}{\left(1 - \frac{1}{{u1}^{2}}\right)} \cdot u1\right) \cdot u1}{\left(-u1\right) - 1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   7. unpow2 N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{\left(\left(1 - \frac{1}{\color{blue}{u1 \cdot u1}}\right) \cdot u1\right) \cdot u1}{\left(-u1\right) - 1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   8. associate-/r* N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{\left(\left(1 - \color{blue}{\frac{\frac{1}{u1}}{u1}}\right) \cdot u1\right) \cdot u1}{\left(-u1\right) - 1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   9. lower-/.f32 N/A

      \[\leadsto \sqrt{\frac{u1}{\frac{\left(\left(1 - \color{blue}{\frac{\frac{1}{u1}}{u1}}\right) \cdot u1\right) \cdot u1}{\left(-u1\right) - 1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   10. lower-/.f32 98.0

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

10. Applied rewrites 98.0%

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

11. Taylor expanded in u1 around -inf

    \[\leadsto \sqrt{\frac{u1}{\color{blue}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]
12. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \sqrt{\frac{u1}{\color{blue}{\mathsf{neg}\left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

    2. *-commutative N/A

       \[\leadsto \sqrt{\frac{u1}{\mathsf{neg}\left(\color{blue}{\left(1 - \frac{1}{u1}\right) \cdot u1}\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

    3. distribute-rgt-neg-in N/A

       \[\leadsto \sqrt{\frac{u1}{\color{blue}{\left(1 - \frac{1}{u1}\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

    4. mul-1-neg N/A

       \[\leadsto \sqrt{\frac{u1}{\left(1 - \frac{1}{u1}\right) \cdot \color{blue}{\left(-1 \cdot u1\right)}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

    5. lower-*.f32 N/A

       \[\leadsto \sqrt{\frac{u1}{\color{blue}{\left(1 - \frac{1}{u1}\right) \cdot \left(-1 \cdot u1\right)}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

    6. lower--.f32 N/A

       \[\leadsto \sqrt{\frac{u1}{\color{blue}{\left(1 - \frac{1}{u1}\right)} \cdot \left(-1 \cdot u1\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

    7. lower-/.f32 N/A

       \[\leadsto \sqrt{\frac{u1}{\left(1 - \color{blue}{\frac{1}{u1}}\right) \cdot \left(-1 \cdot u1\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

    8. mul-1-neg N/A

       \[\leadsto \sqrt{\frac{u1}{\left(1 - \frac{1}{u1}\right) \cdot \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

    9. lower-neg.f32 98.1

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

13. Applied rewrites 98.1%

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

14. Final simplification 98.1%

    \[\leadsto \sqrt{\frac{u1}{\left(-1 - \frac{-1}{u1}\right) \cdot u1}} \cdot \sin \left(u2 \cdot 6.28318530718\right) \]
15. Add Preprocessing

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (sin (* u2 6.28318530718))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * sinf((u2 * 6.28318530718f));
}

Fortran

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((u2 * 6.28318530718e0))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * sin(Float32(u2 * Float32(6.28318530718))))
end

MATLAB

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

Derivation

1. Initial program 98.1%

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

3. Final simplification 98.1%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(u2 \cdot 6.28318530718\right) \]
4. Add Preprocessing

Alternative 4: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.007000000216066837:\\ \;\;\;\;\left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{u1}{u1 \cdot u1 - 1} \cdot \left(-1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(u2 \cdot 6.28318530718\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.007000000216066837)
   (* (* u2 6.28318530718) (sqrt (* (/ u1 (- (* u1 u1) 1.0)) (- -1.0 u1))))
   (* (sqrt u1) (sin (* u2 6.28318530718)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.007000000216066837f) {
		tmp = (u2 * 6.28318530718f) * sqrtf(((u1 / ((u1 * u1) - 1.0f)) * (-1.0f - u1)));
	} else {
		tmp = sqrtf(u1) * sinf((u2 * 6.28318530718f));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((u2 * 6.28318530718e0) <= 0.007000000216066837e0) then
        tmp = (u2 * 6.28318530718e0) * sqrt(((u1 / ((u1 * u1) - 1.0e0)) * ((-1.0e0) - u1)))
    else
        tmp = sqrt(u1) * sin((u2 * 6.28318530718e0))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.007000000216066837))
		tmp = Float32(Float32(u2 * Float32(6.28318530718)) * sqrt(Float32(Float32(u1 / Float32(Float32(u1 * u1) - Float32(1.0))) * Float32(Float32(-1.0) - u1))));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(u2 * Float32(6.28318530718))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.007000000216066837))
		tmp = (u2 * single(6.28318530718)) * sqrt(((u1 / ((u1 * u1) - single(1.0))) * (single(-1.0) - u1)));
	else
		tmp = sqrt(u1) * sin((u2 * single(6.28318530718)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00700000022

