Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 14.8s

Alternatives: 26

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: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. lift--.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sqrt.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-sin.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-sqrt.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. clear-num N/A

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

   10. sqrt-div N/A

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

   11. metadata-eval N/A

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

   12. un-div-inv N/A

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

   13. lower-/.f32 N/A

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

   14. lower-sqrt.f32 N/A

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

   15. lift--.f32 N/A

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

   16. div-sub N/A

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

   17. sub-neg N/A

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

   18. *-inverses N/A

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

   19. metadata-eval N/A

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

   20. lower-+.f32 N/A

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

   21. lower-/.f32 98.3

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

4. Applied egg-rr 98.3%

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

Alternative 2: 97.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.6499999761581421:\\ \;\;\;\;\frac{u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2, u2 \cdot -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718\right)}{\sqrt{\frac{1}{u1} + -1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.6499999761581421)
   (/
    (*
     u2
     (fma
      (* u2 u2)
      (fma
       u2
       (* u2 (fma u2 (* u2 -76.70585975309672) 81.6052492761019))
       -41.341702240407926)
      6.28318530718))
    (sqrt (+ (/ 1.0 u1) -1.0)))
   (* (sin (* 6.28318530718 u2)) (sqrt (fma u1 (fma u1 u1 u1) u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.6499999761581421f) {
		tmp = (u2 * fmaf((u2 * u2), fmaf(u2, (u2 * fmaf(u2, (u2 * -76.70585975309672f), 81.6052492761019f)), -41.341702240407926f), 6.28318530718f)) / sqrtf(((1.0f / u1) + -1.0f));
	} else {
		tmp = sinf((6.28318530718f * u2)) * sqrtf(fmaf(u1, fmaf(u1, u1, u1), u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.6499999761581421))
		tmp = Float32(Float32(u2 * fma(Float32(u2 * u2), fma(u2, Float32(u2 * fma(u2, Float32(u2 * Float32(-76.70585975309672)), Float32(81.6052492761019))), Float32(-41.341702240407926)), Float32(6.28318530718))) / sqrt(Float32(Float32(Float32(1.0) / u1) + Float32(-1.0))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(fma(u1, fma(u1, u1, u1), u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.649999976

   1. Initial program 98.4%

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

      1. lift--.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-sqrt.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-sin.f32 N/A

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

      6. *-commutative N/A

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

      7. lift-sqrt.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. clear-num N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. un-div-inv N/A

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

      13. lower-/.f32 N/A

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

      14. lower-sqrt.f32 N/A

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

      15. lift--.f32 N/A

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

      16. div-sub N/A

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

      17. sub-neg N/A

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

      18. *-inverses N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f32 N/A

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

      21. lower-/.f32 98.5

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

      16. lower-fma.f32 N/A

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

      17. lower-*.f32 98.6

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

   7. Simplified 98.6%

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

   if 0.649999976 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.0%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   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-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f32 83.4

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

   5. Simplified 83.4%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.6499999761581421:\\ \;\;\;\;\frac{u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot \mathsf{fma}\left(u2, u2 \cdot -76.70585975309672, 81.6052492761019\right), -41.341702240407926\right), 6.28318530718\right)}{\sqrt{\frac{1}{u1} + -1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 1

   1. Initial program 98.4%

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

      1. lift--.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-sqrt.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-sin.f32 N/A

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

      6. *-commutative N/A

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

      7. lift-sqrt.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. clear-num N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. un-div-inv N/A

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

      13. lower-/.f32 N/A

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

      14. lower-sqrt.f32 N/A

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

      15. lift--.f32 N/A

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

      16. div-sub N/A

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

      17. sub-neg N/A

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

      18. *-inverses N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f32 N/A

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

      21. lower-/.f32 98.5

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

      16. lower-fma.f32 N/A

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

      17. lower-*.f32 98.3

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

   7. Simplified 98.3%

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

   if 1 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 96.5%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]
   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 77.6

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

   5. Simplified 77.6%

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

      1. distribute-lft1-in N/A

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

      2. +-commutative N/A

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

      3. lower-*.f32 N/A

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

      4. lower-+.f32 77.6

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

   7. Applied egg-rr 77.6%

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

4. Final simplification 96.3%

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

Alternative 4: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(1.0))
		tmp = Float32(Float32(u2 * fma(Float32(u2 * u2), fma(u2, Float32(u2 * fma(u2, Float32(u2 * Float32(-76.70585975309672)), Float32(81.6052492761019))), Float32(-41.341702240407926)), Float32(6.28318530718))) / sqrt(Float32(Float32(Float32(1.0) / u1) + Float32(-1.0))));
	else
		tmp = Float32(sin(Float32(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) < 1

   1. Initial program 98.4%

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

      1. lift--.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-sqrt.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-sin.f32 N/A

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

      6. *-commutative N/A

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

      7. lift-sqrt.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. clear-num N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. un-div-inv N/A

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

      13. lower-/.f32 N/A

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

      14. lower-sqrt.f32 N/A

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

      15. lift--.f32 N/A

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

      16. div-sub N/A

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

      17. sub-neg N/A

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

      18. *-inverses N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f32 N/A

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

      21. lower-/.f32 98.5

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*l* N/A

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

      16. lower-fma.f32 N/A

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

      17. lower-*.f32 98.3

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

   7. Simplified 98.3%

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

   if 1 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 96.5%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1\right)}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]
   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 77.6

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

   5. Simplified 77.6%

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

4. Final simplification 96.3%

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

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Final simplification 98.3%

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

Alternative 6: 93.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.3%

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

   1. lift--.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sqrt.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-sin.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-sqrt.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. clear-num N/A

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

   10. sqrt-div N/A

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

   11. metadata-eval N/A

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

   12. un-div-inv N/A

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

   13. lower-/.f32 N/A

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

   14. lower-sqrt.f32 N/A

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

   15. lift--.f32 N/A

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

   16. div-sub N/A

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

   17. sub-neg N/A

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

   18. *-inverses N/A

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

   19. metadata-eval N/A

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

   20. lower-+.f32 N/A

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

   21. lower-/.f32 98.3

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

4. Applied egg-rr 98.3%

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

5. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. sub-neg N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. +-commutative N/A

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

   13. *-commutative N/A

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

   14. unpow2 N/A

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

   15. associate-*l* N/A

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

   16. lower-fma.f32 N/A

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

   17. lower-*.f32 93.0

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

7. Simplified 93.0%

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

Alternative 7: 93.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. unpow2 N/A

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

   14. associate-*l* N/A

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

   15. lower-fma.f32 N/A

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

   16. lower-*.f32 92.9

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

5. Simplified 92.9%

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

Alternative 8: 91.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.3%

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

   1. lift--.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sqrt.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-sin.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-sqrt.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. clear-num N/A

