Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.1%

Time: 12.3s

Alternatives: 24

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: 98.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf((u2 * 6.28318530718f)) * (sqrtf((-1.0f / (u1 - 1.0f))) * powf((1.0f / u1), -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((u2 * 6.28318530718e0)) * (sqrt(((-1.0e0) / (u1 - 1.0e0))) * ((1.0e0 / u1) ** (-0.5e0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. lift-sqrt.f32 N/A

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

   2. pow1/2 N/A

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

   3. lift-/.f32 N/A

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

   4. clear-num N/A

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

   5. inv-pow N/A

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

   6. pow-pow N/A

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

   7. clear-num N/A

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

   8. associate-/r/ N/A

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

   9. unpow-prod-down N/A

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

   10. pow-pow N/A

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

   11. inv-pow N/A

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

   12. lower-*.f32 N/A

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

   13. lower-pow.f32 N/A

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

   14. lower-/.f32 N/A

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

   15. metadata-eval N/A

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

   16. pow1/2 N/A

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

   17. lower-sqrt.f32 N/A

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

   18. frac-2neg N/A

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

   19. metadata-eval N/A

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

   20. lower-/.f32 N/A

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

   21. neg-sub0 N/A

       \[\leadsto \left({\left(\frac{1}{u1}\right)}^{\frac{-1}{2}} \cdot \sqrt{\frac{-1}{\color{blue}{0 - \left(1 - u1\right)}}}\right) \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   22. lift--.f32 N/A

       \[\leadsto \left({\left(\frac{1}{u1}\right)}^{\frac{-1}{2}} \cdot \sqrt{\frac{-1}{0 - \color{blue}{\left(1 - u1\right)}}}\right) \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   23. sub-neg N/A

       \[\leadsto \left({\left(\frac{1}{u1}\right)}^{\frac{-1}{2}} \cdot \sqrt{\frac{-1}{0 - \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right)\right)}}}\right) \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   24. +-commutative N/A

       \[\leadsto \left({\left(\frac{1}{u1}\right)}^{\frac{-1}{2}} \cdot \sqrt{\frac{-1}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(u1\right)\right) + 1\right)}}}\right) \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   25. associate--r+ N/A

       \[\leadsto \left({\left(\frac{1}{u1}\right)}^{\frac{-1}{2}} \cdot \sqrt{\frac{-1}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(u1\right)\right)\right) - 1}}}\right) \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   26. neg-sub0 N/A

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

   27. remove-double-neg N/A

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

   28. lower--.f32 98.2

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

4. Applied rewrites 98.2%

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

5. Final simplification 98.2%

   \[\leadsto \sin \left(u2 \cdot 6.28318530718\right) \cdot \left(\sqrt{\frac{-1}{u1 - 1}} \cdot {\left(\frac{1}{u1}\right)}^{-0.5}\right) \]
6. Add Preprocessing

Alternative 2: 97.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 1.399999976158142:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-76.70585975309672, u2 \cdot u2, 81.6052492761019\right), u2 \cdot u2, -41.341702240407926\right), u2 \cdot u2, 6.28318530718\right) \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1 + 1, u1, 1\right) \cdot u1} \cdot \sin \left(u2 \cdot 6.28318530718\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 1.399999976158142)
   (*
    (*
     (fma
      (fma
       (fma -76.70585975309672 (* u2 u2) 81.6052492761019)
       (* u2 u2)
       -41.341702240407926)
      (* u2 u2)
      6.28318530718)
     u2)
    (sqrt (/ u1 (- 1.0 u1))))
   (* (sqrt (* (fma (+ u1 1.0) u1 1.0) u1)) (sin (* u2 6.28318530718)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 1.399999976158142f) {
		tmp = (fmaf(fmaf(fmaf(-76.70585975309672f, (u2 * u2), 81.6052492761019f), (u2 * u2), -41.341702240407926f), (u2 * u2), 6.28318530718f) * u2) * sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = sqrtf((fmaf((u1 + 1.0f), u1, 1.0f) * u1)) * sinf((u2 * 6.28318530718f));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(1.399999976158142))
		tmp = Float32(Float32(fma(fma(fma(Float32(-76.70585975309672), Float32(u2 * u2), Float32(81.6052492761019)), Float32(u2 * u2), Float32(-41.341702240407926)), Float32(u2 * u2), Float32(6.28318530718)) * u2) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1))));
	else
		tmp = Float32(sqrt(Float32(fma(Float32(u1 + Float32(1.0)), u1, Float32(1.0)) * u1)) * sin(Float32(u2 * Float32(6.28318530718))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 1.39999998

