Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.2%

Time: 16.6s

Alternatives: 18

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, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (/ (sin (* 6.28318530718 u2)) (pow (/ 1.0 (+ (/ 1.0 u1) -1.0)) -0.5)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf((6.28318530718f * u2)) / powf((1.0f / ((1.0f / u1) + -1.0f)), -0.5f);
}

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((6.28318530718e0 * u2)) / ((1.0e0 / ((1.0e0 / u1) + (-1.0e0))) ** (-0.5e0))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sin(Float32(Float32(6.28318530718) * u2)) / (Float32(Float32(1.0) / Float32(Float32(Float32(1.0) / u1) + Float32(-1.0))) ^ Float32(-0.5)))
end

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. *-lft-identity N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\left(1 \cdot \frac{u1}{1 - u1}\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\color{blue}{\frac{314159265359}{50000000000}}, u2\right)\right)\right) \]

   2. *-inverses N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\left(\frac{u1}{u1} \cdot \frac{u1}{1 - u1}\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   3. times-frac N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\left(\frac{u1 \cdot u1}{u1 \cdot \left(1 - u1\right)}\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\color{blue}{\frac{314159265359}{50000000000}}, u2\right)\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\left(\frac{1 \cdot \left(u1 \cdot u1\right)}{u1 \cdot \left(1 - u1\right)}\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   5. times-frac N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\left(\frac{1}{u1} \cdot \frac{u1 \cdot u1}{1 - u1}\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\color{blue}{\frac{314159265359}{50000000000}}, u2\right)\right)\right) \]

   6. sqr-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\left(\frac{1}{u1} \cdot \frac{\left(\mathsf{neg}\left(u1\right)\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)}{1 - u1}\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\left(\frac{1}{u1} \cdot \left(\left(\mathsf{neg}\left(u1\right)\right) \cdot \frac{\mathsf{neg}\left(u1\right)}{1 - u1}\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   8. distribute-frac-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\left(\frac{1}{u1} \cdot \left(\left(\mathsf{neg}\left(u1\right)\right) \cdot \left(\mathsf{neg}\left(\frac{u1}{1 - u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   9. distribute-frac-neg2 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\left(\frac{1}{u1} \cdot \left(\left(\mathsf{neg}\left(u1\right)\right) \cdot \frac{u1}{\mathsf{neg}\left(\left(1 - u1\right)\right)}\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\left(\frac{1}{u1}\right), \left(\left(\mathsf{neg}\left(u1\right)\right) \cdot \frac{u1}{\mathsf{neg}\left(\left(1 - u1\right)\right)}\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\color{blue}{\frac{314159265359}{50000000000}}, u2\right)\right)\right) \]

   11. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(1, u1\right), \left(\left(\mathsf{neg}\left(u1\right)\right) \cdot \frac{u1}{\mathsf{neg}\left(\left(1 - u1\right)\right)}\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   12. clear-num N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(1, u1\right), \left(\left(\mathsf{neg}\left(u1\right)\right) \cdot \frac{1}{\frac{\mathsf{neg}\left(\left(1 - u1\right)\right)}{u1}}\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   13. distribute-neg-frac N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(1, u1\right), \left(\left(\mathsf{neg}\left(u1\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\frac{1 - u1}{u1}\right)}\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(1, u1\right), \left(\left(\mathsf{neg}\left(u1\right)\right) \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\frac{1 - u1}{u1}\right)}\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   15. frac-2neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(1, u1\right), \left(\left(\mathsf{neg}\left(u1\right)\right) \cdot \frac{-1}{\frac{1 - u1}{u1}}\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   16. associate-*r/ N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(1, u1\right), \left(\frac{\left(\mathsf{neg}\left(u1\right)\right) \cdot -1}{\frac{1 - u1}{u1}}\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(1, u1\right), \left(\frac{-1 \cdot \left(\mathsf{neg}\left(u1\right)\right)}{\frac{1 - u1}{u1}}\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   18. neg-mul-1 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(1, u1\right), \left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(u1\right)\right)\right)}{\frac{1 - u1}{u1}}\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   19. remove-double-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(1, u1\right), \left(\frac{u1}{\frac{1 - u1}{u1}}\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   20. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(1, u1\right), \mathsf{/.f32}\left(u1, \left(\frac{1 - u1}{u1}\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   21. div-sub N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(1, u1\right), \mathsf{/.f32}\left(u1, \left(\frac{1}{u1} - \frac{u1}{u1}\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   22. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(1, u1\right), \mathsf{/.f32}\left(u1, \left(\frac{1}{u1} + \left(\mathsf{neg}\left(\frac{u1}{u1}\right)\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   23. *-inverses N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(1, u1\right), \mathsf{/.f32}\left(u1, \left(\frac{1}{u1} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   24. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(1, u1\right), \mathsf{/.f32}\left(u1, \left(\frac{1}{u1} + -1\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

4. Applied egg-rr 98.6%

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. lft-mult-inverse N/A

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

   4. sqrt-div N/A

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

   5. metadata-eval N/A

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

   6. unpow1/2 N/A

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

   7. div-inv N/A

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

   8. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\sin \left(\frac{314159265359}{50000000000} \cdot u2\right), \color{blue}{\left({\left(-1 + \frac{1}{u1}\right)}^{\frac{1}{2}}\right)}\right) \]

   9. sin-lowering-sin.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{sin.f32}\left(\left(\frac{314159265359}{50000000000} \cdot u2\right)\right), \left({\color{blue}{\left(-1 + \frac{1}{u1}\right)}}^{\frac{1}{2}}\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right), \left({\left(\color{blue}{-1} + \frac{1}{u1}\right)}^{\frac{1}{2}}\right)\right) \]

   11. pow-lowering-pow.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right), \mathsf{pow.f32}\left(\left(-1 + \frac{1}{u1}\right), \color{blue}{\frac{1}{2}}\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right), \mathsf{pow.f32}\left(\left(\frac{1}{u1} + -1\right), \frac{1}{2}\right)\right) \]

   13. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right), \mathsf{pow.f32}\left(\mathsf{+.f32}\left(\left(\frac{1}{u1}\right), -1\right), \frac{1}{2}\right)\right) \]

   14. /-lowering-/.f32 98.5%

       \[\leadsto \mathsf{/.f32}\left(\mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right), \mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{1}{2}\right)\right) \]

6. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\frac{\sin \left(6.28318530718 \cdot u2\right)}{{\left(\frac{1}{u1} + -1\right)}^{0.5}}} \]
7. Step-by-step derivation

   1. remove-double-div N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right), \left({\left(\frac{1}{\frac{1}{\frac{1}{u1} + -1}}\right)}^{\frac{1}{2}}\right)\right) \]

   2. inv-pow N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right), \left({\left({\left(\frac{1}{\frac{1}{u1} + -1}\right)}^{-1}\right)}^{\frac{1}{2}}\right)\right) \]

   3. pow-pow N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right), \left({\left(\frac{1}{\frac{1}{u1} + -1}\right)}^{\color{blue}{\left(-1 \cdot \frac{1}{2}\right)}}\right)\right) \]