   1. Initial program 98.4%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \frac{314159265359}{50000000000}\right)} \]

      2. lower-*.f32 96.8

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

   5. Applied rewrites 96.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot 6.28318530718\right)} \]
   6. Step-by-step derivation

      1. lift-/.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{u1}{1 - u1}}} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]

      2. lift--.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{\color{blue}{1 - u1}}} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]

      3. flip-- N/A

         \[\leadsto \sqrt{\frac{u1}{\color{blue}{\frac{1 \cdot 1 - u1 \cdot u1}{1 + u1}}}} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]

      4. associate-/r/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{u1}{1 \cdot 1 - u1 \cdot u1} \cdot \left(1 + u1\right)}} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{u1}{1 \cdot 1 - u1 \cdot u1} \cdot \left(1 + u1\right)}} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]

      6. lower-/.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{u1}{1 \cdot 1 - u1 \cdot u1}} \cdot \left(1 + u1\right)} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]

      7. metadata-eval N/A

         \[\leadsto \sqrt{\frac{u1}{\color{blue}{1} - u1 \cdot u1} \cdot \left(1 + u1\right)} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]

      8. lift-*.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - \color{blue}{u1 \cdot u1}} \cdot \left(1 + u1\right)} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]

      9. lower--.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{\color{blue}{1 - u1 \cdot u1}} \cdot \left(1 + u1\right)} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]

      10. lower-+.f32 97.0

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

   7. Applied rewrites 97.0%

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

   if 0.00700000022 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.1%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\sqrt{u1}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]
   4. Step-by-step derivation

      1. lower-sqrt.f32 79.1

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

   5. Applied rewrites 79.1%

      \[\leadsto \color{blue}{\sqrt{u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.007000000216066837:\\ \;\;\;\;\left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{u1}{u1 \cdot u1 - 1} \cdot \left(-1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(u2 \cdot 6.28318530718\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.002940000034868717:\\ \;\;\;\;\left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{u1}{u1 \cdot u1 - 1} \cdot \left(-1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-41.341702240407926 \cdot \left(u2 \cdot u2\right) + 6.28318530718\right) \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.002940000034868717)
   (* (* u2 6.28318530718) (sqrt (* (/ u1 (- (* u1 u1) 1.0)) (- -1.0 u1))))
   (* (* (+ (* -41.341702240407926 (* u2 u2)) 6.28318530718) u2) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.002940000034868717f) {
		tmp = (u2 * 6.28318530718f) * sqrtf(((u1 / ((u1 * u1) - 1.0f)) * (-1.0f - u1)));
	} else {
		tmp = (((-41.341702240407926f * (u2 * u2)) + 6.28318530718f) * u2) * sqrtf(u1);
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((u2 * 6.28318530718e0) <= 0.002940000034868717e0) then
        tmp = (u2 * 6.28318530718e0) * sqrt(((u1 / ((u1 * u1) - 1.0e0)) * ((-1.0e0) - u1)))
    else
        tmp = ((((-41.341702240407926e0) * (u2 * u2)) + 6.28318530718e0) * u2) * sqrt(u1)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.002940000034868717))
		tmp = Float32(Float32(u2 * Float32(6.28318530718)) * sqrt(Float32(Float32(u1 / Float32(Float32(u1 * u1) - Float32(1.0))) * Float32(Float32(-1.0) - u1))));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(-41.341702240407926) * Float32(u2 * u2)) + Float32(6.28318530718)) * u2) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.002940000034868717))
		tmp = (u2 * single(6.28318530718)) * sqrt(((u1 / ((u1 * u1) - single(1.0))) * (single(-1.0) - u1)));
	else
		tmp = (((single(-41.341702240407926) * (u2 * u2)) + single(6.28318530718)) * u2) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00294000003