   1. Initial program 98.4%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f32 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f32 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f32 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f32 N/A

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

      16. unpow2 N/A

          \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right), u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \color{blue}{u2 \cdot u2}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \]

      17. lower-*.f32 97.9

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

   5. Applied rewrites 97.9%

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

   if 1.39999998 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 94.4%

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

      1. lift-/.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. associate-/r/ N/A

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

      5. associate-*l/ N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-out N/A

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

      8. frac-2neg N/A

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

      9. lower-/.f32 N/A

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

      10. lower-neg.f32 N/A

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

      11. *-lft-identity N/A

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

      12. lower-fma.f32 N/A

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

      15. distribute-neg-in N/A

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

      16. metadata-eval N/A

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

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

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

      18. distribute-lft-neg-out N/A

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

      19. sqr-neg N/A

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

      20. lower-+.f32 N/A

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

      21. lower-*.f32 94.6

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

   4. Applied rewrites 94.6%

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

   6. Applied rewrites 94.4%

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

   7. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-+.f32 91.7

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

   9. Applied rewrites 91.7%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 1.399999976158142:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-76.70585975309672, u2 \cdot u2, 81.6052492761019\right), u2 \cdot u2, -41.341702240407926\right), u2 \cdot u2, 6.28318530718\right) \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1 + 1, u1, 1\right) \cdot u1} \cdot \sin \left(u2 \cdot 6.28318530718\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 1.399999976158142:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-76.70585975309672, u2 \cdot u2, 81.6052492761019\right), u2 \cdot u2, -41.341702240407926\right), u2 \cdot u2, 6.28318530718\right) \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(u1, u1, 1\right) + u1\right) \cdot u1} \cdot \sin \left(u2 \cdot 6.28318530718\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 1.399999976158142)
   (*
    (*
     (fma
      (fma
       (fma -76.70585975309672 (* u2 u2) 81.6052492761019)
       (* u2 u2)
       -41.341702240407926)
      (* u2 u2)
      6.28318530718)
     u2)
    (sqrt (/ u1 (- 1.0 u1))))
   (* (sqrt (* (+ (fma u1 u1 1.0) u1) u1)) (sin (* u2 6.28318530718)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 1.399999976158142f) {
		tmp = (fmaf(fmaf(fmaf(-76.70585975309672f, (u2 * u2), 81.6052492761019f), (u2 * u2), -41.341702240407926f), (u2 * u2), 6.28318530718f) * u2) * sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = sqrtf(((fmaf(u1, u1, 1.0f) + u1) * u1)) * sinf((u2 * 6.28318530718f));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(1.399999976158142))
		tmp = Float32(Float32(fma(fma(fma(Float32(-76.70585975309672), Float32(u2 * u2), Float32(81.6052492761019)), Float32(u2 * u2), Float32(-41.341702240407926)), Float32(u2 * u2), Float32(6.28318530718)) * u2) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1))));
	else
		tmp = Float32(sqrt(Float32(Float32(fma(u1, u1, Float32(1.0)) + u1) * u1)) * sin(Float32(u2 * Float32(6.28318530718))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 1.39999998

   1. Initial program 98.4%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f32 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f32 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f32 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f32 N/A