   4. pow-lowering-pow.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right), \mathsf{pow.f32}\left(\left(\frac{1}{\frac{1}{u1} + -1}\right), \color{blue}{\left(-1 \cdot \frac{1}{2}\right)}\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right), \mathsf{pow.f32}\left(\mathsf{/.f32}\left(1, \left(\frac{1}{u1} + -1\right)\right), \left(\color{blue}{-1} \cdot \frac{1}{2}\right)\right)\right) \]

   6. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right), \mathsf{pow.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(\frac{1}{u1}\right), -1\right)\right), \left(-1 \cdot \frac{1}{2}\right)\right)\right) \]

   7. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right), \mathsf{pow.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right)\right), \left(-1 \cdot \frac{1}{2}\right)\right)\right) \]

   8. metadata-eval 98.7%

      \[\leadsto \mathsf{/.f32}\left(\mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right), \mathsf{pow.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right)\right), \frac{-1}{2}\right)\right) \]

8. Applied egg-rr 98.7%

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

Alternative 2: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.5:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot \left(-41.341702240407926 + u2 \cdot \left(u2 \cdot \left(81.6052492761019 + \left(u2 \cdot u2\right) \cdot -76.70585975309672\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.5)
   (*
    (sqrt (/ u1 (- 1.0 u1)))
    (*
     u2
     (+
      6.28318530718
      (*
       (* u2 u2)
       (+
        -41.341702240407926
        (*
         u2
         (* u2 (+ 81.6052492761019 (* (* u2 u2) -76.70585975309672)))))))))
   (* (sin (* 6.28318530718 u2)) (sqrt (* u1 (+ 1.0 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.5f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * (u2 * (6.28318530718f + ((u2 * u2) * (-41.341702240407926f + (u2 * (u2 * (81.6052492761019f + ((u2 * u2) * -76.70585975309672f))))))));
	} else {
		tmp = sinf((6.28318530718f * u2)) * sqrtf((u1 * (1.0f + 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 ((6.28318530718e0 * u2) <= 0.5e0) then
        tmp = sqrt((u1 / (1.0e0 - u1))) * (u2 * (6.28318530718e0 + ((u2 * u2) * ((-41.341702240407926e0) + (u2 * (u2 * (81.6052492761019e0 + ((u2 * u2) * (-76.70585975309672e0)))))))))
    else
        tmp = sin((6.28318530718e0 * u2)) * sqrt((u1 * (1.0e0 + u1)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.5))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(u2 * Float32(Float32(6.28318530718) + Float32(Float32(u2 * u2) * Float32(Float32(-41.341702240407926) + Float32(u2 * Float32(u2 * Float32(Float32(81.6052492761019) + Float32(Float32(u2 * u2) * Float32(-76.70585975309672))))))))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(Float32(u1 * Float32(Float32(1.0) + u1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.5))
		tmp = sqrt((u1 / (single(1.0) - u1))) * (u2 * (single(6.28318530718) + ((u2 * u2) * (single(-41.341702240407926) + (u2 * (u2 * (single(81.6052492761019) + ((u2 * u2) * single(-76.70585975309672)))))))));
	else
		tmp = sin((single(6.28318530718) * u2)) * sqrt((u1 * (single(1.0) + u1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.5

   1. Initial program 98.7%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \color{blue}{\left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)}\right)\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \color{blue}{\left({u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)}\right)\right)\right) \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\left({u2}^{2}\right), \color{blue}{\left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)}\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\left(u2 \cdot u2\right), \left(\color{blue}{{u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \left(\color{blue}{{u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)}\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} + \color{blue}{{u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)}\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \color{blue}{\left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \left(\left(u2 \cdot u2\right) \cdot \left(\color{blue}{\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \left(u2 \cdot \color{blue}{\left(u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \mathsf{*.f32}\left(u2, \color{blue}{\left(u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \color{blue}{\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \color{blue}{\left(\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \left({u2}^{2} \cdot \color{blue}{\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \mathsf{*.f32}\left(\left({u2}^{2}\right), \color{blue}{\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \mathsf{*.f32}\left(\left(u2 \cdot u2\right), \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f32 98.7%

          \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 98.7%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot \left(-41.341702240407926 + u2 \cdot \left(u2 \cdot \left(81.6052492761019 + \left(u2 \cdot u2\right) \cdot -76.70585975309672\right)\right)\right)\right)\right)} \]

   if 0.5 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.6%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\color{blue}{\left(u1 \cdot \left(1 + u1\right)\right)}\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(u1, \left(1 + u1\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\color{blue}{\frac{314159265359}{50000000000}}, u2\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(u1, \left(u1 + 1\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

      3. +-lowering-+.f32 91.9%

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{+.f32}\left(u1, 1\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   5. Simplified 91.9%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(u1 + 1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.5:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot \left(-41.341702240407926 + u2 \cdot \left(u2 \cdot \left(81.6052492761019 + \left(u2 \cdot u2\right) \cdot -76.70585975309672\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf((6.28318530718f * u2)) * sqrtf((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((6.28318530718e0 * u2)) * sqrt((u1 / (1.0e0 - u1)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Final simplification 98.6%

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

Alternative 4: 94.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot \left(-41.341702240407926 + u2 \cdot \left(u2 \cdot \left(81.6052492761019 + \left(u2 \cdot u2\right) \cdot -76.70585975309672\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ u1 (- 1.0 u1)))
  (*
   u2
   (+
    6.28318530718
    (*
     (* u2 u2)
     (+
      -41.341702240407926
      (* u2 (* u2 (+ 81.6052492761019 (* (* u2 u2) -76.70585975309672))))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * (u2 * (6.28318530718f + ((u2 * u2) * (-41.341702240407926f + (u2 * (u2 * (81.6052492761019f + ((u2 * u2) * -76.70585975309672f))))))));
}

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))) * (u2 * (6.28318530718e0 + ((u2 * u2) * ((-41.341702240407926e0) + (u2 * (u2 * (81.6052492761019e0 + ((u2 * u2) * (-76.70585975309672e0)))))))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(u2 * Float32(Float32(6.28318530718) + Float32(Float32(u2 * u2) * Float32(Float32(-41.341702240407926) + Float32(u2 * Float32(u2 * Float32(Float32(81.6052492761019) + Float32(Float32(u2 * u2) * Float32(-76.70585975309672))))))))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * (u2 * (single(6.28318530718) + ((u2 * u2) * (single(-41.341702240407926) + (u2 * (u2 * (single(81.6052492761019) + ((u2 * u2) * single(-76.70585975309672)))))))));
end

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)}\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \color{blue}{\left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)}\right)\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \color{blue}{\left({u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)}\right)\right)\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\left({u2}^{2}\right), \color{blue}{\left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)}\right)\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\left(u2 \cdot u2\right), \left(\color{blue}{{u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \left(\color{blue}{{u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)}\right)\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} + \color{blue}{{u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)}\right)\right)\right)\right)\right) \]

   9. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \color{blue}{\left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \left(\left(u2 \cdot u2\right) \cdot \left(\color{blue}{\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)\right)\right)\right)\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \left(u2 \cdot \color{blue}{\left(u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \mathsf{*.f32}\left(u2, \color{blue}{\left(u2 \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \color{blue}{\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   14. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \color{blue}{\left(\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \left({u2}^{2} \cdot \color{blue}{\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   16. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \mathsf{*.f32}\left(\left({u2}^{2}\right), \color{blue}{\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   17. unpow2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \mathsf{*.f32}\left(\left(u2 \cdot u2\right), \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   18. *-lowering-*.f32 94.6%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

