Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 14.0s

Alternatives: 21

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

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

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

Derivation

1. Initial program 98.3%

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

Alternative 2: 97.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.7549999952316284:\\ \;\;\;\;\frac{u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot \left(-41.341702240407926 + \left(u2 \cdot u2\right) \cdot \left(81.6052492761019 + u2 \cdot \left(u2 \cdot -76.70585975309672\right)\right)\right)\right)}{\sqrt{\frac{1}{u1} + -1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.7549999952316284)
   (/
    (*
     u2
     (+
      6.28318530718
      (*
       (* u2 u2)
       (+
        -41.341702240407926
        (* (* u2 u2) (+ 81.6052492761019 (* u2 (* u2 -76.70585975309672))))))))
    (sqrt (+ (/ 1.0 u1) -1.0)))
   (* (sin (* 6.28318530718 u2)) (sqrt (* u1 (+ u1 1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.7549999952316284f) {
		tmp = (u2 * (6.28318530718f + ((u2 * u2) * (-41.341702240407926f + ((u2 * u2) * (81.6052492761019f + (u2 * (u2 * -76.70585975309672f)))))))) / sqrtf(((1.0f / u1) + -1.0f));
	} else {
		tmp = sinf((6.28318530718f * u2)) * sqrtf((u1 * (u1 + 1.0f)));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((6.28318530718e0 * u2) <= 0.7549999952316284e0) then
        tmp = (u2 * (6.28318530718e0 + ((u2 * u2) * ((-41.341702240407926e0) + ((u2 * u2) * (81.6052492761019e0 + (u2 * (u2 * (-76.70585975309672e0))))))))) / sqrt(((1.0e0 / u1) + (-1.0e0)))
    else
        tmp = sin((6.28318530718e0 * u2)) * sqrt((u1 * (u1 + 1.0e0)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.7549999952316284))
		tmp = Float32(Float32(u2 * Float32(Float32(6.28318530718) + Float32(Float32(u2 * u2) * Float32(Float32(-41.341702240407926) + Float32(Float32(u2 * u2) * Float32(Float32(81.6052492761019) + Float32(u2 * Float32(u2 * Float32(-76.70585975309672))))))))) / sqrt(Float32(Float32(Float32(1.0) / u1) + Float32(-1.0))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(Float32(u1 * Float32(u1 + Float32(1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.7549999952316284))
		tmp = (u2 * (single(6.28318530718) + ((u2 * u2) * (single(-41.341702240407926) + ((u2 * u2) * (single(81.6052492761019) + (u2 * (u2 * single(-76.70585975309672))))))))) / sqrt(((single(1.0) / u1) + single(-1.0)));
	else
		tmp = sin((single(6.28318530718) * u2)) * sqrt((u1 * (u1 + single(1.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.754999995

   1. Initial program 98.5%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

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

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

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

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

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

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

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

      9. pow1/2 N/A

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

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

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

      11. div-sub N/A

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

      12. sub-neg N/A

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

      13. *-inverses N/A

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

      14. metadata-eval N/A

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

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

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

      16. /-lowering-/.f32 98.5%

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

   4. Applied egg-rr 98.5%

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

      1. unpow1/2 N/A

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

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

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

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

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

      4. /-lowering-/.f32 98.5%

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

   6. Applied egg-rr 98.5%

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

   7. Taylor expanded in u2 around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-lowering-*.f32 98.6%

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

   9. Simplified 98.6%

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

   if 0.754999995 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 96.7%

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

   3. Taylor expanded in u1 around 0

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

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

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

      2. +-commutative N/A

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

      3. +-lowering-+.f32 87.8%

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

   5. Simplified 87.8%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.7549999952316284:\\ \;\;\;\;\frac{u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot \left(-41.341702240407926 + \left(u2 \cdot u2\right) \cdot \left(81.6052492761019 + u2 \cdot \left(u2 \cdot -76.70585975309672\right)\right)\right)\right)}{\sqrt{\frac{1}{u1} + -1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return (u2 * (6.28318530718f + ((u2 * u2) * (-41.341702240407926f + ((u2 * u2) * (81.6052492761019f + (u2 * (u2 * -76.70585975309672f)))))))) / 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 = (u2 * (6.28318530718e0 + ((u2 * u2) * ((-41.341702240407926e0) + ((u2 * u2) * (81.6052492761019e0 + (u2 * (u2 * (-76.70585975309672e0))))))))) / sqrt(((1.0e0 / u1) + (-1.0e0)))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (u2 * (single(6.28318530718) + ((u2 * u2) * (single(-41.341702240407926) + ((u2 * u2) * (single(81.6052492761019) + (u2 * (u2 * single(-76.70585975309672))))))))) / 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. *-commutative N/A