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

      16. unpow2 N/A

          \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right), u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \color{blue}{u2 \cdot u2}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \]

      17. lower-*.f32 97.9

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

   5. Applied rewrites 97.9%

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

   if 1.39999998 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 94.4%

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

   3. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f32 91.5

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

   5. Applied rewrites 91.5%

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

   7. Applied rewrites 91.6%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 1.399999976158142:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-76.70585975309672, u2 \cdot u2, 81.6052492761019\right), u2 \cdot u2, -41.341702240407926\right), u2 \cdot u2, 6.28318530718\right) \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(u1, u1, 1\right) + u1\right) \cdot u1} \cdot \sin \left(u2 \cdot 6.28318530718\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.1%

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

   1. lift-/.f32 N/A

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

   2. lift--.f32 N/A

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

   3. flip-- N/A

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

   4. associate-/r/ N/A

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

   5. associate-*l/ N/A

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

   6. +-commutative N/A

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

   7. distribute-rgt-out N/A

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

   8. frac-2neg N/A

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

   9. lower-/.f32 N/A

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

   10. lower-neg.f32 N/A

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

   11. *-lft-identity N/A

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

   12. lower-fma.f32 N/A

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

   13. metadata-eval N/A

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

   14. sub-neg N/A

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

   15. distribute-neg-in N/A

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

   16. metadata-eval N/A

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

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

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

   18. distribute-lft-neg-out N/A

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

   19. sqr-neg N/A

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

   20. lower-+.f32 N/A

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

   21. lower-*.f32 98.1

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

4. Applied rewrites 98.1%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lower-fma.f32 98.2

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

6. Applied rewrites 98.2%

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

7. Final simplification 98.2%

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

Alternative 5: 97.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 1.399999976158142:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-76.70585975309672, u2 \cdot u2, 81.6052492761019\right), u2 \cdot u2, -41.341702240407926\right), u2 \cdot u2, 6.28318530718\right) \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(u1, u1, u1\right), u1, u1\right)} \cdot \sin \left(u2 \cdot 6.28318530718\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 1.399999976158142)
   (*
    (*
     (fma
      (fma
       (fma -76.70585975309672 (* u2 u2) 81.6052492761019)
       (* u2 u2)
       -41.341702240407926)
      (* u2 u2)
      6.28318530718)
     u2)
    (sqrt (/ u1 (- 1.0 u1))))
   (* (sqrt (fma (fma u1 u1 u1) u1 u1)) (sin (* u2 6.28318530718)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 1.399999976158142f) {
		tmp = (fmaf(fmaf(fmaf(-76.70585975309672f, (u2 * u2), 81.6052492761019f), (u2 * u2), -41.341702240407926f), (u2 * u2), 6.28318530718f) * u2) * sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = sqrtf(fmaf(fmaf(u1, u1, u1), u1, u1)) * sinf((u2 * 6.28318530718f));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(1.399999976158142))
		tmp = Float32(Float32(fma(fma(fma(Float32(-76.70585975309672), Float32(u2 * u2), Float32(81.6052492761019)), Float32(u2 * u2), Float32(-41.341702240407926)), Float32(u2 * u2), Float32(6.28318530718)) * u2) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1))));
	else
		tmp = Float32(sqrt(fma(fma(u1, u1, u1), u1, u1)) * sin(Float32(u2 * Float32(6.28318530718))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 1.39999998

   1. Initial program 98.4%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f32 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f32 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f32 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f32 N/A

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

      16. unpow2 N/A

          \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right), u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \color{blue}{u2 \cdot u2}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \]

      17. lower-*.f32 97.9

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

   5. Applied rewrites 97.9%

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

   if 1.39999998 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 94.4%

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

   3. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f32 91.5

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

   5. Applied rewrites 91.5%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 1.399999976158142:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-76.70585975309672, u2 \cdot u2, 81.6052492761019\right), u2 \cdot u2, -41.341702240407926\right), u2 \cdot u2, 6.28318530718\right) \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(u1, u1, u1\right), u1, u1\right)} \cdot \sin \left(u2 \cdot 6.28318530718\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