5. Simplified 94.6%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot \left(-41.341702240407926 + u2 \cdot \left(u2 \cdot \left(81.6052492761019 + \left(u2 \cdot u2\right) \cdot -76.70585975309672\right)\right)\right)\right)\right)} \]
6. Add Preprocessing

Alternative 5: 86.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{u1}{1 - u1} \leq 0.0020000000949949026:\\ \;\;\;\;u2 \cdot \left(\sqrt{u1 \cdot \left(1 + u1\right)} \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot -41.341702240407926\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6.28318530718 \cdot u2}{{\left(\frac{1}{u1} + -1\right)}^{0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (/ u1 (- 1.0 u1)) 0.0020000000949949026)
   (*
    u2
    (*
     (sqrt (* u1 (+ 1.0 u1)))
     (+ 6.28318530718 (* (* u2 u2) -41.341702240407926))))
   (/ (* 6.28318530718 u2) (pow (+ (/ 1.0 u1) -1.0) 0.5))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u1 / (1.0f - u1)) <= 0.0020000000949949026f) {
		tmp = u2 * (sqrtf((u1 * (1.0f + u1))) * (6.28318530718f + ((u2 * u2) * -41.341702240407926f)));
	} else {
		tmp = (6.28318530718f * u2) / powf(((1.0f / u1) + -1.0f), 0.5f);
	}
	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 ((u1 / (1.0e0 - u1)) <= 0.0020000000949949026e0) then
        tmp = u2 * (sqrt((u1 * (1.0e0 + u1))) * (6.28318530718e0 + ((u2 * u2) * (-41.341702240407926e0))))
    else
        tmp = (6.28318530718e0 * u2) / (((1.0e0 / u1) + (-1.0e0)) ** 0.5e0)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u1 / Float32(Float32(1.0) - u1)) <= Float32(0.0020000000949949026))
		tmp = Float32(u2 * Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + u1))) * Float32(Float32(6.28318530718) + Float32(Float32(u2 * u2) * Float32(-41.341702240407926)))));
	else
		tmp = Float32(Float32(Float32(6.28318530718) * u2) / (Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)) ^ Float32(0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u1 / (single(1.0) - u1)) <= single(0.0020000000949949026))
		tmp = u2 * (sqrt((u1 * (single(1.0) + u1))) * (single(6.28318530718) + ((u2 * u2) * single(-41.341702240407926))));
	else
		tmp = (single(6.28318530718) * u2) / (((single(1.0) / u1) + single(-1.0)) ^ single(0.5));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 u1 (-.f32 #s(literal 1 binary32) u1)) < 0.00200000009

   1. Initial program 98.6%

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

      1. *-lft-identity N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\left(1 \cdot \frac{u1}{1 - u1}\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\color{blue}{\frac{314159265359}{50000000000}}, u2\right)\right)\right) \]

      2. *-inverses N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\left(\frac{u1}{u1} \cdot \frac{u1}{1 - u1}\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

      3. times-frac N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\left(\frac{u1 \cdot u1}{u1 \cdot \left(1 - u1\right)}\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\color{blue}{\frac{314159265359}{50000000000}}, u2\right)\right)\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\left(\frac{1 \cdot \left(u1 \cdot u1\right)}{u1 \cdot \left(1 - u1\right)}\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

      5. times-frac N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\left(\frac{1}{u1} \cdot \frac{u1 \cdot u1}{1 - u1}\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\color{blue}{\frac{314159265359}{50000000000}}, u2\right)\right)\right) \]

      6. sqr-neg N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\left(\frac{1}{u1} \cdot \frac{\left(\mathsf{neg}\left(u1\right)\right) \cdot \left(\mathsf{neg}\left(u1\right)\right)}{1 - u1}\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\left(\frac{1}{u1} \cdot \left(\left(\mathsf{neg}\left(u1\right)\right) \cdot \frac{\mathsf{neg}\left(u1\right)}{1 - u1}\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

      8. distribute-frac-neg N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\left(\frac{1}{u1} \cdot \left(\left(\mathsf{neg}\left(u1\right)\right) \cdot \left(\mathsf{neg}\left(\frac{u1}{1 - u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

      9. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\left(\frac{1}{u1} \cdot \left(\left(\mathsf{neg}\left(u1\right)\right) \cdot \frac{u1}{\mathsf{neg}\left(\left(1 - u1\right)\right)}\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

      10. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\left(\frac{1}{u1}\right), \left(\left(\mathsf{neg}\left(u1\right)\right) \cdot \frac{u1}{\mathsf{neg}\left(\left(1 - u1\right)\right)}\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\color{blue}{\frac{314159265359}{50000000000}}, u2\right)\right)\right) \]

      11. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(1, u1\right), \left(\left(\mathsf{neg}\left(u1\right)\right) \cdot \frac{u1}{\mathsf{neg}\left(\left(1 - u1\right)\right)}\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

      12. clear-num N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(1, u1\right), \left(\left(\mathsf{neg}\left(u1\right)\right) \cdot \frac{1}{\frac{\mathsf{neg}\left(\left(1 - u1\right)\right)}{u1}}\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

      13. distribute-neg-frac N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(1, u1\right), \left(\left(\mathsf{neg}\left(u1\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\frac{1 - u1}{u1}\right)}\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(1, u1\right), \left(\left(\mathsf{neg}\left(u1\right)\right) \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\frac{1 - u1}{u1}\right)}\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

      15. frac-2neg N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(1, u1\right), \left(\left(\mathsf{neg}\left(u1\right)\right) \cdot \frac{-1}{\frac{1 - u1}{u1}}\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

      16. associate-*r/ N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(1, u1\right), \left(\frac{\left(\mathsf{neg}\left(u1\right)\right) \cdot -1}{\frac{1 - u1}{u1}}\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(1, u1\right), \left(\frac{-1 \cdot \left(\mathsf{neg}\left(u1\right)\right)}{\frac{1 - u1}{u1}}\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

      18. neg-mul-1 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(1, u1\right), \left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(u1\right)\right)\right)}{\frac{1 - u1}{u1}}\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

      19. remove-double-neg N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(1, u1\right), \left(\frac{u1}{\frac{1 - u1}{u1}}\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

      20. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(1, u1\right), \mathsf{/.f32}\left(u1, \left(\frac{1 - u1}{u1}\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

      21. div-sub N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(1, u1\right), \mathsf{/.f32}\left(u1, \left(\frac{1}{u1} - \frac{u1}{u1}\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

      22. sub-neg N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(1, u1\right), \mathsf{/.f32}\left(u1, \left(\frac{1}{u1} + \left(\mathsf{neg}\left(\frac{u1}{u1}\right)\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

      23. *-inverses N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(1, u1\right), \mathsf{/.f32}\left(u1, \left(\frac{1}{u1} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