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

   10. sqrt-div N/A

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

   11. metadata-eval N/A

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

   12. un-div-inv N/A

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

   13. lower-/.f32 N/A

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

   14. lower-sqrt.f32 N/A

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

   15. lift--.f32 N/A

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

   16. div-sub N/A

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

   17. sub-neg N/A

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

   18. *-inverses N/A

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

   19. metadata-eval N/A

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

   20. lower-+.f32 N/A

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

   21. lower-/.f32 98.3

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

4. Applied egg-rr 98.3%

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

5. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. sub-neg N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 90.7

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

7. Simplified 90.7%

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

Alternative 9: 86.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{u1}{1 - u1}\\ \mathbf{if}\;t\_0 \leq 0.0006000000284984708:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot -41.341702240407926, 6.28318530718\right)\right)\\ \mathbf{else}:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot \sqrt{t\_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (/ u1 (- 1.0 u1))))
   (if (<= t_0 0.0006000000284984708)
     (*
      (sqrt (fma u1 u1 u1))
      (* u2 (fma u2 (* u2 -41.341702240407926) 6.28318530718)))
     (* 6.28318530718 (* u2 (sqrt t_0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u1 / (1.0f - u1);
	float tmp;
	if (t_0 <= 0.0006000000284984708f) {
		tmp = sqrtf(fmaf(u1, u1, u1)) * (u2 * fmaf(u2, (u2 * -41.341702240407926f), 6.28318530718f));
	} else {
		tmp = 6.28318530718f * (u2 * sqrtf(t_0));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u1 / Float32(Float32(1.0) - u1))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0006000000284984708))
		tmp = Float32(sqrt(fma(u1, u1, u1)) * Float32(u2 * fma(u2, Float32(u2 * Float32(-41.341702240407926)), Float32(6.28318530718))));
	else
		tmp = Float32(Float32(6.28318530718) * Float32(u2 * sqrt(t_0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 u1 (-.f32 #s(literal 1 binary32) u1)) < 6.00000028e-4

   1. Initial program 98.3%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   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 98.3

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

   5. Simplified 98.3%

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

   6. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 89.0

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

   8. Simplified 89.0%

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

   if 6.00000028e-4 < (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 98.2%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. *-rgt-identity N/A

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

      6. sub-neg N/A

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

      7. rgt-mult-inverse N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. mul-1-neg N/A

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

      11. *-rgt-identity N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. associate-*r* N/A

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

      16. lower-sqrt.f32 N/A

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

      17. *-rgt-identity N/A

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

      18. lower-/.f32 N/A

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

      19. associate-*r* N/A

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

      20. sub-neg N/A

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

      21. +-commutative N/A

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

   5. Simplified 79.7%

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

      1. lift--.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-sqrt.f32 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 79.8

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

   7. Applied egg-rr 79.8%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{u1}{1 - u1} \leq 0.0006000000284984708:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(u2, u2 \cdot -41.341702240407926, 6.28318530718\right)\right)\\ \mathbf{else}:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{u1}{1 - u1}\\ \mathbf{if}\;t\_0 \leq 0.0006000000284984708:\\ \;\;\;\;u2 \cdot \left(\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot -41.341702240407926, 6.28318530718\right)\right)\\ \mathbf{else}:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot \sqrt{t\_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (/ u1 (- 1.0 u1))))
   (if (<= t_0 0.0006000000284984708)
     (*
      u2
      (*
       (sqrt (fma u1 u1 u1))
       (fma u2 (* u2 -41.341702240407926) 6.28318530718)))
     (* 6.28318530718 (* u2 (sqrt t_0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u1 / (1.0f - u1);
	float tmp;
	if (t_0 <= 0.0006000000284984708f) {
		tmp = u2 * (sqrtf(fmaf(u1, u1, u1)) * fmaf(u2, (u2 * -41.341702240407926f), 6.28318530718f));
	} else {
		tmp = 6.28318530718f * (u2 * sqrtf(t_0));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u1 / Float32(Float32(1.0) - u1))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0006000000284984708))
		tmp = Float32(u2 * Float32(sqrt(fma(u1, u1, u1)) * fma(u2, Float32(u2 * Float32(-41.341702240407926)), Float32(6.28318530718))));
	else
		tmp = Float32(Float32(6.28318530718) * Float32(u2 * sqrt(t_0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 u1 (-.f32 #s(literal 1 binary32) u1)) < 6.00000028e-4

   1. Initial program 98.3%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   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 98.3

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

   5. Simplified 98.3%

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

   6. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-*.f32 N/A

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

   8. Simplified 88.9%

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

   if 6.00000028e-4 < (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 98.2%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. *-rgt-identity N/A

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

      6. sub-neg N/A

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

      7. rgt-mult-inverse N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. mul-1-neg N/A

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

      11. *-rgt-identity N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. associate-*r* N/A

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

      16. lower-sqrt.f32 N/A

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

      17. *-rgt-identity N/A

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

      18. lower-/.f32 N/A

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

      19. associate-*r* N/A

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

      20. sub-neg N/A

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

      21. +-commutative N/A

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

   5. Simplified 79.7%

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

      1. lift--.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-sqrt.f32 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 79.8

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

   7. Applied egg-rr 79.8%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{u1}{1 - u1} \leq 0.0006000000284984708:\\ \;\;\;\;u2 \cdot \left(\sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot -41.341702240407926, 6.28318530718\right)\right)\\ \mathbf{else}:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.5% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0

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

5. Simplified 90.7%

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

Alternative 12: 86.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.0003499999875202775:\\ \;\;\;\;\frac{6.28318530718 \cdot u2}{\sqrt{\frac{1}{u1} + -1}}\\ \mathbf{else}:\\ \;\;\;\;u2 \cdot \left(\sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)} \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.0003499999875202775)
   (/ (* 6.28318530718 u2) (sqrt (+ (/ 1.0 u1) -1.0)))
   (*
    u2
    (*
     (sqrt (fma u1 (fma u1 u1 u1) u1))
     (fma -41.341702240407926 (* u2 u2) 6.28318530718)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.0003499999875202775f) {
		tmp = (6.28318530718f * u2) / sqrtf(((1.0f / u1) + -1.0f));
	} else {
		tmp = u2 * (sqrtf(fmaf(u1, fmaf(u1, u1, u1), u1)) * fmaf(-41.341702240407926f, (u2 * u2), 6.28318530718f));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.0003499999875202775))
		tmp = Float32(Float32(Float32(6.28318530718) * u2) / sqrt(Float32(Float32(Float32(1.0) / u1) + Float32(-1.0))));
	else
		tmp = Float32(u2 * Float32(sqrt(fma(u1, fma(u1, u1, u1), u1)) * fma(Float32(-41.341702240407926), Float32(u2 * u2), Float32(6.28318530718))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 3.49999988e-4