   5. frac-2neg N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. neg-sub0 N/A

      \[\leadsto \sqrt{\frac{-1}{\color{blue}{0 - \left(1 - u1\right)}} \cdot u1} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   9. lift--.f32 N/A

      \[\leadsto \sqrt{\frac{-1}{0 - \color{blue}{\left(1 - u1\right)}} \cdot u1} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   10. sub-neg N/A

       \[\leadsto \sqrt{\frac{-1}{0 - \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right)\right)}} \cdot u1} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   11. +-commutative N/A

       \[\leadsto \sqrt{\frac{-1}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(u1\right)\right) + 1\right)}} \cdot u1} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   12. associate--r+ N/A

       \[\leadsto \sqrt{\frac{-1}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(u1\right)\right)\right) - 1}} \cdot u1} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

   13. neg-sub0 N/A

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

   14. remove-double-neg N/A

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

   15. lower--.f32 98.1

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

4. Applied rewrites 98.1%

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

5. Final simplification 98.1%

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

Alternative 7: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 1.399999976158142:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-76.70585975309672, u2 \cdot u2, 81.6052492761019\right), u2 \cdot u2, -41.341702240407926\right), u2 \cdot u2, 6.28318530718\right) \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \sin \left(u2 \cdot 6.28318530718\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 1.399999976158142)
   (*
    (*
     (fma
      (fma
       (fma -76.70585975309672 (* u2 u2) 81.6052492761019)
       (* u2 u2)
       -41.341702240407926)
      (* u2 u2)
      6.28318530718)
     u2)
    (sqrt (/ u1 (- 1.0 u1))))
   (* (sqrt (fma u1 u1 u1)) (sin (* u2 6.28318530718)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 1.399999976158142f) {
		tmp = (fmaf(fmaf(fmaf(-76.70585975309672f, (u2 * u2), 81.6052492761019f), (u2 * u2), -41.341702240407926f), (u2 * u2), 6.28318530718f) * u2) * sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = sqrtf(fmaf(u1, u1, u1)) * sinf((u2 * 6.28318530718f));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(1.399999976158142))
		tmp = Float32(Float32(fma(fma(fma(Float32(-76.70585975309672), Float32(u2 * u2), Float32(81.6052492761019)), Float32(u2 * u2), Float32(-41.341702240407926)), Float32(u2 * u2), Float32(6.28318530718)) * u2) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1))));
	else
		tmp = Float32(sqrt(fma(u1, u1, u1)) * sin(Float32(u2 * Float32(6.28318530718))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 1.39999998

   1. Initial program 98.4%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f32 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f32 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f32 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f32 N/A

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

      16. unpow2 N/A

          \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right), u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \color{blue}{u2 \cdot u2}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \]

      17. lower-*.f32 97.9

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

   5. Applied rewrites 97.9%

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

   if 1.39999998 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 94.4%

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

   3. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f32 88.8

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

   5. Applied rewrites 88.8%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 1.399999976158142:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-76.70585975309672, u2 \cdot u2, 81.6052492761019\right), u2 \cdot u2, -41.341702240407926\right), u2 \cdot u2, 6.28318530718\right) \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \sin \left(u2 \cdot 6.28318530718\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

3. Final simplification 98.1%

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

Alternative 9: 93.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. sub-neg N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f32 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f32 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f32 N/A

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

   16. unpow2 N/A

       \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}, u2 \cdot u2, \frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}\right), u2 \cdot u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right), \color{blue}{u2 \cdot u2}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \]