      24. metadata-eval N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(1, u1\right), \mathsf{/.f32}\left(u1, \left(\frac{1}{u1} + -1\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   4. Applied egg-rr 98.6%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left({u2}^{2} \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(u2, \color{blue}{\left(\frac{314159265359}{50000000000} \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}} + {u2}^{2} \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}\right)\right)}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f32}\left(u2, \left(\frac{314159265359}{50000000000} \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}} + \left({u2}^{2} \cdot \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right) \cdot \color{blue}{\sqrt{\frac{1}{\frac{1}{u1} - 1}}}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(u2, \left(\frac{314159265359}{50000000000} \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}} + \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{1}{u1} - 1}}}\right)\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f32}\left(u2, \left(\sqrt{\frac{1}{\frac{1}{u1} - 1}} \cdot \color{blue}{\left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)}\right)\right) \]

      9. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(\left(\sqrt{\frac{1}{\frac{1}{u1} - 1}}\right), \color{blue}{\left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)}\right)\right) \]

   7. Simplified 89.6%

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

   8. Taylor expanded in u1 around 0

      \[\leadsto \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\color{blue}{\left(u1 \cdot \left(1 + u1\right)\right)}\right), \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \mathsf{*.f32}\left(u2, u2\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(u1, \left(1 + u1\right)\right)\right), \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \mathsf{*.f32}\left(u2, u2\right)\right)\right)\right)\right) \]

      2. +-lowering-+.f32 88.9%

         \[\leadsto \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{+.f32}\left(1, u1\right)\right)\right), \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \mathsf{*.f32}\left(u2, u2\right)\right)\right)\right)\right) \]

   10. Simplified 88.9%

       \[\leadsto u2 \cdot \left(\sqrt{\color{blue}{u1 \cdot \left(1 + u1\right)}} \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right) \]

   if 0.00200000009 < (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 98.6%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\left(\frac{314159265359}{50000000000} \cdot u2\right), \color{blue}{\left(\sqrt{\frac{u1}{1 - u1}}\right)}\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\color{blue}{\frac{u1}{1 - u1}}}\right)\right) \]

      5. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{1 - u1}}\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{1 + \left(\mathsf{neg}\left(u1\right)\right)}}\right)\right) \]

      7. rgt-mult-inverse N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \frac{1}{-1 \cdot u1} + \left(\mathsf{neg}\left(u1\right)\right)}}\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \frac{1}{\mathsf{neg}\left(u1\right)} + \left(\mathsf{neg}\left(u1\right)\right)}}\right)\right) \]

      9. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \left(\mathsf{neg}\left(u1\right)\right)}}\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + -1 \cdot u1}}\right)\right) \]

      11. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \left(-1 \cdot u1\right) \cdot 1}}\right)\right) \]

      12. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}}\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(1 - \frac{1}{u1}\right)}}\right)\right) \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}\right)\right) \]

      16. sqrt-lowering-sqrt.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \mathsf{sqrt.f32}\left(\left(\frac{u1 \cdot 1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}\right)\right)\right) \]

      17. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \mathsf{sqrt.f32}\left(\left(\frac{u1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}\right)\right)\right) \]

      18. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)\right)\right)\right)\right) \]

      19. associate-*r* N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\left(-1 \cdot u1\right) \cdot \left(1 - \frac{1}{u1}\right)\right)\right)\right)\right) \]

      20. sub-neg N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\left(-1 \cdot u1\right) \cdot \left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)\right)\right)\right)\right) \]

      21. +-commutative N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\left(-1 \cdot u1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)\right)\right)\right)\right) \]

   5. Simplified 83.9%

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

      1. clear-num N/A

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

      2. div-sub N/A

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

      3. *-lft-identity N/A

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

      4. associate-*l/ N/A

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

      5. lft-mult-inverse N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. sqrt-div N/A

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

      10. metadata-eval N/A

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

      11. unpow1/2 N/A

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

      12. un-div-inv N/A

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

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\left(\frac{314159265359}{50000000000} \cdot u2\right), \color{blue}{\left({\left(-1 + \frac{1}{u1}\right)}^{\frac{1}{2}}\right)}\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left({\color{blue}{\left(-1 + \frac{1}{u1}\right)}}^{\frac{1}{2}}\right)\right) \]

      15. pow-lowering-pow.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \mathsf{pow.f32}\left(\left(-1 + \frac{1}{u1}\right), \color{blue}{\frac{1}{2}}\right)\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \mathsf{pow.f32}\left(\left(\frac{1}{u1} + -1\right), \frac{1}{2}\right)\right) \]

      17. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \mathsf{pow.f32}\left(\mathsf{+.f32}\left(\left(\frac{1}{u1}\right), -1\right), \frac{1}{2}\right)\right) \]

      18. /-lowering-/.f32 84.0%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \mathsf{pow.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \frac{1}{2}\right)\right) \]

   7. Applied egg-rr 84.0%

      \[\leadsto \color{blue}{\frac{6.28318530718 \cdot u2}{{\left(\frac{1}{u1} + -1\right)}^{0.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{u1}{1 - u1} \leq 0.0020000000949949026:\\ \;\;\;\;u2 \cdot \left(\sqrt{u1 \cdot \left(1 + u1\right)} \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot -41.341702240407926\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{6.28318530718 \cdot u2}{{\left(\frac{1}{u1} + -1\right)}^{0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot \left(-41.341702240407926 + u2 \cdot \left(u2 \cdot 81.6052492761019\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ u1 (- 1.0 u1)))
  (*
   u2
   (+
    6.28318530718
    (* (* u2 u2) (+ -41.341702240407926 (* u2 (* u2 81.6052492761019))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * (u2 * (6.28318530718f + ((u2 * u2) * (-41.341702240407926f + (u2 * (u2 * 81.6052492761019f))))));
}

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))) * (u2 * (6.28318530718e0 + ((u2 * u2) * ((-41.341702240407926e0) + (u2 * (u2 * 81.6052492761019e0))))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(u2 * Float32(Float32(6.28318530718) + Float32(Float32(u2 * u2) * Float32(Float32(-41.341702240407926) + Float32(u2 * Float32(u2 * Float32(81.6052492761019))))))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * (u2 * (single(6.28318530718) + ((u2 * u2) * (single(-41.341702240407926) + (u2 * (u2 * single(81.6052492761019)))))));
end

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)}\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \color{blue}{\left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)}\right)\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \color{blue}{\left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)}\right)\right)\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\left({u2}^{2}\right), \color{blue}{\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)}\right)\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\left(u2 \cdot u2\right), \left(\color{blue}{\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \left(\color{blue}{\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)}\right)\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} + \color{blue}{\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}}\right)\right)\right)\right)\right) \]

   9. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \color{blue}{\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)}\right)\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot \left(u2 \cdot \color{blue}{u2}\right)\right)\right)\right)\right)\right)\right) \]

   11. associate-*r* N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \left(\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot u2\right) \cdot \color{blue}{u2}\right)\right)\right)\right)\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \left(u2 \cdot \color{blue}{\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot u2\right)}\right)\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \mathsf{*.f32}\left(u2, \color{blue}{\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} \cdot u2\right)}\right)\right)\right)\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \mathsf{*.f32}\left(u2, \left(u2 \cdot \color{blue}{\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}}\right)\right)\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f32 92.5%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \color{blue}{\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}}\right)\right)\right)\right)\right)\right)\right) \]

5. Simplified 92.5%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot \left(-41.341702240407926 + u2 \cdot \left(u2 \cdot 81.6052492761019\right)\right)\right)\right)} \]
6. Add Preprocessing