   1. Initial program 98.6%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. *-rgt-identity N/A

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

      6. sub-neg N/A

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

      7. rgt-mult-inverse N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. mul-1-neg N/A

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

      11. *-rgt-identity N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. associate-*r* N/A

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

      16. lower-sqrt.f32 N/A

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

      17. *-rgt-identity N/A

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

      18. lower-/.f32 N/A

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

      19. associate-*r* N/A

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

      20. sub-neg N/A

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

      21. +-commutative N/A

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

   5. Simplified 98.6%

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

      1. lift-*.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sqrt-div N/A

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

      4. clear-num N/A

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

      5. sqrt-div N/A

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

      6. un-div-inv N/A

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

      7. lift--.f32 N/A

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

      8. div-sub N/A

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

      9. lift-/.f32 N/A

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

      10. sub-neg N/A

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

      11. *-inverses N/A

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

      12. metadata-eval N/A

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

      13. lift-+.f32 N/A

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

      14. lift-sqrt.f32 N/A

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

      15. lower-/.f32 98.6

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

   7. Applied egg-rr 98.6%

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

   if 3.49999988e-4 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.9%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   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-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f32 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f32 90.3

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

   5. Simplified 90.3%

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

   6. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f32 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

   8. Simplified 73.5%

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

Alternative 13: 88.9% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.3%

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

   1. lift--.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sqrt.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-sin.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-sqrt.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. clear-num N/A

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

   10. sqrt-div N/A

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

   11. metadata-eval N/A

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

   12. un-div-inv N/A

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

   13. lower-/.f32 N/A

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

   14. lower-sqrt.f32 N/A

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

   15. lift--.f32 N/A

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

   16. div-sub N/A

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

   17. sub-neg N/A

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

   18. *-inverses N/A

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

   19. metadata-eval N/A

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

   20. lower-+.f32 N/A

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

   21. lower-/.f32 98.3

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

4. Applied egg-rr 98.3%

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

5. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 88.3

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

7. Simplified 88.3%

   \[\leadsto \frac{\color{blue}{u2 \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)}}{\sqrt{\frac{1}{u1} + -1}} \]
8. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. metadata-eval N/A

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

   3. *-inverses N/A

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

   4. sub-neg N/A

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

   5. lift-/.f32 N/A

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

   6. div-sub N/A

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

   7. lift--.f32 N/A

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

   8. lower-/.f32 88.4

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

9. Applied egg-rr 88.4%

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

Alternative 14: 88.9% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.3%

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

   1. lift--.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sqrt.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-sin.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-sqrt.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. clear-num N/A

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

   10. sqrt-div N/A

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

   11. metadata-eval N/A

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

   12. un-div-inv N/A

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

   13. lower-/.f32 N/A

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

   14. lower-sqrt.f32 N/A

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

   15. lift--.f32 N/A

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

   16. div-sub N/A

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

   17. sub-neg N/A

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

   18. *-inverses N/A

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

   19. metadata-eval N/A

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

   20. lower-+.f32 N/A

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

   21. lower-/.f32 98.3

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

4. Applied egg-rr 98.3%

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

5. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 88.3

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

7. Simplified 88.3%

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

Alternative 15: 89.0% accurate, 2.6× speedup

Math

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

FPCore

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

Julia

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

Derivation

1. Initial program 98.3%

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

   1. lift--.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sqrt.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-sin.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-sqrt.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. clear-num N/A

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

   10. sqrt-div N/A

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

   11. metadata-eval N/A

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

   12. un-div-inv N/A

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

   13. lower-/.f32 N/A

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

   14. lower-sqrt.f32 N/A

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

   15. lift--.f32 N/A

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

   16. div-sub N/A

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

   17. sub-neg N/A

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

   18. *-inverses N/A

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

   19. metadata-eval N/A

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

   20. lower-+.f32 N/A

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

   21. lower-/.f32 98.3

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

4. Applied egg-rr 98.3%

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

5. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 88.3

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

7. Simplified 88.3%

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

9. Applied egg-rr 88.3%

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

11. Applied egg-rr 88.3%

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

Alternative 16: 84.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.014000000432133675:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u2, u2 \cdot -41.341702240407926, 6.28318530718\right) \cdot \left(u2 \cdot \sqrt{u1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.014000000432133675)
   (* (* 6.28318530718 u2) (sqrt (/ u1 (- 1.0 u1))))
   (* (fma u2 (* u2 -41.341702240407926) 6.28318530718) (* u2 (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.014000000432133675f) {
		tmp = (6.28318530718f * u2) * sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = fmaf(u2, (u2 * -41.341702240407926f), 6.28318530718f) * (u2 * sqrtf(u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.014000000432133675))
		tmp = Float32(Float32(Float32(6.28318530718) * u2) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1))));
	else
		tmp = Float32(fma(u2, Float32(u2 * Float32(-41.341702240407926)), Float32(6.28318530718)) * Float32(u2 * sqrt(u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.0140000004

   1. Initial program 98.5%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. *-rgt-identity N/A

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

      6. sub-neg N/A

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

      7. rgt-mult-inverse N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. mul-1-neg N/A

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

      11. *-rgt-identity N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. associate-*r* N/A

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

      16. lower-sqrt.f32 N/A

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

      17. *-rgt-identity N/A

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

      18. lower-/.f32 N/A

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

      19. associate-*r* N/A

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

      20. sub-neg N/A

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

      21. +-commutative N/A

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

   5. Simplified 96.3%

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

   if 0.0140000004 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.7%

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

      1. lift--.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-sqrt.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-sin.f32 N/A

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

      6. *-commutative N/A

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

      7. lift-sqrt.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. clear-num N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. un-div-inv N/A

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

      13. lower-/.f32 N/A

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

      14. lower-sqrt.f32 N/A

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

      15. lift--.f32 N/A

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

      16. div-sub N/A

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

      17. sub-neg N/A

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

      18. *-inverses N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f32 N/A

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

      21. lower-/.f32 97.8

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

   4. Applied egg-rr 97.8%

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

   5. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 63.7

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

   7. Simplified 63.7%

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

   9. Applied egg-rr 63.8%

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

   10. Taylor expanded in u1 around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. lower-*.f32 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f32 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f32 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f32 N/A

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

       14. lower-sqrt.f32 54.9

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

   12. Simplified 54.9%

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

Alternative 17: 89.0% accurate, 2.9× speedup

Math

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

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(fma(u2, Float32(u2 * Float32(-41.341702240407926)), Float32(6.28318530718)) * Float32(u2 * sqrt(Float32(u1 / Float32(Float32(1.0) - u1)))))
end

Derivation

1. Initial program 98.3%

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

   1. lift--.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sqrt.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-sin.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-sqrt.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. clear-num N/A