   1. Initial program 98.4%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \frac{314159265359}{50000000000}\right)} \]

      2. lower-*.f32 97.9

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

   5. Applied rewrites 97.9%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot 6.28318530718\right)} \]
   6. Step-by-step derivation

      1. lift-/.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{u1}{1 - u1}}} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]

      2. lift--.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{\color{blue}{1 - u1}}} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]

      3. flip-- N/A

         \[\leadsto \sqrt{\frac{u1}{\color{blue}{\frac{1 \cdot 1 - u1 \cdot u1}{1 + u1}}}} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]

      4. associate-/r/ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{u1}{1 \cdot 1 - u1 \cdot u1} \cdot \left(1 + u1\right)}} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{u1}{1 \cdot 1 - u1 \cdot u1} \cdot \left(1 + u1\right)}} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]

      6. lower-/.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{u1}{1 \cdot 1 - u1 \cdot u1}} \cdot \left(1 + u1\right)} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]

      7. metadata-eval N/A

         \[\leadsto \sqrt{\frac{u1}{\color{blue}{1} - u1 \cdot u1} \cdot \left(1 + u1\right)} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]

      8. lift-*.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - \color{blue}{u1 \cdot u1}} \cdot \left(1 + u1\right)} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]

      9. lower--.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{\color{blue}{1 - u1 \cdot u1}} \cdot \left(1 + u1\right)} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]

      10. lower-+.f32 98.0

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

   7. Applied rewrites 98.0%

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

   if 0.00294000003 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.3%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \frac{314159265359}{50000000000}\right)} \]

      2. lower-*.f32 48.9

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

   5. Applied rewrites 48.9%

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

   6. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\sqrt{u1}} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]
   7. Step-by-step derivation

      1. lower-sqrt.f32 46.1

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

   8. Applied rewrites 46.1%

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

   9. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(\left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right) \cdot u2\right)} \]

       2. lower-*.f32 N/A

          \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(\left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right) \cdot u2\right)} \]

       3. +-commutative N/A

          \[\leadsto \sqrt{u1} \cdot \left(\color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2} + \frac{314159265359}{50000000000}\right)} \cdot u2\right) \]

       4. lower-fma.f32 N/A

          \[\leadsto \sqrt{u1} \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, {u2}^{2}, \frac{314159265359}{50000000000}\right)} \cdot u2\right) \]

       5. unpow2 N/A

          \[\leadsto \sqrt{u1} \cdot \left(\mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \color{blue}{u2 \cdot u2}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \]

       6. lower-*.f32 46.1

          \[\leadsto \sqrt{u1} \cdot \left(\mathsf{fma}\left(-41.341702240407926, \color{blue}{u2 \cdot u2}, 6.28318530718\right) \cdot u2\right) \]

   11. Applied rewrites 46.1%

       \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(\mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right) \cdot u2\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 59.4%

       \[\leadsto \sqrt{u1} \cdot \left(\left(\left(u2 \cdot u2\right) \cdot -41.341702240407926 + 6.28318530718\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.002940000034868717:\\ \;\;\;\;\left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{u1}{u1 \cdot u1 - 1} \cdot \left(-1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-41.341702240407926 \cdot \left(u2 \cdot u2\right) + 6.28318530718\right) \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.002940000034868717:\\ \;\;\;\;\frac{1}{\sqrt{\frac{1 - u1}{u1}}} \cdot \left(u2 \cdot 6.28318530718\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-41.341702240407926 \cdot \left(u2 \cdot u2\right) + 6.28318530718\right) \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.002940000034868717)
   (* (/ 1.0 (sqrt (/ (- 1.0 u1) u1))) (* u2 6.28318530718))
   (* (* (+ (* -41.341702240407926 (* u2 u2)) 6.28318530718) u2) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.002940000034868717f) {
		tmp = (1.0f / sqrtf(((1.0f - u1) / u1))) * (u2 * 6.28318530718f);
	} else {
		tmp = (((-41.341702240407926f * (u2 * u2)) + 6.28318530718f) * u2) * sqrtf(u1);
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((u2 * 6.28318530718e0) <= 0.002940000034868717e0) then
        tmp = (1.0e0 / sqrt(((1.0e0 - u1) / u1))) * (u2 * 6.28318530718e0)
    else
        tmp = ((((-41.341702240407926e0) * (u2 * u2)) + 6.28318530718e0) * u2) * sqrt(u1)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.002940000034868717))
		tmp = Float32(Float32(Float32(1.0) / sqrt(Float32(Float32(Float32(1.0) - u1) / u1))) * Float32(u2 * Float32(6.28318530718)));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(-41.341702240407926) * Float32(u2 * u2)) + Float32(6.28318530718)) * u2) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.002940000034868717))
		tmp = (single(1.0) / sqrt(((single(1.0) - u1) / u1))) * (u2 * single(6.28318530718));
	else
		tmp = (((single(-41.341702240407926) * (u2 * u2)) + single(6.28318530718)) * u2) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00294000003