Alternative 7: 89.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot -41.341702240407926\right)\right) \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * (u2 * (6.28318530718f + ((u2 * u2) * -41.341702240407926f)));
}

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))) * (u2 * (6.28318530718e0 + ((u2 * u2) * (-41.341702240407926e0))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(u2 * Float32(Float32(6.28318530718) + Float32(Float32(u2 * u2) * Float32(-41.341702240407926)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * (u2 * (single(6.28318530718) + ((u2 * u2) * single(-41.341702240407926))));
end

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)}\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \color{blue}{\left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)}\right)\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)}\right)\right)\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \color{blue}{\left({u2}^{2}\right)}\right)\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \left(u2 \cdot \color{blue}{u2}\right)\right)\right)\right)\right) \]

   5. *-lowering-*.f32 90.2%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \mathsf{*.f32}\left(u2, \color{blue}{u2}\right)\right)\right)\right)\right) \]

5. Simplified 90.2%

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

6. Final simplification 90.2%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot -41.341702240407926\right)\right) \]
7. Add Preprocessing

Alternative 8: 89.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot -41.341702240407926\right)\right) \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 * (sqrtf((u1 / (1.0f - u1))) * (6.28318530718f + ((u2 * u2) * -41.341702240407926f)));
}

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 = u2 * (sqrt((u1 / (1.0e0 - u1))) * (6.28318530718e0 + ((u2 * u2) * (-41.341702240407926e0))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u2 * Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(6.28318530718) + Float32(Float32(u2 * u2) * Float32(-41.341702240407926)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u2 * (sqrt((u1 / (single(1.0) - u1))) * (single(6.28318530718) + ((u2 * u2) * single(-41.341702240407926))));
end

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(u2, \color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u2, \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left({u2}^{2} \cdot \sqrt{\frac{u1}{1 - u1}}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{*.f32}\left(u2, \left(\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right) \cdot \sqrt{\frac{u1}{1 - u1}} + \color{blue}{\frac{314159265359}{50000000000}} \cdot \sqrt{\frac{u1}{1 - u1}}\right)\right) \]

   4. distribute-rgt-out N/A

      \[\leadsto \mathsf{*.f32}\left(u2, \left(\sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2} + \frac{314159265359}{50000000000}\right)}\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(u2, \left(\sqrt{\frac{u1}{1 - u1}} \cdot \left(\frac{314159265359}{50000000000} + \color{blue}{\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}}\right)\right)\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(\left(\sqrt{\frac{u1}{1 - u1}}\right), \color{blue}{\left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)}\right)\right) \]

5. Simplified 90.1%

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

6. Final simplification 90.1%

   \[\leadsto u2 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot -41.341702240407926\right)\right) \]
7. Add Preprocessing

Alternative 9: 81.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return (6.28318530718f * u2) * sqrtf((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 = (6.28318530718e0 * u2) * sqrt((u1 / (1.0e0 - u1)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\frac{314159265359}{50000000000} \cdot u2\right), \color{blue}{\left(\sqrt{\frac{u1}{1 - u1}}\right)}\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\color{blue}{\frac{u1}{1 - u1}}}\right)\right) \]

   5. *-rgt-identity N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{1 - u1}}\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{1 + \left(\mathsf{neg}\left(u1\right)\right)}}\right)\right) \]

   7. rgt-mult-inverse N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \frac{1}{-1 \cdot u1} + \left(\mathsf{neg}\left(u1\right)\right)}}\right)\right) \]

   8. mul-1-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \frac{1}{\mathsf{neg}\left(u1\right)} + \left(\mathsf{neg}\left(u1\right)\right)}}\right)\right) \]

   9. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \left(\mathsf{neg}\left(u1\right)\right)}}\right)\right) \]

   10. mul-1-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + -1 \cdot u1}}\right)\right) \]

   11. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \left(-1 \cdot u1\right) \cdot 1}}\right)\right) \]

   12. distribute-lft-in N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}}\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}}\right)\right) \]

   14. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(1 - \frac{1}{u1}\right)}}\right)\right) \]

   15. associate-*r* N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}\right)\right) \]

   16. sqrt-lowering-sqrt.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \mathsf{sqrt.f32}\left(\left(\frac{u1 \cdot 1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}\right)\right)\right) \]

   17. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \mathsf{sqrt.f32}\left(\left(\frac{u1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}\right)\right)\right) \]

   18. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)\right)\right)\right)\right) \]

   19. associate-*r* N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\left(-1 \cdot u1\right) \cdot \left(1 - \frac{1}{u1}\right)\right)\right)\right)\right) \]

   20. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\left(-1 \cdot u1\right) \cdot \left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)\right)\right)\right)\right) \]

   21. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\left(-1 \cdot u1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)\right)\right)\right)\right) \]

5. Simplified 82.7%

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

Alternative 10: 81.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 * (6.28318530718f * sqrtf((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 = u2 * (6.28318530718e0 * sqrt((u1 / (1.0e0 - u1))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
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. Simplified 92.4%

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

6. Taylor expanded in u2 around 0

   \[\leadsto \mathsf{*.f32}\left(u2, \color{blue}{\left(\frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)}\right) \]
7. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \color{blue}{\left(\sqrt{\frac{u1}{1 - u1}}\right)}\right)\right) \]

   2. sqrt-lowering-sqrt.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \mathsf{sqrt.f32}\left(\left(\frac{u1}{1 - u1}\right)\right)\right)\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \mathsf{sqrt.f32}\left(\left(\frac{u1}{1 + \left(\mathsf{neg}\left(u1\right)\right)}\right)\right)\right)\right) \]

   4. mul-1-neg N/A

      \[\leadsto \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \mathsf{sqrt.f32}\left(\left(\frac{u1}{1 + -1 \cdot u1}\right)\right)\right)\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(1 + -1 \cdot u1\right)\right)\right)\right)\right) \]

   6. mul-1-neg N/A

      \[\leadsto \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(1 + \left(\mathsf{neg}\left(u1\right)\right)\right)\right)\right)\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(1 - u1\right)\right)\right)\right)\right) \]

   8. --lowering--.f32 82.6%

      \[\leadsto \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \mathsf{\_.f32}\left(1, u1\right)\right)\right)\right)\right) \]

8. Simplified 82.6%

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

Alternative 11: 64.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return (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 = (6.28318530718e0 * u2) * sqrt(u1)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\frac{314159265359}{50000000000} \cdot u2\right), \color{blue}{\left(\sqrt{\frac{u1}{1 - u1}}\right)}\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\color{blue}{\frac{u1}{1 - u1}}}\right)\right) \]

   5. *-rgt-identity N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{1 - u1}}\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{1 + \left(\mathsf{neg}\left(u1\right)\right)}}\right)\right) \]

   7. rgt-mult-inverse N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \frac{1}{-1 \cdot u1} + \left(\mathsf{neg}\left(u1\right)\right)}}\right)\right) \]

   8. mul-1-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \frac{1}{\mathsf{neg}\left(u1\right)} + \left(\mathsf{neg}\left(u1\right)\right)}}\right)\right) \]

   9. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \left(\mathsf{neg}\left(u1\right)\right)}}\right)\right) \]

   10. mul-1-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + -1 \cdot u1}}\right)\right) \]