   17. lower-*.f32 93.9

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

5. Applied rewrites 93.9%

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

6. Final simplification 93.9%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-76.70585975309672, u2 \cdot u2, 81.6052492761019\right), u2 \cdot u2, -41.341702240407926\right), u2 \cdot u2, 6.28318530718\right) \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}} \]
7. Add Preprocessing

Alternative 10: 91.1% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 91.7

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

5. Applied rewrites 91.7%

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

6. Final simplification 91.7%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(81.6052492761019, u2 \cdot u2, -41.341702240407926\right), u2 \cdot u2, 6.28318530718\right) \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}} \]
7. Add Preprocessing

Alternative 11: 91.1% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in u2 around 0

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

5. Applied rewrites 91.6%

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

6. Final simplification 91.6%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(81.6052492761019, u2 \cdot u2, -41.341702240407926\right), u2 \cdot u2, 6.28318530718\right) \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot u2 \]
7. Add Preprocessing

Alternative 12: 86.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.0017999999690800905:\\ \;\;\;\;\left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{-1}{u1 - 1} \cdot u1}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(u2 \cdot u2, -41.341702240407926, 6.28318530718\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(u1, u1, u1\right), u1, u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.0017999999690800905)
   (* (* u2 6.28318530718) (sqrt (* (/ -1.0 (- u1 1.0)) u1)))
   (*
    (* (fma (* u2 u2) -41.341702240407926 6.28318530718) u2)
    (sqrt (fma (fma u1 u1 u1) u1 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.0017999999690800905f) {
		tmp = (u2 * 6.28318530718f) * sqrtf(((-1.0f / (u1 - 1.0f)) * u1));
	} else {
		tmp = (fmaf((u2 * u2), -41.341702240407926f, 6.28318530718f) * u2) * sqrtf(fmaf(fmaf(u1, u1, u1), u1, u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.0017999999690800905))
		tmp = Float32(Float32(u2 * Float32(6.28318530718)) * sqrt(Float32(Float32(Float32(-1.0) / Float32(u1 - Float32(1.0))) * u1)));
	else
		tmp = Float32(Float32(fma(Float32(u2 * u2), Float32(-41.341702240407926), Float32(6.28318530718)) * u2) * sqrt(fma(fma(u1, u1, u1), u1, u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00179999997

   1. Initial program 98.4%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 98.1

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

   5. Applied rewrites 98.1%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift--.f32 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. lift--.f32 N/A

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

      12. metadata-eval N/A

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

      13. frac-2neg N/A

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

      14. lift-/.f32 N/A

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

      15. lower-*.f32 98.2

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

   7. Applied rewrites 98.2%

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

   if 0.00179999997 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.5%

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

   3. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f32 89.2

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

   5. Applied rewrites 89.2%

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

   6. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 68.1

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

   8. Applied rewrites 68.1%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.0017999999690800905:\\ \;\;\;\;\left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{-1}{u1 - 1} \cdot u1}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(u2 \cdot u2, -41.341702240407926, 6.28318530718\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(u1, u1, u1\right), u1, u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 85.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.0017999999690800905:\\ \;\;\;\;\left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{-1}{u1 - 1} \cdot u1}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(u2 \cdot u2, -41.341702240407926, 6.28318530718\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1, u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.0017999999690800905)
   (* (* u2 6.28318530718) (sqrt (* (/ -1.0 (- u1 1.0)) u1)))
   (*
    (* (fma (* u2 u2) -41.341702240407926 6.28318530718) u2)
    (sqrt (fma u1 u1 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.0017999999690800905f) {
		tmp = (u2 * 6.28318530718f) * sqrtf(((-1.0f / (u1 - 1.0f)) * u1));
	} else {
		tmp = (fmaf((u2 * u2), -41.341702240407926f, 6.28318530718f) * u2) * sqrtf(fmaf(u1, u1, u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.0017999999690800905))
		tmp = Float32(Float32(u2 * Float32(6.28318530718)) * sqrt(Float32(Float32(Float32(-1.0) / Float32(u1 - Float32(1.0))) * u1)));
	else
		tmp = Float32(Float32(fma(Float32(u2 * u2), Float32(-41.341702240407926), Float32(6.28318530718)) * u2) * sqrt(fma(u1, u1, u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00179999997