   1. Initial program 98.4%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \frac{314159265359}{50000000000}\right)} \]

      2. lower-*.f32 97.9

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

   5. Applied rewrites 97.9%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot 6.28318530718\right)} \]
   6. Step-by-step derivation

      1. lift-sqrt.f32 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]

      2. lift-/.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{u1}{1 - u1}}} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]

      3. frac-2neg N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\mathsf{neg}\left(u1\right)}{\mathsf{neg}\left(\left(1 - u1\right)\right)}}} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]

      4. lift-neg.f32 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{-u1}}{\mathsf{neg}\left(\left(1 - u1\right)\right)}} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]

      5. div-inv N/A

         \[\leadsto \sqrt{\color{blue}{\left(-u1\right) \cdot \frac{1}{\mathsf{neg}\left(\left(1 - u1\right)\right)}}} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]

      6. remove-double-div N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{1}{-u1}}} \cdot \frac{1}{\mathsf{neg}\left(\left(1 - u1\right)\right)}} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]

      7. lift-neg.f32 N/A

         \[\leadsto \sqrt{\frac{1}{\frac{1}{\color{blue}{\mathsf{neg}\left(u1\right)}}} \cdot \frac{1}{\mathsf{neg}\left(\left(1 - u1\right)\right)}} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\mathsf{neg}\left(\frac{1}{u1}\right)}} \cdot \frac{1}{\mathsf{neg}\left(\left(1 - u1\right)\right)}} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]

      9. lift-/.f32 N/A

         \[\leadsto \sqrt{\frac{1}{\mathsf{neg}\left(\color{blue}{\frac{1}{u1}}\right)} \cdot \frac{1}{\mathsf{neg}\left(\left(1 - u1\right)\right)}} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]

      10. lift-neg.f32 N/A

          \[\leadsto \sqrt{\frac{1}{\color{blue}{-\frac{1}{u1}}} \cdot \frac{1}{\mathsf{neg}\left(\left(1 - u1\right)\right)}} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]

      11. metadata-eval N/A

          \[\leadsto \sqrt{\frac{1}{-\frac{1}{u1}} \cdot \frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\mathsf{neg}\left(\left(1 - u1\right)\right)}} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]

      12. frac-2neg N/A

          \[\leadsto \sqrt{\frac{1}{-\frac{1}{u1}} \cdot \color{blue}{\frac{-1}{1 - u1}}} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]

      13. lift-/.f32 N/A

          \[\leadsto \sqrt{\frac{1}{-\frac{1}{u1}} \cdot \color{blue}{\frac{-1}{1 - u1}}} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]

      14. associate-/r/ N/A

          \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{-\frac{1}{u1}}{\frac{-1}{1 - u1}}}}} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]

      15. sqrt-div N/A

          \[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{-\frac{1}{u1}}{\frac{-1}{1 - u1}}}}} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]

      16. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{1}}{\sqrt{\frac{-\frac{1}{u1}}{\frac{-1}{1 - u1}}}} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]

      17. lower-/.f32 N/A

          \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{-\frac{1}{u1}}{\frac{-1}{1 - u1}}}}} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]

      18. lower-sqrt.f32 N/A

          \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{-\frac{1}{u1}}{\frac{-1}{1 - u1}}}}} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]