   11. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \left(-1 \cdot u1\right) \cdot 1}}\right)\right) \]

   12. distribute-lft-in N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}}\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}}\right)\right) \]

   14. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(1 - \frac{1}{u1}\right)}}\right)\right) \]

   15. associate-*r* N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}\right)\right) \]

   16. sqrt-lowering-sqrt.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \mathsf{sqrt.f32}\left(\left(\frac{u1 \cdot 1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}\right)\right)\right) \]

   17. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \mathsf{sqrt.f32}\left(\left(\frac{u1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}\right)\right)\right) \]

   18. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)\right)\right)\right)\right) \]

   19. associate-*r* N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\left(-1 \cdot u1\right) \cdot \left(1 - \frac{1}{u1}\right)\right)\right)\right)\right) \]

   20. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\left(-1 \cdot u1\right) \cdot \left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)\right)\right)\right)\right) \]

   21. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\left(-1 \cdot u1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)\right)\right)\right)\right) \]

5. Simplified 82.7%

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

6. Taylor expanded in u1 around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \color{blue}{\left(\sqrt{u1}\right)}\right) \]
7. Step-by-step derivation

   1. sqrt-lowering-sqrt.f32 65.7%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \mathsf{sqrt.f32}\left(u1\right)\right) \]

8. Simplified 65.7%

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

Alternative 12: 64.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return 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 = 6.28318530718e0 * (u2 * sqrt(u1))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. sqrt-div N/A

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

   2. associate-*l/ N/A

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

   3. clear-num N/A

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

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{\sqrt{1 - u1}}{\sqrt{u1} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}\right)}\right) \]

   5. associate-/r* N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{\sqrt{1 - u1}}{\sqrt{u1}}}{\color{blue}{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}}\right)\right) \]

   6. sqrt-div N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\sqrt{\frac{1 - u1}{u1}}}{\sin \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}}\right)\right) \]

   7. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left(\sqrt{\frac{1 - u1}{u1}}\right), \color{blue}{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}\right)\right) \]

   8. pow1/2 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left({\left(\frac{1 - u1}{u1}\right)}^{\frac{1}{2}}\right), \sin \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}\right)\right) \]

   9. pow-lowering-pow.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\left(\frac{1 - u1}{u1}\right), \frac{1}{2}\right), \sin \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}\right)\right) \]

   10. div-sub N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\left(\frac{1}{u1} - \frac{u1}{u1}\right), \frac{1}{2}\right), \sin \left(\color{blue}{\frac{314159265359}{50000000000}} \cdot u2\right)\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\left(\frac{1}{u1} + \left(\mathsf{neg}\left(\frac{u1}{u1}\right)\right)\right), \frac{1}{2}\right), \sin \left(\color{blue}{\frac{314159265359}{50000000000}} \cdot u2\right)\right)\right) \]

   12. *-inverses N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\left(\frac{1}{u1} + \left(\mathsf{neg}\left(1\right)\right)\right), \frac{1}{2}\right), \sin \left(\frac{314159265359}{50000000000} \cdot u2\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\left(\frac{1}{u1} + -1\right), \frac{1}{2}\right), \sin \left(\frac{314159265359}{50000000000} \cdot u2\right)\right)\right) \]

   14. +-commutative N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\left(-1 + \frac{1}{u1}\right), \frac{1}{2}\right), \sin \left(\color{blue}{\frac{314159265359}{50000000000}} \cdot u2\right)\right)\right) \]

   15. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(-1, \left(\frac{1}{u1}\right)\right), \frac{1}{2}\right), \sin \left(\color{blue}{\frac{314159265359}{50000000000}} \cdot u2\right)\right)\right) \]

   16. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(-1, \mathsf{/.f32}\left(1, u1\right)\right), \frac{1}{2}\right), \sin \left(\frac{314159265359}{50000000000} \cdot u2\right)\right)\right) \]

   17. sin-lowering-sin.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(-1, \mathsf{/.f32}\left(1, u1\right)\right), \frac{1}{2}\right), \mathsf{sin.f32}\left(\left(\frac{314159265359}{50000000000} \cdot u2\right)\right)\right)\right) \]

   18. *-lowering-*.f32 98.3%

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{+.f32}\left(-1, \mathsf{/.f32}\left(1, u1\right)\right), \frac{1}{2}\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right)\right) \]

4. Applied egg-rr 98.3%

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

5. Taylor expanded in u2 around 0

   \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{50000000000}{314159265359} \cdot \left(\frac{1}{u2} \cdot \sqrt{\frac{1}{u1} - 1}\right)\right)}\right) \]
6. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(\frac{50000000000}{314159265359}, \color{blue}{\left(\frac{1}{u2} \cdot \sqrt{\frac{1}{u1} - 1}\right)}\right)\right) \]

   2. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(\frac{50000000000}{314159265359}, \left(\frac{1 \cdot \sqrt{\frac{1}{u1} - 1}}{\color{blue}{u2}}\right)\right)\right) \]

   3. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(\frac{50000000000}{314159265359}, \left(\frac{\sqrt{\frac{1}{u1} - 1}}{u2}\right)\right)\right) \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(\frac{50000000000}{314159265359}, \mathsf{/.f32}\left(\left(\sqrt{\frac{1}{u1} - 1}\right), \color{blue}{u2}\right)\right)\right) \]

   5. sqrt-lowering-sqrt.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(\frac{50000000000}{314159265359}, \mathsf{/.f32}\left(\mathsf{sqrt.f32}\left(\left(\frac{1}{u1} - 1\right)\right), u2\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(\frac{50000000000}{314159265359}, \mathsf{/.f32}\left(\mathsf{sqrt.f32}\left(\left(\frac{1}{u1} + \left(\mathsf{neg}\left(1\right)\right)\right)\right), u2\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(\frac{50000000000}{314159265359}, \mathsf{/.f32}\left(\mathsf{sqrt.f32}\left(\left(\frac{1}{u1} + -1\right)\right), u2\right)\right)\right) \]

   8. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(\frac{50000000000}{314159265359}, \mathsf{/.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{+.f32}\left(\left(\frac{1}{u1}\right), -1\right)\right), u2\right)\right)\right) \]

   9. /-lowering-/.f32 82.4%

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{*.f32}\left(\frac{50000000000}{314159265359}, \mathsf{/.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right)\right), u2\right)\right)\right) \]

7. Simplified 82.4%

   \[\leadsto \frac{1}{\color{blue}{0.15915494309188485 \cdot \frac{\sqrt{\frac{1}{u1} + -1}}{u2}}} \]

8. Taylor expanded in u1 around 0

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

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \color{blue}{\left(\sqrt{u1} \cdot u2\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \left(u2 \cdot \color{blue}{\sqrt{u1}}\right)\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(u2, \color{blue}{\left(\sqrt{u1}\right)}\right)\right) \]

   4. sqrt-lowering-sqrt.f32 65.7%

      \[\leadsto \mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(u2, \mathsf{sqrt.f32}\left(u1\right)\right)\right) \]

10. Simplified 65.7%

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

Alternative 13: 19.7% accurate, 19.0× speedup

Math

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

FPCore

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

C

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

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in u1 around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\color{blue}{\left(u1 \cdot \left(1 + u1\right)\right)}\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(u1, \left(1 + u1\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\color{blue}{\frac{314159265359}{50000000000}}, u2\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(u1, \left(u1 + 1\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   3. +-lowering-+.f32 87.0%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{+.f32}\left(u1, 1\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