   1. Initial program 98.4%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 98.1

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

   5. Applied rewrites 98.1%

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

      1. lift-/.f32 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift--.f32 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. lift--.f32 N/A

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

      12. metadata-eval N/A

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

      13. frac-2neg N/A

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

      14. lift-/.f32 N/A

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

      15. lower-*.f32 98.2

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

   7. Applied rewrites 98.2%

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

   if 0.00179999997 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.5%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 50.6

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f32 49.6

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

   8. Applied rewrites 49.6%

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

   9. Taylor expanded in u2 around 0

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f32 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f32 66.0

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

   11. Applied rewrites 66.0%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.0017999999690800905:\\ \;\;\;\;\left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{-1}{u1 - 1} \cdot u1}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(u2 \cdot u2, -41.341702240407926, 6.28318530718\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1, u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 88.6% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.1%

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

   1. lift-/.f32 N/A

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

   2. lift--.f32 N/A

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

   3. flip-- N/A

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

   4. associate-/r/ N/A

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

   5. associate-*l/ N/A

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

   6. +-commutative N/A

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

   7. distribute-rgt-out N/A

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

   8. frac-2neg N/A

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

   9. lower-/.f32 N/A

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

   10. lower-neg.f32 N/A

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

   11. *-lft-identity N/A

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

   12. lower-fma.f32 N/A

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

   13. metadata-eval N/A

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

   14. sub-neg N/A

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

   15. distribute-neg-in N/A

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

   16. metadata-eval N/A

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

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

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

   18. distribute-lft-neg-out N/A

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

   19. sqr-neg N/A

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

   20. lower-+.f32 N/A

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

   21. lower-*.f32 98.1

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

4. Applied rewrites 98.1%

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

6. Applied rewrites 98.1%

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

7. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. unpow2 N/A

      \[\leadsto \sqrt{\frac{1}{1 - u1} \cdot u1} \cdot \left(\mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \]

   7. lower-*.f32 89.5

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

9. Applied rewrites 89.5%

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

10. Final simplification 89.5%

    \[\leadsto \left(\mathsf{fma}\left(u2 \cdot u2, -41.341702240407926, 6.28318530718\right) \cdot u2\right) \cdot \sqrt{\frac{1}{1 - u1} \cdot u1} \]
11. Add Preprocessing

Alternative 15: 85.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.0017999999690800905:\\ \;\;\;\;\left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(u2 \cdot u2, -41.341702240407926, 6.28318530718\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1, u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.0017999999690800905)
   (* (* u2 6.28318530718) (sqrt (/ u1 (- 1.0 u1))))
   (*
    (* (fma (* u2 u2) -41.341702240407926 6.28318530718) u2)
    (sqrt (fma u1 u1 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.0017999999690800905f) {
		tmp = (u2 * 6.28318530718f) * sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = (fmaf((u2 * u2), -41.341702240407926f, 6.28318530718f) * u2) * sqrtf(fmaf(u1, u1, u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.0017999999690800905))
		tmp = Float32(Float32(u2 * Float32(6.28318530718)) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1))));
	else
		tmp = Float32(Float32(fma(Float32(u2 * u2), Float32(-41.341702240407926), Float32(6.28318530718)) * u2) * sqrt(fma(u1, u1, u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00179999997

   1. Initial program 98.4%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 98.1

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

   5. Applied rewrites 98.1%

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

   if 0.00179999997 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.5%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 50.6

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

   5. Applied rewrites 50.6%

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

   6. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f32 49.6

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

   8. Applied rewrites 49.6%

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

   9. Taylor expanded in u2 around 0

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f32 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f32 66.0