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

   10. sqrt-div N/A

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

   11. metadata-eval N/A

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

   12. un-div-inv N/A

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

   13. lower-/.f32 N/A

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

   14. lower-sqrt.f32 N/A

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

   15. lift--.f32 N/A

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

   16. div-sub N/A

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

   17. sub-neg N/A

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

   18. *-inverses N/A

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

   19. metadata-eval N/A

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

   20. lower-+.f32 N/A

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

   21. lower-/.f32 98.3

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

4. Applied egg-rr 98.3%

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

5. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 88.3

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

7. Simplified 88.3%

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

9. Applied egg-rr 88.3%

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

Alternative 18: 88.9% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.3%

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

   1. lift--.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-sqrt.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-sin.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-sqrt.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. clear-num N/A

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

   10. sqrt-div N/A

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

   11. metadata-eval N/A

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

   12. un-div-inv N/A

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

   13. lower-/.f32 N/A

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

   14. lower-sqrt.f32 N/A

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

   15. lift--.f32 N/A

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

   16. div-sub N/A

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

   17. sub-neg N/A

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

   18. *-inverses N/A

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

   19. metadata-eval N/A

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

   20. lower-+.f32 N/A

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

   21. lower-/.f32 98.3

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

4. Applied egg-rr 98.3%

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

5. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 88.3

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

7. Simplified 88.3%

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

9. Applied egg-rr 88.3%

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

10. Final simplification 88.3%

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

Alternative 19: 88.9% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0

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

5. Simplified 88.3%

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

Alternative 20: 78.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.014000000432133675:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1, u1\right), u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u2, u2 \cdot -41.341702240407926, 6.28318530718\right) \cdot \left(u2 \cdot \sqrt{u1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.014000000432133675)
   (* (* 6.28318530718 u2) (sqrt (fma u1 (fma u1 u1 u1) u1)))
   (* (fma u2 (* u2 -41.341702240407926) 6.28318530718) (* u2 (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.014000000432133675f) {
		tmp = (6.28318530718f * u2) * sqrtf(fmaf(u1, fmaf(u1, u1, u1), u1));
	} else {
		tmp = fmaf(u2, (u2 * -41.341702240407926f), 6.28318530718f) * (u2 * sqrtf(u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.014000000432133675))
		tmp = Float32(Float32(Float32(6.28318530718) * u2) * sqrt(fma(u1, fma(u1, u1, u1), u1)));
	else
		tmp = Float32(fma(u2, Float32(u2 * Float32(-41.341702240407926)), Float32(6.28318530718)) * Float32(u2 * sqrt(u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.0140000004

   1. Initial program 98.5%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. *-rgt-identity N/A

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

      6. sub-neg N/A

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

      7. rgt-mult-inverse N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. mul-1-neg N/A

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

      11. *-rgt-identity N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. associate-*r* N/A

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

      16. lower-sqrt.f32 N/A

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

      17. *-rgt-identity N/A

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

      18. lower-/.f32 N/A

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

      19. associate-*r* N/A

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

      20. sub-neg N/A

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

      21. +-commutative N/A

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

   5. Simplified 96.3%

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

   6. Taylor expanded in u1 around 0

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. lower-fma.f32 85.2

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

   8. Simplified 85.2%

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

   if 0.0140000004 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.7%

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

      1. lift--.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-sqrt.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-sin.f32 N/A

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

      6. *-commutative N/A

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

      7. lift-sqrt.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. clear-num N/A

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

      10. sqrt-div N/A

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

      11. metadata-eval N/A

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

      12. un-div-inv N/A

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

      13. lower-/.f32 N/A

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

      14. lower-sqrt.f32 N/A

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

      15. lift--.f32 N/A

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

      16. div-sub N/A

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

      17. sub-neg N/A

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

      18. *-inverses N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f32 N/A

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

      21. lower-/.f32 97.8

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

   4. Applied egg-rr 97.8%

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

   5. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 63.7

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

   7. Simplified 63.7%

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

   9. Applied egg-rr 63.8%

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

   10. Taylor expanded in u1 around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. lower-*.f32 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f32 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f32 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f32 N/A

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

       14. lower-sqrt.f32 54.9

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

   12. Simplified 54.9%

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

Alternative 21: 76.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.0018700000364333391:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1, u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(u2, u2 \cdot -41.341702240407926, 6.28318530718\right) \cdot \left(u2 \cdot \sqrt{u1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.0018700000364333391)
   (* (* 6.28318530718 u2) (sqrt (fma u1 u1 u1)))
   (* (fma u2 (* u2 -41.341702240407926) 6.28318530718) (* u2 (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.0018700000364333391f) {
		tmp = (6.28318530718f * u2) * sqrtf(fmaf(u1, u1, u1));
	} else {
		tmp = fmaf(u2, (u2 * -41.341702240407926f), 6.28318530718f) * (u2 * sqrtf(u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.0018700000364333391))
		tmp = Float32(Float32(Float32(6.28318530718) * u2) * sqrt(fma(u1, u1, u1)));
	else
		tmp = Float32(fma(u2, Float32(u2 * Float32(-41.341702240407926)), Float32(6.28318530718)) * Float32(u2 * sqrt(u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00187000004

   1. Initial program 98.5%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. *-rgt-identity N/A

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

      6. sub-neg N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{1 + \left(\mathsf{neg}\left(u1\right)\right)}}} \]

      7. rgt-mult-inverse N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \frac{1}{-1 \cdot u1}} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

      8. mul-1-neg N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(u1\right)}} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

      9. distribute-neg-frac2 N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

      10. mul-1-neg N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \color{blue}{-1 \cdot u1}}} \]

      11. *-rgt-identity N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \color{blue}{\left(-1 \cdot u1\right) \cdot 1}}} \]

      12. distribute-lft-in N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}}} \]

      13. +-commutative N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}}} \]

      14. sub-neg N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 - \frac{1}{u1}\right)}}} \]

      15. associate-*r* N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

      16. lower-sqrt.f32 N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\sqrt{\frac{u1 \cdot 1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

      17. *-rgt-identity N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{\color{blue}{u1}}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}} \]

      18. lower-/.f32 N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{u1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

      19. associate-*r* N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \left(1 - \frac{1}{u1}\right)}}} \]

      20. sub-neg N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}}} \]

      21. +-commutative N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}}} \]

   5. Simplified 97.9%

      \[\leadsto \color{blue}{\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   6. Taylor expanded in u1 around 0

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{u1 \cdot \left(1 + u1\right)}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\left(1 + u1\right) \cdot u1}} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\left(u1 + 1\right)} \cdot u1} \]

      3. distribute-lft1-in N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{u1 \cdot u1 + u1}} \]

      4. lower-fma.f32 82.4

         \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}} \]