5. Simplified 87.0%

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

6. Taylor expanded in u1 around inf

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

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(u1, \color{blue}{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}\right) \]

   2. sin-lowering-sin.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(u1, \mathsf{sin.f32}\left(\left(\frac{314159265359}{50000000000} \cdot u2\right)\right)\right) \]

   3. *-lowering-*.f32 19.7%

      \[\leadsto \mathsf{*.f32}\left(u1, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

8. Simplified 19.7%

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

9. Taylor expanded in u2 around 0

   \[\leadsto \mathsf{*.f32}\left(u1, \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)}\right) \]
10. Step-by-step derivation

    1. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(u1, \mathsf{*.f32}\left(u2, \color{blue}{\left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)}\right)\right) \]

    2. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(u1, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)}\right)\right)\right) \]

    3. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(u1, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \color{blue}{\left({u2}^{2}\right)}\right)\right)\right)\right) \]

    4. unpow2 N/A

       \[\leadsto \mathsf{*.f32}\left(u1, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \left(u2 \cdot \color{blue}{u2}\right)\right)\right)\right)\right) \]

    5. *-lowering-*.f32 19.8%

       \[\leadsto \mathsf{*.f32}\left(u1, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \mathsf{*.f32}\left(u2, \color{blue}{u2}\right)\right)\right)\right)\right) \]

11. Simplified 19.8%

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

12. Final simplification 19.8%

    \[\leadsto u1 \cdot \left(u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot -41.341702240407926\right)\right) \]
13. Add Preprocessing

Alternative 14: 19.4% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\frac{1}{u1} \cdot 0.15915494309188485}{u2}} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (/ 1.0 (/ (* (/ 1.0 u1) 0.15915494309188485) u2)))

C

float code(float cosTheta_i, float u1, float u2) {
	return 1.0f / (((1.0f / u1) * 0.15915494309188485f) / 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 = 1.0e0 / (((1.0e0 / u1) * 0.15915494309188485e0) / u2)
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(1.0) / Float32(Float32(Float32(Float32(1.0) / u1) * Float32(0.15915494309188485)) / u2))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(1.0) / (((single(1.0) / u1) * single(0.15915494309188485)) / u2);
end

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in u1 around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\color{blue}{\left(u1 \cdot \left(1 + u1\right)\right)}\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(u1, \left(1 + u1\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\color{blue}{\frac{314159265359}{50000000000}}, u2\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(u1, \left(u1 + 1\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   3. +-lowering-+.f32 87.0%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{+.f32}\left(u1, 1\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

5. Simplified 87.0%

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

6. Taylor expanded in u1 around inf

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

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(u1, \color{blue}{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}\right) \]

   2. sin-lowering-sin.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(u1, \mathsf{sin.f32}\left(\left(\frac{314159265359}{50000000000} \cdot u2\right)\right)\right) \]

   3. *-lowering-*.f32 19.7%

      \[\leadsto \mathsf{*.f32}\left(u1, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

8. Simplified 19.7%

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

9. Taylor expanded in u2 around 0

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

    1. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \color{blue}{\left(u1 \cdot u2\right)}\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \left(u2 \cdot \color{blue}{u1}\right)\right) \]

    3. *-lowering-*.f32 19.4%

       \[\leadsto \mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(u2, \color{blue}{u1}\right)\right) \]

11. Simplified 19.4%

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

    1. *-commutative N/A

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

    2. *-commutative N/A

       \[\leadsto \left(u1 \cdot u2\right) \cdot \frac{314159265359}{50000000000} \]

    3. associate-*l* N/A

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

    4. remove-double-div N/A

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

    5. metadata-eval N/A

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

    6. div-inv N/A

       \[\leadsto \frac{1}{\frac{1}{u1}} \cdot \frac{u2}{\color{blue}{\frac{50000000000}{314159265359}}} \]

    7. frac-times N/A

       \[\leadsto \frac{1 \cdot u2}{\color{blue}{\frac{1}{u1} \cdot \frac{50000000000}{314159265359}}} \]

    8. *-commutative N/A

       \[\leadsto \frac{u2 \cdot 1}{\color{blue}{\frac{1}{u1}} \cdot \frac{50000000000}{314159265359}} \]

    9. *-rgt-identity N/A

       \[\leadsto \frac{u2}{\color{blue}{\frac{1}{u1}} \cdot \frac{50000000000}{314159265359}} \]

    10. clear-num N/A

        \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{u1} \cdot \frac{50000000000}{314159265359}}{u2}}} \]

    11. /-lowering-/.f32 N/A

        \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{\frac{1}{u1} \cdot \frac{50000000000}{314159265359}}{u2}\right)}\right) \]

    12. /-lowering-/.f32 N/A

        \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left(\frac{1}{u1} \cdot \frac{50000000000}{314159265359}\right), \color{blue}{u2}\right)\right) \]

    13. *-lowering-*.f32 N/A

        \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(\frac{1}{u1}\right), \frac{50000000000}{314159265359}\right), u2\right)\right) \]

    14. /-lowering-/.f32 19.4%

        \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(1, u1\right), \frac{50000000000}{314159265359}\right), u2\right)\right) \]

13. Applied egg-rr 19.4%

    \[\leadsto \color{blue}{\frac{1}{\frac{\frac{1}{u1} \cdot 0.15915494309188485}{u2}}} \]
14. Add Preprocessing

Alternative 15: 19.4% accurate, 29.9× speedup

Math

\[\begin{array}{l} \\ \frac{u2}{\frac{1}{u1} \cdot 0.15915494309188485} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (/ u2 (* (/ 1.0 u1) 0.15915494309188485)))

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 / ((1.0f / u1) * 0.15915494309188485f);
}

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 = u2 / ((1.0e0 / u1) * 0.15915494309188485e0)
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u2 / Float32(Float32(Float32(1.0) / u1) * Float32(0.15915494309188485)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u2 / ((single(1.0) / u1) * single(0.15915494309188485));
end

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in u1 around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\color{blue}{\left(u1 \cdot \left(1 + u1\right)\right)}\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(u1, \left(1 + u1\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\color{blue}{\frac{314159265359}{50000000000}}, u2\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(u1, \left(u1 + 1\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   3. +-lowering-+.f32 87.0%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{+.f32}\left(u1, 1\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

5. Simplified 87.0%

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

6. Taylor expanded in u1 around inf

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

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(u1, \color{blue}{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}\right) \]

   2. sin-lowering-sin.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(u1, \mathsf{sin.f32}\left(\left(\frac{314159265359}{50000000000} \cdot u2\right)\right)\right) \]

   3. *-lowering-*.f32 19.7%

      \[\leadsto \mathsf{*.f32}\left(u1, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

8. Simplified 19.7%

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

9. Taylor expanded in u2 around 0

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

    1. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \color{blue}{\left(u1 \cdot u2\right)}\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \left(u2 \cdot \color{blue}{u1}\right)\right) \]

    3. *-lowering-*.f32 19.4%

       \[\leadsto \mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(u2, \color{blue}{u1}\right)\right) \]