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

   11. Applied rewrites 66.0%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.0017999999690800905:\\ \;\;\;\;\left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(u2 \cdot u2, -41.341702240407926, 6.28318530718\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1, u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 83.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.004900000058114529:\\ \;\;\;\;\left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right) \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.004900000058114529)
   (* (* u2 6.28318530718) (sqrt (/ u1 (- 1.0 u1))))
   (* (* (fma -41.341702240407926 (* u2 u2) 6.28318530718) u2) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.004900000058114529f) {
		tmp = (u2 * 6.28318530718f) * sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = (fmaf(-41.341702240407926f, (u2 * u2), 6.28318530718f) * u2) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.004900000058114529))
		tmp = Float32(Float32(u2 * Float32(6.28318530718)) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1))));
	else
		tmp = Float32(Float32(fma(Float32(-41.341702240407926), Float32(u2 * u2), Float32(6.28318530718)) * u2) * sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00490000006

   1. Initial program 98.4%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 97.5

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

   5. Applied rewrites 97.5%

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

   if 0.00490000006 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.4%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 45.8

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

   5. Applied rewrites 45.8%

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

   6. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 43.6

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

   8. Applied rewrites 43.6%

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

   9. Taylor expanded in u2 around 0

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f32 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f32 56.7

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

   11. Applied rewrites 56.7%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.004900000058114529:\\ \;\;\;\;\left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right) \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 83.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.004900000058114529:\\ \;\;\;\;\left(\sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718\right) \cdot u2\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right) \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.004900000058114529)
   (* (* (sqrt (/ u1 (- 1.0 u1))) 6.28318530718) u2)
   (* (* (fma -41.341702240407926 (* u2 u2) 6.28318530718) u2) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.004900000058114529f) {
		tmp = (sqrtf((u1 / (1.0f - u1))) * 6.28318530718f) * u2;
	} else {
		tmp = (fmaf(-41.341702240407926f, (u2 * u2), 6.28318530718f) * u2) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.004900000058114529))
		tmp = Float32(Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(6.28318530718)) * u2);
	else
		tmp = Float32(Float32(fma(Float32(-41.341702240407926), Float32(u2 * u2), Float32(6.28318530718)) * u2) * sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00490000006

   1. Initial program 98.4%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 97.5

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

   5. Applied rewrites 97.5%

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

   6. Taylor expanded in u2 around 0

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

   8. Applied rewrites 98.4%

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

   9. Taylor expanded in u2 around 0

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

   11. Applied rewrites 97.5%

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

   if 0.00490000006 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.4%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 45.8

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

   5. Applied rewrites 45.8%

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

   6. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 43.6

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

   8. Applied rewrites 43.6%

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

   9. Taylor expanded in u2 around 0

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f32 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f32 56.7

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

   11. Applied rewrites 56.7%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.004900000058114529:\\ \;\;\;\;\left(\sqrt{\frac{u1}{1 - u1}} \cdot 6.28318530718\right) \cdot u2\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right) \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 88.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. unpow2 N/A

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(\mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot u2\right) \]

   7. lower-*.f32 89.5

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

5. Applied rewrites 89.5%

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

6. Final simplification 89.5%

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

Alternative 19: 88.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 82.1

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

5. Applied rewrites 82.1%

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

6. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

   6. +-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-sqrt.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lower--.f32 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f32 N/A

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

   14. unpow2 N/A

       \[\leadsto \left(\sqrt{\frac{u1}{1 - u1}} \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right)\right) \cdot u2 \]

   15. lower-*.f32 89.5

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

8. Applied rewrites 89.5%

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

9. Final simplification 89.5%

   \[\leadsto \left(\mathsf{fma}\left(u2 \cdot u2, -41.341702240407926, 6.28318530718\right) \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot u2 \]
10. Add Preprocessing