   8. Simplified 82.4%

      \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}} \]

   if 0.00187000004 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.9%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{\color{blue}{1 - u1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      2. lift-/.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{u1}{1 - u1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      3. lift-sqrt.f32 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      4. lift-*.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)} \]

      5. lift-sin.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

      7. lift-sqrt.f32 N/A

         \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]

      8. lift-/.f32 N/A

         \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{u1}{1 - u1}}} \]

      9. clear-num N/A

         \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{1 - u1}{u1}}}} \]

      10. sqrt-div N/A

          \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{1 - u1}{u1}}}} \]

      11. metadata-eval N/A

          \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \frac{\color{blue}{1}}{\sqrt{\frac{1 - u1}{u1}}} \]

      12. un-div-inv N/A

          \[\leadsto \color{blue}{\frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\frac{1 - u1}{u1}}}} \]

      13. lower-/.f32 N/A

          \[\leadsto \color{blue}{\frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\frac{1 - u1}{u1}}}} \]

      14. lower-sqrt.f32 N/A

          \[\leadsto \frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\color{blue}{\sqrt{\frac{1 - u1}{u1}}}} \]

      15. lift--.f32 N/A

          \[\leadsto \frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\frac{\color{blue}{1 - u1}}{u1}}} \]

      16. div-sub N/A

          \[\leadsto \frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\color{blue}{\frac{1}{u1} - \frac{u1}{u1}}}} \]

      17. sub-neg N/A

          \[\leadsto \frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\color{blue}{\frac{1}{u1} + \left(\mathsf{neg}\left(\frac{u1}{u1}\right)\right)}}} \]

      18. *-inverses N/A

          \[\leadsto \frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\frac{1}{u1} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)}} \]

      19. metadata-eval N/A

          \[\leadsto \frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\frac{1}{u1} + \color{blue}{-1}}} \]

      20. lower-+.f32 N/A

          \[\leadsto \frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\color{blue}{\frac{1}{u1} + -1}}} \]

      21. lower-/.f32 97.9

          \[\leadsto \frac{\sin \left(6.28318530718 \cdot u2\right)}{\sqrt{\color{blue}{\frac{1}{u1}} + -1}} \]

   4. Applied egg-rr 97.9%

      \[\leadsto \color{blue}{\frac{\sin \left(6.28318530718 \cdot u2\right)}{\sqrt{\frac{1}{u1} + -1}}} \]

   5. Taylor expanded in u2 around 0

      \[\leadsto \frac{\color{blue}{u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)}}{\sqrt{\frac{1}{u1} + -1}} \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \frac{\color{blue}{u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)}}{\sqrt{\frac{1}{u1} + -1}} \]

      2. +-commutative N/A

         \[\leadsto \frac{u2 \cdot \color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2} + \frac{314159265359}{50000000000}\right)}}{\sqrt{\frac{1}{u1} + -1}} \]

      3. lower-fma.f32 N/A

         \[\leadsto \frac{u2 \cdot \color{blue}{\mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, {u2}^{2}, \frac{314159265359}{50000000000}\right)}}{\sqrt{\frac{1}{u1} + -1}} \]

      4. unpow2 N/A

         \[\leadsto \frac{u2 \cdot \mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \color{blue}{u2 \cdot u2}, \frac{314159265359}{50000000000}\right)}{\sqrt{\frac{1}{u1} + -1}} \]

      5. lower-*.f32 70.6

         \[\leadsto \frac{u2 \cdot \mathsf{fma}\left(-41.341702240407926, \color{blue}{u2 \cdot u2}, 6.28318530718\right)}{\sqrt{\frac{1}{u1} + -1}} \]

   7. Simplified 70.6%

      \[\leadsto \frac{\color{blue}{u2 \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)}}{\sqrt{\frac{1}{u1} + -1}} \]

   9. Applied egg-rr 70.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(u2, u2 \cdot -41.341702240407926, 6.28318530718\right) \cdot \left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   10. Taylor expanded in u1 around 0

       \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\sqrt{u1} \cdot u2\right) \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right) \cdot \left(\sqrt{u1} \cdot u2\right)} \]

       3. lower-*.f32 N/A

          \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right) \cdot \left(\sqrt{u1} \cdot u2\right)} \]

       4. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2} + \frac{314159265359}{50000000000}\right)} \cdot \left(\sqrt{u1} \cdot u2\right) \]

       5. *-commutative N/A

          \[\leadsto \left(\color{blue}{{u2}^{2} \cdot \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}} + \frac{314159265359}{50000000000}\right) \cdot \left(\sqrt{u1} \cdot u2\right) \]

       6. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(u2 \cdot u2\right)} \cdot \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} + \frac{314159265359}{50000000000}\right) \cdot \left(\sqrt{u1} \cdot u2\right) \]

       7. associate-*l* N/A

          \[\leadsto \left(\color{blue}{u2 \cdot \left(u2 \cdot \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right)} + \frac{314159265359}{50000000000}\right) \cdot \left(\sqrt{u1} \cdot u2\right) \]

       8. *-commutative N/A

          \[\leadsto \left(u2 \cdot \color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot u2\right)} + \frac{314159265359}{50000000000}\right) \cdot \left(\sqrt{u1} \cdot u2\right) \]

       9. lower-fma.f32 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(u2, \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot u2, \frac{314159265359}{50000000000}\right)} \cdot \left(\sqrt{u1} \cdot u2\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(u2, \color{blue}{u2 \cdot \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}}, \frac{314159265359}{50000000000}\right) \cdot \left(\sqrt{u1} \cdot u2\right) \]

       11. lower-*.f32 N/A

           \[\leadsto \mathsf{fma}\left(u2, \color{blue}{u2 \cdot \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}}, \frac{314159265359}{50000000000}\right) \cdot \left(\sqrt{u1} \cdot u2\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(u2, u2 \cdot \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot \color{blue}{\left(u2 \cdot \sqrt{u1}\right)} \]

       13. lower-*.f32 N/A

           \[\leadsto \mathsf{fma}\left(u2, u2 \cdot \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \frac{314159265359}{50000000000}\right) \cdot \color{blue}{\left(u2 \cdot \sqrt{u1}\right)} \]

       14. lower-sqrt.f32 59.4

           \[\leadsto \mathsf{fma}\left(u2, u2 \cdot -41.341702240407926, 6.28318530718\right) \cdot \left(u2 \cdot \color{blue}{\sqrt{u1}}\right) \]