   7. Applied rewrites 97.9%

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

   if 0.00294000003 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.3%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \frac{314159265359}{50000000000}\right)} \]

      2. lower-*.f32 48.9

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

   5. Applied rewrites 48.9%

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

   6. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\sqrt{u1}} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]
   7. Step-by-step derivation

      1. lower-sqrt.f32 46.1

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

   8. Applied rewrites 46.1%

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

   9. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(\left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right) \cdot u2\right)} \]

       2. lower-*.f32 N/A

          \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(\left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right) \cdot u2\right)} \]

       3. +-commutative N/A

          \[\leadsto \sqrt{u1} \cdot \left(\color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2} + \frac{314159265359}{50000000000}\right)} \cdot u2\right) \]

       4. lower-fma.f32 N/A

          \[\leadsto \sqrt{u1} \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, {u2}^{2}, \frac{314159265359}{50000000000}\right)} \cdot u2\right) \]

       5. unpow2 N/A

          \[\leadsto \sqrt{u1} \cdot \left(\mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \color{blue}{u2 \cdot u2}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \]

       6. lower-*.f32 46.1

          \[\leadsto \sqrt{u1} \cdot \left(\mathsf{fma}\left(-41.341702240407926, \color{blue}{u2 \cdot u2}, 6.28318530718\right) \cdot u2\right) \]

   11. Applied rewrites 46.1%

       \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(\mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right) \cdot u2\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 59.4%

       \[\leadsto \sqrt{u1} \cdot \left(\left(\left(u2 \cdot u2\right) \cdot -41.341702240407926 + 6.28318530718\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.002940000034868717:\\ \;\;\;\;\frac{1}{\sqrt{\frac{1 - u1}{u1}}} \cdot \left(u2 \cdot 6.28318530718\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-41.341702240407926 \cdot \left(u2 \cdot u2\right) + 6.28318530718\right) \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.002940000034868717:\\ \;\;\;\;\left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-41.341702240407926 \cdot \left(u2 \cdot u2\right) + 6.28318530718\right) \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.002940000034868717)
   (* (* u2 6.28318530718) (sqrt (/ u1 (- 1.0 u1))))
   (* (* (+ (* -41.341702240407926 (* u2 u2)) 6.28318530718) u2) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.002940000034868717f) {
		tmp = (u2 * 6.28318530718f) * sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = (((-41.341702240407926f * (u2 * u2)) + 6.28318530718f) * u2) * sqrtf(u1);
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((u2 * 6.28318530718e0) <= 0.002940000034868717e0) then
        tmp = (u2 * 6.28318530718e0) * sqrt((u1 / (1.0e0 - u1)))
    else
        tmp = ((((-41.341702240407926e0) * (u2 * u2)) + 6.28318530718e0) * u2) * sqrt(u1)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.002940000034868717))
		tmp = Float32(Float32(u2 * Float32(6.28318530718)) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1))));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(-41.341702240407926) * Float32(u2 * u2)) + Float32(6.28318530718)) * u2) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.002940000034868717))
		tmp = (u2 * single(6.28318530718)) * sqrt((u1 / (single(1.0) - u1)));
	else
		tmp = (((single(-41.341702240407926) * (u2 * u2)) + single(6.28318530718)) * u2) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00294000003

   1. Initial program 98.4%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \frac{314159265359}{50000000000}\right)} \]

      2. lower-*.f32 97.9

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

   5. Applied rewrites 97.9%

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

   if 0.00294000003 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.3%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \frac{314159265359}{50000000000}\right)} \]

      2. lower-*.f32 48.9

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

   5. Applied rewrites 48.9%

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

   6. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\sqrt{u1}} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]
   7. Step-by-step derivation

      1. lower-sqrt.f32 46.1

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

   8. Applied rewrites 46.1%

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

   9. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(\left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right) \cdot u2\right)} \]

       2. lower-*.f32 N/A

          \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(\left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right) \cdot u2\right)} \]

       3. +-commutative N/A

          \[\leadsto \sqrt{u1} \cdot \left(\color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2} + \frac{314159265359}{50000000000}\right)} \cdot u2\right) \]

       4. lower-fma.f32 N/A

          \[\leadsto \sqrt{u1} \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, {u2}^{2}, \frac{314159265359}{50000000000}\right)} \cdot u2\right) \]