Alternative 20: 78.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.002199999988079071:\\ \;\;\;\;\left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(u1, u1, u1\right), u1, u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right) \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.002199999988079071)
   (* (* u2 6.28318530718) (sqrt (fma (fma u1 u1 u1) u1 u1)))
   (* (* (fma -41.341702240407926 (* u2 u2) 6.28318530718) u2) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.002199999988079071f) {
		tmp = (u2 * 6.28318530718f) * sqrtf(fmaf(fmaf(u1, u1, u1), u1, u1));
	} else {
		tmp = (fmaf(-41.341702240407926f, (u2 * u2), 6.28318530718f) * u2) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.002199999988079071))
		tmp = Float32(Float32(u2 * Float32(6.28318530718)) * sqrt(fma(fma(u1, u1, u1), u1, u1)));
	else
		tmp = Float32(Float32(fma(Float32(-41.341702240407926), Float32(u2 * u2), Float32(6.28318530718)) * u2) * sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.0022

   1. Initial program 98.4%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 98.0

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f32 89.3

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

   8. Applied rewrites 89.3%

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

   if 0.0022 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.5%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 49.2

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

   5. Applied rewrites 49.2%

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

   6. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 46.3

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

   8. Applied rewrites 46.3%

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

   9. Taylor expanded in u2 around 0

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f32 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f32 58.8

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

   11. Applied rewrites 58.8%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.002199999988079071:\\ \;\;\;\;\left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(u1, u1, u1\right), u1, u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right) \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 75.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.002199999988079071:\\ \;\;\;\;\sqrt{\left(u1 + 1\right) \cdot u1} \cdot \left(u2 \cdot 6.28318530718\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right) \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.002199999988079071)
   (* (sqrt (* (+ u1 1.0) u1)) (* u2 6.28318530718))
   (* (* (fma -41.341702240407926 (* u2 u2) 6.28318530718) u2) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.002199999988079071f) {
		tmp = sqrtf(((u1 + 1.0f) * u1)) * (u2 * 6.28318530718f);
	} else {
		tmp = (fmaf(-41.341702240407926f, (u2 * u2), 6.28318530718f) * u2) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.002199999988079071))
		tmp = Float32(sqrt(Float32(Float32(u1 + Float32(1.0)) * u1)) * Float32(u2 * Float32(6.28318530718)));
	else
		tmp = Float32(Float32(fma(Float32(-41.341702240407926), Float32(u2 * u2), Float32(6.28318530718)) * u2) * sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.0022

   1. Initial program 98.4%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 98.0

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f32 85.3

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

   8. Applied rewrites 85.3%

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

   10. Applied rewrites 85.4%

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

   if 0.0022 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.5%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 49.2

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

   5. Applied rewrites 49.2%

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

   6. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 46.3

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

   8. Applied rewrites 46.3%

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

   9. Taylor expanded in u2 around 0

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f32 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f32 58.8

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

   11. Applied rewrites 58.8%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.002199999988079071:\\ \;\;\;\;\sqrt{\left(u1 + 1\right) \cdot u1} \cdot \left(u2 \cdot 6.28318530718\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right) \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 72.3% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 82.1

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

5. Applied rewrites 82.1%

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

6. Taylor expanded in u1 around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f32 73.3

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

8. Applied rewrites 73.3%

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

10. Applied rewrites 73.3%

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

11. Final simplification 73.3%

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

Alternative 23: 72.3% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 82.1

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

5. Applied rewrites 82.1%

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

6. Taylor expanded in u1 around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f32 73.3

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

8. Applied rewrites 73.3%

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

9. Final simplification 73.3%

   \[\leadsto \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \]
10. Add Preprocessing

Alternative 24: 64.1% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt(u1) * (u2 * 6.28318530718e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 82.1

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

5. Applied rewrites 82.1%

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

6. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 65.8

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

8. Applied rewrites 65.8%

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

Reproduce

herbie shell --seed 2024240 
(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))))