   12. Simplified 59.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(u2, u2 \cdot -41.341702240407926, 6.28318530718\right) \cdot \left(u2 \cdot \sqrt{u1}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 76.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.004399999976158142:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1, u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(u2 \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.004399999976158142)
   (* (* 6.28318530718 u2) (sqrt (fma u1 u1 u1)))
   (* (* u2 (fma -41.341702240407926 (* u2 u2) 6.28318530718)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.004399999976158142f) {
		tmp = (6.28318530718f * u2) * sqrtf(fmaf(u1, u1, u1));
	} else {
		tmp = (u2 * fmaf(-41.341702240407926f, (u2 * u2), 6.28318530718f)) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.004399999976158142))
		tmp = Float32(Float32(Float32(6.28318530718) * u2) * sqrt(fma(u1, u1, u1)));
	else
		tmp = Float32(Float32(u2 * fma(Float32(-41.341702240407926), Float32(u2 * u2), Float32(6.28318530718))) * sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.0044

   1. Initial program 98.5%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\frac{314159265359}{50000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{314159265359}{50000000000} \cdot \color{blue}{\left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

      3. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

      4. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)} \cdot \sqrt{\frac{u1}{1 - u1}} \]

      5. *-rgt-identity N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{\color{blue}{u1 \cdot 1}}{1 - u1}} \]

      6. sub-neg N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{1 + \left(\mathsf{neg}\left(u1\right)\right)}}} \]

      7. rgt-mult-inverse N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \frac{1}{-1 \cdot u1}} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

      8. mul-1-neg N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(u1\right)}} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

      9. distribute-neg-frac2 N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

      10. mul-1-neg N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \color{blue}{-1 \cdot u1}}} \]

      11. *-rgt-identity N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \color{blue}{\left(-1 \cdot u1\right) \cdot 1}}} \]

      12. distribute-lft-in N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}}} \]

      13. +-commutative N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}}} \]

      14. sub-neg N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 - \frac{1}{u1}\right)}}} \]

      15. associate-*r* N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

      16. lower-sqrt.f32 N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\sqrt{\frac{u1 \cdot 1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

      17. *-rgt-identity N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{\color{blue}{u1}}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}} \]

      18. lower-/.f32 N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{u1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

      19. associate-*r* N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \left(1 - \frac{1}{u1}\right)}}} \]

      20. sub-neg N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}}} \]

      21. +-commutative N/A

          \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}}} \]

   5. Simplified 97.5%

      \[\leadsto \color{blue}{\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   6. Taylor expanded in u1 around 0

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{u1 \cdot \left(1 + u1\right)}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\left(1 + u1\right) \cdot u1}} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\left(u1 + 1\right)} \cdot u1} \]

      3. distribute-lft1-in N/A

         \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{u1 \cdot u1 + u1}} \]

      4. lower-fma.f32 82.3

         \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}} \]

   8. Simplified 82.3%

      \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}} \]

   if 0.0044 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.8%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{\color{blue}{1 - u1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      2. lift-/.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{u1}{1 - u1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      3. lift-sqrt.f32 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \cdot \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \]

      4. lift-*.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)} \]

      5. lift-sin.f32 N/A

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

      7. lift-sqrt.f32 N/A

         \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]

      8. lift-/.f32 N/A

         \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{u1}{1 - u1}}} \]

      9. clear-num N/A

         \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{1}{\frac{1 - u1}{u1}}}} \]

      10. sqrt-div N/A

          \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{1 - u1}{u1}}}} \]

      11. metadata-eval N/A

          \[\leadsto \sin \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \frac{\color{blue}{1}}{\sqrt{\frac{1 - u1}{u1}}} \]

      12. un-div-inv N/A

          \[\leadsto \color{blue}{\frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\frac{1 - u1}{u1}}}} \]

      13. lower-/.f32 N/A

          \[\leadsto \color{blue}{\frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\frac{1 - u1}{u1}}}} \]

      14. lower-sqrt.f32 N/A

          \[\leadsto \frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\color{blue}{\sqrt{\frac{1 - u1}{u1}}}} \]

      15. lift--.f32 N/A

          \[\leadsto \frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\frac{\color{blue}{1 - u1}}{u1}}} \]

      16. div-sub N/A

          \[\leadsto \frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\color{blue}{\frac{1}{u1} - \frac{u1}{u1}}}} \]

      17. sub-neg N/A

          \[\leadsto \frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\color{blue}{\frac{1}{u1} + \left(\mathsf{neg}\left(\frac{u1}{u1}\right)\right)}}} \]

      18. *-inverses N/A

          \[\leadsto \frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\frac{1}{u1} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)}} \]

      19. metadata-eval N/A

          \[\leadsto \frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\frac{1}{u1} + \color{blue}{-1}}} \]

      20. lower-+.f32 N/A

          \[\leadsto \frac{\sin \left(\frac{314159265359}{50000000000} \cdot u2\right)}{\sqrt{\color{blue}{\frac{1}{u1} + -1}}} \]

      21. lower-/.f32 97.8

          \[\leadsto \frac{\sin \left(6.28318530718 \cdot u2\right)}{\sqrt{\color{blue}{\frac{1}{u1}} + -1}} \]

   4. Applied egg-rr 97.8%

      \[\leadsto \color{blue}{\frac{\sin \left(6.28318530718 \cdot u2\right)}{\sqrt{\frac{1}{u1} + -1}}} \]

   5. Taylor expanded in u2 around 0

      \[\leadsto \frac{\color{blue}{u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)}}{\sqrt{\frac{1}{u1} + -1}} \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \frac{\color{blue}{u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)}}{\sqrt{\frac{1}{u1} + -1}} \]

      2. +-commutative N/A

         \[\leadsto \frac{u2 \cdot \color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2} + \frac{314159265359}{50000000000}\right)}}{\sqrt{\frac{1}{u1} + -1}} \]

      3. lower-fma.f32 N/A

         \[\leadsto \frac{u2 \cdot \color{blue}{\mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, {u2}^{2}, \frac{314159265359}{50000000000}\right)}}{\sqrt{\frac{1}{u1} + -1}} \]

      4. unpow2 N/A

         \[\leadsto \frac{u2 \cdot \mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \color{blue}{u2 \cdot u2}, \frac{314159265359}{50000000000}\right)}{\sqrt{\frac{1}{u1} + -1}} \]

      5. lower-*.f32 68.4

         \[\leadsto \frac{u2 \cdot \mathsf{fma}\left(-41.341702240407926, \color{blue}{u2 \cdot u2}, 6.28318530718\right)}{\sqrt{\frac{1}{u1} + -1}} \]

   7. Simplified 68.4%

      \[\leadsto \frac{\color{blue}{u2 \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)}}{\sqrt{\frac{1}{u1} + -1}} \]

   8. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)} \]

      2. lower-sqrt.f32 N/A

         \[\leadsto \color{blue}{\sqrt{u1}} \cdot \left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2} + \frac{314159265359}{50000000000}\right)}\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, {u2}^{2}, \frac{314159265359}{50000000000}\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \color{blue}{u2 \cdot u2}, \frac{314159265359}{50000000000}\right)\right) \]

      7. lower-*.f32 57.8

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.341702240407926, \color{blue}{u2 \cdot u2}, 6.28318530718\right)\right) \]