       5. unpow2 N/A

          \[\leadsto \sqrt{u1} \cdot \left(\mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \color{blue}{u2 \cdot u2}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \]

       6. lower-*.f32 46.1

          \[\leadsto \sqrt{u1} \cdot \left(\mathsf{fma}\left(-41.341702240407926, \color{blue}{u2 \cdot u2}, 6.28318530718\right) \cdot u2\right) \]

   11. Applied rewrites 46.1%

       \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(\mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right) \cdot u2\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 59.4%

       \[\leadsto \sqrt{u1} \cdot \left(\left(\left(u2 \cdot u2\right) \cdot -41.341702240407926 + 6.28318530718\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.002940000034868717:\\ \;\;\;\;\left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-41.341702240407926 \cdot \left(u2 \cdot u2\right) + 6.28318530718\right) \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.002940000034868717:\\ \;\;\;\;\left(6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot u2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-41.341702240407926 \cdot \left(u2 \cdot u2\right) + 6.28318530718\right) \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.002940000034868717)
   (* (* 6.28318530718 (sqrt (/ u1 (- 1.0 u1)))) u2)
   (* (* (+ (* -41.341702240407926 (* u2 u2)) 6.28318530718) u2) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.002940000034868717f) {
		tmp = (6.28318530718f * sqrtf((u1 / (1.0f - u1)))) * u2;
	} else {
		tmp = (((-41.341702240407926f * (u2 * u2)) + 6.28318530718f) * u2) * sqrtf(u1);
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((u2 * 6.28318530718e0) <= 0.002940000034868717e0) then
        tmp = (6.28318530718e0 * sqrt((u1 / (1.0e0 - u1)))) * u2
    else
        tmp = ((((-41.341702240407926e0) * (u2 * u2)) + 6.28318530718e0) * u2) * sqrt(u1)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.002940000034868717))
		tmp = Float32(Float32(Float32(6.28318530718) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))) * u2);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(-41.341702240407926) * Float32(u2 * u2)) + Float32(6.28318530718)) * u2) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.002940000034868717))
		tmp = (single(6.28318530718) * sqrt((u1 / (single(1.0) - u1)))) * u2;
	else
		tmp = (((single(-41.341702240407926) * (u2 * u2)) + single(6.28318530718)) * u2) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00294000003

   1. Initial program 98.4%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + {u2}^{2} \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \sqrt{\frac{u1}{1 - u1}} + \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right)\right)\right)} \]

   5. Applied rewrites 97.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(81.6052492761019, u2 \cdot u2, -41.341702240407926\right), u2 \cdot u2, 6.28318530718\right)\right) \cdot u2} \]

   6. Taylor expanded in u2 around 0

      \[\leadsto \left(\sqrt{\frac{u1}{1 - u1}} \cdot \frac{314159265359}{50000000000}\right) \cdot u2 \]
   7. Step-by-step derivation

   8. Applied rewrites 97.8%

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

   if 0.00294000003 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.3%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \frac{314159265359}{50000000000}\right)} \]

      2. lower-*.f32 48.9

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

   5. Applied rewrites 48.9%

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

   6. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\sqrt{u1}} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]
   7. Step-by-step derivation

      1. lower-sqrt.f32 46.1

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

   8. Applied rewrites 46.1%

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

   9. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(\left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right) \cdot u2\right)} \]

       2. lower-*.f32 N/A

          \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(\left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right) \cdot u2\right)} \]

       3. +-commutative N/A

          \[\leadsto \sqrt{u1} \cdot \left(\color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2} + \frac{314159265359}{50000000000}\right)} \cdot u2\right) \]

       4. lower-fma.f32 N/A

          \[\leadsto \sqrt{u1} \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, {u2}^{2}, \frac{314159265359}{50000000000}\right)} \cdot u2\right) \]

       5. unpow2 N/A

          \[\leadsto \sqrt{u1} \cdot \left(\mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \color{blue}{u2 \cdot u2}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \]

       6. lower-*.f32 46.1

          \[\leadsto \sqrt{u1} \cdot \left(\mathsf{fma}\left(-41.341702240407926, \color{blue}{u2 \cdot u2}, 6.28318530718\right) \cdot u2\right) \]