11. Simplified 19.4%

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

    1. *-commutative N/A

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

    2. *-commutative N/A

       \[\leadsto \left(u1 \cdot u2\right) \cdot \frac{314159265359}{50000000000} \]

    3. associate-*l* N/A

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

    4. remove-double-div N/A

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

    5. metadata-eval N/A

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

    6. div-inv N/A

       \[\leadsto \frac{1}{\frac{1}{u1}} \cdot \frac{u2}{\color{blue}{\frac{50000000000}{314159265359}}} \]

    7. frac-times N/A

       \[\leadsto \frac{1 \cdot u2}{\color{blue}{\frac{1}{u1} \cdot \frac{50000000000}{314159265359}}} \]

    8. *-commutative N/A

       \[\leadsto \frac{u2 \cdot 1}{\color{blue}{\frac{1}{u1}} \cdot \frac{50000000000}{314159265359}} \]

    9. *-rgt-identity N/A

       \[\leadsto \frac{u2}{\color{blue}{\frac{1}{u1}} \cdot \frac{50000000000}{314159265359}} \]

    10. /-lowering-/.f32 N/A

        \[\leadsto \mathsf{/.f32}\left(u2, \color{blue}{\left(\frac{1}{u1} \cdot \frac{50000000000}{314159265359}\right)}\right) \]

    11. *-lowering-*.f32 N/A

        \[\leadsto \mathsf{/.f32}\left(u2, \mathsf{*.f32}\left(\left(\frac{1}{u1}\right), \color{blue}{\frac{50000000000}{314159265359}}\right)\right) \]

    12. /-lowering-/.f32 19.4%

        \[\leadsto \mathsf{/.f32}\left(u2, \mathsf{*.f32}\left(\mathsf{/.f32}\left(1, u1\right), \frac{50000000000}{314159265359}\right)\right) \]

13. Applied egg-rr 19.4%

    \[\leadsto \color{blue}{\frac{u2}{\frac{1}{u1} \cdot 0.15915494309188485}} \]
14. Add Preprocessing

Alternative 16: 19.4% accurate, 41.8× speedup

Math

\[\begin{array}{l} \\ u1 \cdot \frac{u2}{0.15915494309188485} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* u1 (/ u2 0.15915494309188485)))

C

float code(float cosTheta_i, float u1, float u2) {
	return u1 * (u2 / 0.15915494309188485f);
}

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 = u1 * (u2 / 0.15915494309188485e0)
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u1 * Float32(u2 / Float32(0.15915494309188485)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u1 * (u2 / single(0.15915494309188485));
end

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in u1 around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\color{blue}{\left(u1 \cdot \left(1 + u1\right)\right)}\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(u1, \left(1 + u1\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\color{blue}{\frac{314159265359}{50000000000}}, u2\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(u1, \left(u1 + 1\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   3. +-lowering-+.f32 87.0%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{+.f32}\left(u1, 1\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

5. Simplified 87.0%

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

6. Taylor expanded in u1 around inf

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

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(u1, \color{blue}{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}\right) \]

   2. sin-lowering-sin.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(u1, \mathsf{sin.f32}\left(\left(\frac{314159265359}{50000000000} \cdot u2\right)\right)\right) \]

   3. *-lowering-*.f32 19.7%

      \[\leadsto \mathsf{*.f32}\left(u1, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

8. Simplified 19.7%

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

9. Taylor expanded in u2 around 0

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

    1. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \color{blue}{\left(u1 \cdot u2\right)}\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \left(u2 \cdot \color{blue}{u1}\right)\right) \]

    3. *-lowering-*.f32 19.4%

       \[\leadsto \mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(u2, \color{blue}{u1}\right)\right) \]

11. Simplified 19.4%

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

    1. associate-*r* N/A

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

    2. *-commutative N/A

       \[\leadsto \left(u2 \cdot \frac{314159265359}{50000000000}\right) \cdot u1 \]

    3. metadata-eval N/A

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

    4. div-inv N/A

       \[\leadsto \frac{u2}{\frac{50000000000}{314159265359}} \cdot u1 \]

    5. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\left(\frac{u2}{\frac{50000000000}{314159265359}}\right), \color{blue}{u1}\right) \]

    6. /-lowering-/.f32 19.4%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(u2, \frac{50000000000}{314159265359}\right), u1\right) \]

13. Applied egg-rr 19.4%

    \[\leadsto \color{blue}{\frac{u2}{0.15915494309188485} \cdot u1} \]

14. Final simplification 19.4%

    \[\leadsto u1 \cdot \frac{u2}{0.15915494309188485} \]
15. Add Preprocessing

Alternative 17: 19.4% accurate, 41.8× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return (6.28318530718f * u2) * 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 = (6.28318530718e0 * u2) * u1
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in u1 around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\color{blue}{\left(u1 \cdot \left(1 + u1\right)\right)}\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(u1, \left(1 + u1\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\color{blue}{\frac{314159265359}{50000000000}}, u2\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(u1, \left(u1 + 1\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   3. +-lowering-+.f32 87.0%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{+.f32}\left(u1, 1\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

5. Simplified 87.0%

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

6. Taylor expanded in u1 around inf

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

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(u1, \color{blue}{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}\right) \]

   2. sin-lowering-sin.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(u1, \mathsf{sin.f32}\left(\left(\frac{314159265359}{50000000000} \cdot u2\right)\right)\right) \]

   3. *-lowering-*.f32 19.7%

      \[\leadsto \mathsf{*.f32}\left(u1, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

8. Simplified 19.7%

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

9. Taylor expanded in u2 around 0

   \[\leadsto \mathsf{*.f32}\left(u1, \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}\right) \]
10. Step-by-step derivation

    1. *-lowering-*.f32 19.4%

       \[\leadsto \mathsf{*.f32}\left(u1, \mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \color{blue}{u2}\right)\right) \]

11. Simplified 19.4%

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

12. Final simplification 19.4%

    \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot u1 \]
13. Add Preprocessing

Alternative 18: 19.4% accurate, 41.8× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return 6.28318530718f * (u2 * 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 = 6.28318530718e0 * (u2 * u1)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in u1 around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\color{blue}{\left(u1 \cdot \left(1 + u1\right)\right)}\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(u1, \left(1 + u1\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\color{blue}{\frac{314159265359}{50000000000}}, u2\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(u1, \left(u1 + 1\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   3. +-lowering-+.f32 87.0%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{+.f32}\left(u1, 1\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

5. Simplified 87.0%

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

6. Taylor expanded in u1 around inf

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

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(u1, \color{blue}{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}\right) \]

   2. sin-lowering-sin.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(u1, \mathsf{sin.f32}\left(\left(\frac{314159265359}{50000000000} \cdot u2\right)\right)\right) \]

   3. *-lowering-*.f32 19.7%

      \[\leadsto \mathsf{*.f32}\left(u1, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

8. Simplified 19.7%

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

9. Taylor expanded in u2 around 0

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

    1. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \color{blue}{\left(u1 \cdot u2\right)}\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \left(u2 \cdot \color{blue}{u1}\right)\right) \]

    3. *-lowering-*.f32 19.4%

       \[\leadsto \mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(u2, \color{blue}{u1}\right)\right) \]

11. Simplified 19.4%

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

Reproduce

herbie shell --seed 2024164 
(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))))