   10. Simplified 57.8%

       \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.004399999976158142:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1, u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(u2 \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 73.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1, u1\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (* 6.28318530718 u2) (sqrt (fma u1 u1 u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	return (6.28318530718f * u2) * sqrtf(fmaf(u1, u1, u1));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(Float32(6.28318530718) * u2) * sqrt(fma(u1, u1, u1)))
end

Derivation

1. Initial program 98.3%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\frac{314159265359}{50000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{314159265359}{50000000000} \cdot \color{blue}{\left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   3. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   4. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)} \cdot \sqrt{\frac{u1}{1 - u1}} \]

   5. *-rgt-identity N/A

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{\color{blue}{u1 \cdot 1}}{1 - u1}} \]

   6. sub-neg N/A

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{1 + \left(\mathsf{neg}\left(u1\right)\right)}}} \]

   7. rgt-mult-inverse N/A

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \frac{1}{-1 \cdot u1}} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

   8. mul-1-neg N/A

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(u1\right)}} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

   9. distribute-neg-frac2 N/A

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

   10. mul-1-neg N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \color{blue}{-1 \cdot u1}}} \]

   11. *-rgt-identity N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \color{blue}{\left(-1 \cdot u1\right) \cdot 1}}} \]

   12. distribute-lft-in N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}}} \]

   13. +-commutative N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}}} \]

   14. sub-neg N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 - \frac{1}{u1}\right)}}} \]

   15. associate-*r* N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

   16. lower-sqrt.f32 N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\sqrt{\frac{u1 \cdot 1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

   17. *-rgt-identity N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{\color{blue}{u1}}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}} \]

   18. lower-/.f32 N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{u1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

   19. associate-*r* N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \left(1 - \frac{1}{u1}\right)}}} \]

   20. sub-neg N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}}} \]

   21. +-commutative N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}}} \]

5. Simplified 80.4%

   \[\leadsto \color{blue}{\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

6. Taylor expanded in u1 around 0

   \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{u1 \cdot \left(1 + u1\right)}} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\left(1 + u1\right) \cdot u1}} \]

   2. +-commutative N/A

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\left(u1 + 1\right)} \cdot u1} \]

   3. distribute-lft1-in N/A

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{u1 \cdot u1 + u1}} \]

   4. lower-fma.f32 70.0

      \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}} \]

8. Simplified 70.0%

   \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1, u1\right)}} \]
9. Add Preprocessing

Alternative 24: 64.8% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ u2 \cdot \left(6.28318530718 \cdot \sqrt{u1}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* u2 (* 6.28318530718 (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 * (6.28318530718f * sqrtf(u1));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = u2 * (6.28318530718e0 * sqrt(u1))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u2 * Float32(Float32(6.28318530718) * sqrt(u1)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u2 * (single(6.28318530718) * sqrt(u1));
end

Derivation

1. Initial program 98.3%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\frac{314159265359}{50000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{314159265359}{50000000000} \cdot \color{blue}{\left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   3. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   4. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)} \cdot \sqrt{\frac{u1}{1 - u1}} \]

   5. *-rgt-identity N/A

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{\color{blue}{u1 \cdot 1}}{1 - u1}} \]

   6. sub-neg N/A

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{1 + \left(\mathsf{neg}\left(u1\right)\right)}}} \]

   7. rgt-mult-inverse N/A

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \frac{1}{-1 \cdot u1}} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

   8. mul-1-neg N/A

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(u1\right)}} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

   9. distribute-neg-frac2 N/A

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)} + \left(\mathsf{neg}\left(u1\right)\right)}} \]

   10. mul-1-neg N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \color{blue}{-1 \cdot u1}}} \]

   11. *-rgt-identity N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \color{blue}{\left(-1 \cdot u1\right) \cdot 1}}} \]

   12. distribute-lft-in N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}}} \]

   13. +-commutative N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}}} \]

   14. sub-neg N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 - \frac{1}{u1}\right)}}} \]

   15. associate-*r* N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot 1}{\color{blue}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

   16. lower-sqrt.f32 N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\sqrt{\frac{u1 \cdot 1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

   17. *-rgt-identity N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{\color{blue}{u1}}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}} \]

   18. lower-/.f32 N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{u1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}} \]

   19. associate-*r* N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\color{blue}{\left(-1 \cdot u1\right) \cdot \left(1 - \frac{1}{u1}\right)}}} \]

   20. sub-neg N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}}} \]

   21. +-commutative N/A

       \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \sqrt{\frac{u1}{\left(-1 \cdot u1\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}}} \]

5. Simplified 80.4%

   \[\leadsto \color{blue}{\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

6. Taylor expanded in u1 around 0

   \[\leadsto \color{blue}{\frac{314159265359}{50000000000} \cdot \left(\sqrt{u1} \cdot u2\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sqrt{u1} \cdot u2\right) \cdot \frac{314159265359}{50000000000}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(u2 \cdot \sqrt{u1}\right)} \cdot \frac{314159265359}{50000000000} \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{u2 \cdot \left(\sqrt{u1} \cdot \frac{314159265359}{50000000000}\right)} \]

   4. lower-*.f32 N/A

      \[\leadsto \color{blue}{u2 \cdot \left(\sqrt{u1} \cdot \frac{314159265359}{50000000000}\right)} \]

   5. lower-*.f32 N/A

      \[\leadsto u2 \cdot \color{blue}{\left(\sqrt{u1} \cdot \frac{314159265359}{50000000000}\right)} \]

   6. lower-sqrt.f32 62.9

      \[\leadsto u2 \cdot \left(\color{blue}{\sqrt{u1}} \cdot 6.28318530718\right) \]

8. Simplified 62.9%

   \[\leadsto \color{blue}{u2 \cdot \left(\sqrt{u1} \cdot 6.28318530718\right)} \]

9. Final simplification 62.9%

   \[\leadsto u2 \cdot \left(6.28318530718 \cdot \sqrt{u1}\right) \]
10. Add Preprocessing

Alternative 25: 19.8% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ u2 \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* u2 (fma -41.341702240407926 (* u2 u2) 6.28318530718)))

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 * fmaf(-41.341702240407926f, (u2 * u2), 6.28318530718f);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u2 * fma(Float32(-41.341702240407926), Float32(u2 * u2), Float32(6.28318530718)))
end

Derivation

1. Initial program 98.3%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

4. Applied egg-rr 71.0%

   \[\leadsto \color{blue}{\left(\sqrt{\sqrt{\frac{u1}{u1 + 1}}} \cdot \sqrt{\sqrt{\frac{u1}{u1 + 1}}}\right)} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