   11. Applied rewrites 46.1%

       \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(\mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right) \cdot u2\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 59.4%

       \[\leadsto \sqrt{u1} \cdot \left(\left(\left(u2 \cdot u2\right) \cdot -41.341702240407926 + 6.28318530718\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.002940000034868717:\\ \;\;\;\;\left(6.28318530718 \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot u2\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-41.341702240407926 \cdot \left(u2 \cdot u2\right) + 6.28318530718\right) \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(-41.341702240407926 \cdot \left(u2 \cdot u2\right) + 6.28318530718\right) \cdot u2\right) \cdot \sqrt{u1} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (* (+ (* -41.341702240407926 (* u2 u2)) 6.28318530718) u2) (sqrt u1)))

C

float code(float cosTheta_i, float u1, float u2) {
	return (((-41.341702240407926f * (u2 * u2)) + 6.28318530718f) * u2) * sqrtf(u1);
}

Fortran

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 = ((((-41.341702240407926e0) * (u2 * u2)) + 6.28318530718e0) * u2) * sqrt(u1)
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(Float32(Float32(Float32(-41.341702240407926) * Float32(u2 * u2)) + Float32(6.28318530718)) * u2) * sqrt(u1))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (((single(-41.341702240407926) * (u2 * u2)) + single(6.28318530718)) * u2) * sqrt(u1);
end

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \frac{314159265359}{50000000000}\right)} \]

   2. lower-*.f32 80.8

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

5. Applied rewrites 80.8%

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

6. Taylor expanded in u1 around 0

   \[\leadsto \color{blue}{\sqrt{u1}} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]
7. Step-by-step derivation

   1. lower-sqrt.f32 65.1

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

8. Applied rewrites 65.1%

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

9. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(\left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right) \cdot u2\right)} \]

    2. lower-*.f32 N/A

       \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(\left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right) \cdot u2\right)} \]

    3. +-commutative N/A

       \[\leadsto \sqrt{u1} \cdot \left(\color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2} + \frac{314159265359}{50000000000}\right)} \cdot u2\right) \]

    4. lower-fma.f32 N/A

       \[\leadsto \sqrt{u1} \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, {u2}^{2}, \frac{314159265359}{50000000000}\right)} \cdot u2\right) \]

    5. unpow2 N/A

       \[\leadsto \sqrt{u1} \cdot \left(\mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \color{blue}{u2 \cdot u2}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \]

    6. lower-*.f32 65.1

       \[\leadsto \sqrt{u1} \cdot \left(\mathsf{fma}\left(-41.341702240407926, \color{blue}{u2 \cdot u2}, 6.28318530718\right) \cdot u2\right) \]

11. Applied rewrites 65.1%

    \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(\mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right) \cdot u2\right)} \]
12. Step-by-step derivation

13. Applied rewrites 69.8%

    \[\leadsto \sqrt{u1} \cdot \left(\left(\left(u2 \cdot u2\right) \cdot -41.341702240407926 + 6.28318530718\right) \cdot u2\right) \]

14. Final simplification 69.8%

    \[\leadsto \left(\left(-41.341702240407926 \cdot \left(u2 \cdot u2\right) + 6.28318530718\right) \cdot u2\right) \cdot \sqrt{u1} \]
15. Add Preprocessing

Alternative 10: 64.5% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1} \cdot \left(u2 \cdot 6.28318530718\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt u1) (* u2 6.28318530718)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(u1) * (u2 * 6.28318530718f);
}

Fortran

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) * (u2 * 6.28318530718e0)
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(u1) * Float32(u2 * Float32(6.28318530718)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(u1) * (u2 * single(6.28318530718));
end

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \frac{314159265359}{50000000000}\right)} \]

   2. lower-*.f32 80.8

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

5. Applied rewrites 80.8%

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

6. Taylor expanded in u1 around 0

   \[\leadsto \color{blue}{\sqrt{u1}} \cdot \left(u2 \cdot \frac{314159265359}{50000000000}\right) \]
7. Step-by-step derivation

   1. lower-sqrt.f32 65.1

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

8. Applied rewrites 65.1%

   \[\leadsto \color{blue}{\sqrt{u1}} \cdot \left(u2 \cdot 6.28318530718\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024248 
(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))))