5. Taylor expanded in u2 around 0

   \[\leadsto \left(\sqrt{\sqrt{\frac{u1}{u1 + 1}}} \cdot \sqrt{\sqrt{\frac{u1}{u1 + 1}}}\right) \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)} \]
6. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \left(\sqrt{\sqrt{\frac{u1}{u1 + 1}}} \cdot \sqrt{\sqrt{\frac{u1}{u1 + 1}}}\right) \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \left(\sqrt{\sqrt{\frac{u1}{u1 + 1}}} \cdot \sqrt{\sqrt{\frac{u1}{u1 + 1}}}\right) \cdot \left(u2 \cdot \color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2} + \frac{314159265359}{50000000000}\right)}\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \left(\sqrt{\sqrt{\frac{u1}{u1 + 1}}} \cdot \sqrt{\sqrt{\frac{u1}{u1 + 1}}}\right) \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, {u2}^{2}, \frac{314159265359}{50000000000}\right)}\right) \]

   4. unpow2 N/A

      \[\leadsto \left(\sqrt{\sqrt{\frac{u1}{u1 + 1}}} \cdot \sqrt{\sqrt{\frac{u1}{u1 + 1}}}\right) \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \color{blue}{u2 \cdot u2}, \frac{314159265359}{50000000000}\right)\right) \]

   5. lower-*.f32 65.8

      \[\leadsto \left(\sqrt{\sqrt{\frac{u1}{u1 + 1}}} \cdot \sqrt{\sqrt{\frac{u1}{u1 + 1}}}\right) \cdot \left(u2 \cdot \mathsf{fma}\left(-41.341702240407926, \color{blue}{u2 \cdot u2}, 6.28318530718\right)\right) \]

7. Simplified 65.8%

   \[\leadsto \left(\sqrt{\sqrt{\frac{u1}{u1 + 1}}} \cdot \sqrt{\sqrt{\frac{u1}{u1 + 1}}}\right) \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right)} \]

8. Taylor expanded in u1 around inf

   \[\leadsto \color{blue}{u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)} \]
9. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)} \]

   2. +-commutative N/A

      \[\leadsto u2 \cdot \color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2} + \frac{314159265359}{50000000000}\right)} \]

   3. lower-fma.f32 N/A

      \[\leadsto u2 \cdot \color{blue}{\mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, {u2}^{2}, \frac{314159265359}{50000000000}\right)} \]

   4. unpow2 N/A

      \[\leadsto u2 \cdot \mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \color{blue}{u2 \cdot u2}, \frac{314159265359}{50000000000}\right) \]

   5. lower-*.f32 20.2

      \[\leadsto u2 \cdot \mathsf{fma}\left(-41.341702240407926, \color{blue}{u2 \cdot u2}, 6.28318530718\right) \]

10. Simplified 20.2%

    \[\leadsto \color{blue}{u2 \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)} \]
11. Add Preprocessing

Alternative 26: 19.5% accurate, 22.5× speedup

Math

\[\begin{array}{l} \\ 6.28318530718 \cdot u2 \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (* 6.28318530718 u2))

C

float code(float cosTheta_i, float u1, float u2) {
	return 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 = 6.28318530718e0 * u2
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(6.28318530718) * u2)
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * u2;
end

Derivation

1. Initial program 98.3%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

4. Applied egg-rr 71.0%

   \[\leadsto \color{blue}{\left(\sqrt{\sqrt{\frac{u1}{u1 + 1}}} \cdot \sqrt{\sqrt{\frac{u1}{u1 + 1}}}\right)} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

5. Taylor expanded in u2 around 0

   \[\leadsto \left(\sqrt{\sqrt{\frac{u1}{u1 + 1}}} \cdot \sqrt{\sqrt{\frac{u1}{u1 + 1}}}\right) \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)} \]
6. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \left(\sqrt{\sqrt{\frac{u1}{u1 + 1}}} \cdot \sqrt{\sqrt{\frac{u1}{u1 + 1}}}\right) \cdot \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \left(\sqrt{\sqrt{\frac{u1}{u1 + 1}}} \cdot \sqrt{\sqrt{\frac{u1}{u1 + 1}}}\right) \cdot \left(u2 \cdot \color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2} + \frac{314159265359}{50000000000}\right)}\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \left(\sqrt{\sqrt{\frac{u1}{u1 + 1}}} \cdot \sqrt{\sqrt{\frac{u1}{u1 + 1}}}\right) \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, {u2}^{2}, \frac{314159265359}{50000000000}\right)}\right) \]

   4. unpow2 N/A

      \[\leadsto \left(\sqrt{\sqrt{\frac{u1}{u1 + 1}}} \cdot \sqrt{\sqrt{\frac{u1}{u1 + 1}}}\right) \cdot \left(u2 \cdot \mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \color{blue}{u2 \cdot u2}, \frac{314159265359}{50000000000}\right)\right) \]

   5. lower-*.f32 65.8

      \[\leadsto \left(\sqrt{\sqrt{\frac{u1}{u1 + 1}}} \cdot \sqrt{\sqrt{\frac{u1}{u1 + 1}}}\right) \cdot \left(u2 \cdot \mathsf{fma}\left(-41.341702240407926, \color{blue}{u2 \cdot u2}, 6.28318530718\right)\right) \]

7. Simplified 65.8%

   \[\leadsto \left(\sqrt{\sqrt{\frac{u1}{u1 + 1}}} \cdot \sqrt{\sqrt{\frac{u1}{u1 + 1}}}\right) \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)\right)} \]

8. Taylor expanded in u1 around inf

   \[\leadsto \color{blue}{u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)} \]
9. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \color{blue}{u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)} \]

   2. +-commutative N/A

      \[\leadsto u2 \cdot \color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2} + \frac{314159265359}{50000000000}\right)} \]

   3. lower-fma.f32 N/A

      \[\leadsto u2 \cdot \color{blue}{\mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, {u2}^{2}, \frac{314159265359}{50000000000}\right)} \]

   4. unpow2 N/A

      \[\leadsto u2 \cdot \mathsf{fma}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \color{blue}{u2 \cdot u2}, \frac{314159265359}{50000000000}\right) \]

   5. lower-*.f32 20.2

      \[\leadsto u2 \cdot \mathsf{fma}\left(-41.341702240407926, \color{blue}{u2 \cdot u2}, 6.28318530718\right) \]

10. Simplified 20.2%

    \[\leadsto \color{blue}{u2 \cdot \mathsf{fma}\left(-41.341702240407926, u2 \cdot u2, 6.28318530718\right)} \]

11. Taylor expanded in u2 around 0

    \[\leadsto u2 \cdot \color{blue}{\frac{314159265359}{50000000000}} \]
12. Step-by-step derivation

13. Simplified 19.7%

    \[\leadsto u2 \cdot \color{blue}{6.28318530718} \]

14. Final simplification 19.7%

    \[\leadsto 6.28318530718 \cdot u2 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(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))))