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

   2. clear-num N/A

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

   3. sqrt-div N/A

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

   4. metadata-eval N/A

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

   5. un-div-inv N/A

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

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

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

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

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

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

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

   9. pow1/2 N/A

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

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

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

   11. div-sub N/A

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

   12. sub-neg N/A

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

   13. *-inverses N/A

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

   14. metadata-eval N/A

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

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

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

   16. /-lowering-/.f32 98.3%

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

4. Applied egg-rr 98.3%

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

   1. unpow1/2 N/A

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

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

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

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

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

   4. /-lowering-/.f32 98.3%

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

6. Applied egg-rr 98.3%

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

7. Taylor expanded in u2 around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

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

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

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

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

   11. unpow2 N/A

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

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

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

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

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

   14. *-commutative N/A

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

   15. unpow2 N/A

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

   16. associate-*l* N/A

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

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

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

   18. *-lowering-*.f32 93.4%

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

9. Simplified 93.4%

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

Alternative 4: 93.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 / (1.0e0 - u1))) * (u2 * (6.28318530718e0 + (u2 * (u2 * ((-41.341702240407926e0) + ((u2 * u2) * (81.6052492761019e0 + ((u2 * u2) * (-76.70585975309672e0)))))))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

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

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

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

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

   17. unpow2 N/A

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

   18. *-lowering-*.f32 93.3%

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

5. Simplified 93.3%

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

   1. *-commutative N/A

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

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

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

7. Applied egg-rr 93.3%

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

8. Final simplification 93.3%

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

Alternative 5: 93.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 / (1.0e0 - u1))) * (u2 * (6.28318530718e0 + ((u2 * u2) * ((-41.341702240407926e0) + (u2 * (u2 * (81.6052492761019e0 + ((u2 * u2) * (-76.70585975309672e0)))))))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

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

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

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

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

   17. unpow2 N/A

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

   18. *-lowering-*.f32 93.3%

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

5. Simplified 93.3%

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

Alternative 6: 84.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{u1}{1 - u1}\\ \mathbf{if}\;t\_0 \leq 7.999999979801942 \cdot 10^{-6}:\\ \;\;\;\;\frac{u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot -41.341702240407926\right)}{\sqrt{\frac{1}{u1}}}\\ \mathbf{else}:\\ \;\;\;\;u2 \cdot \left(6.28318530718 \cdot {t\_0}^{0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (/ u1 (- 1.0 u1))))
   (if (<= t_0 7.999999979801942e-6)
     (/
      (* u2 (+ 6.28318530718 (* (* u2 u2) -41.341702240407926)))
      (sqrt (/ 1.0 u1)))
     (* u2 (* 6.28318530718 (pow t_0 0.5))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u1 / (1.0f - u1);
	float tmp;
	if (t_0 <= 7.999999979801942e-6f) {
		tmp = (u2 * (6.28318530718f + ((u2 * u2) * -41.341702240407926f))) / sqrtf((1.0f / u1));
	} else {
		tmp = u2 * (6.28318530718f * powf(t_0, 0.5f));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: t_0
    real(4) :: tmp
    t_0 = u1 / (1.0e0 - u1)
    if (t_0 <= 7.999999979801942e-6) then
        tmp = (u2 * (6.28318530718e0 + ((u2 * u2) * (-41.341702240407926e0)))) / sqrt((1.0e0 / u1))
    else
        tmp = u2 * (6.28318530718e0 * (t_0 ** 0.5e0))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u1 / Float32(Float32(1.0) - u1))
	tmp = Float32(0.0)
	if (t_0 <= Float32(7.999999979801942e-6))
		tmp = Float32(Float32(u2 * Float32(Float32(6.28318530718) + Float32(Float32(u2 * u2) * Float32(-41.341702240407926)))) / sqrt(Float32(Float32(1.0) / u1)));
	else
		tmp = Float32(u2 * Float32(Float32(6.28318530718) * (t_0 ^ Float32(0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = u1 / (single(1.0) - u1);
	tmp = single(0.0);
	if (t_0 <= single(7.999999979801942e-6))
		tmp = (u2 * (single(6.28318530718) + ((u2 * u2) * single(-41.341702240407926)))) / sqrt((single(1.0) / u1));
	else
		tmp = u2 * (single(6.28318530718) * (t_0 ^ single(0.5)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 u1 (-.f32 #s(literal 1 binary32) u1)) < 7.99999998e-6

   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. *-commutative N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. un-div-inv N/A

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

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

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

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

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

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

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

      9. pow1/2 N/A

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

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

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

      11. div-sub N/A

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

      12. sub-neg N/A

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

      13. *-inverses N/A

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

      14. metadata-eval N/A

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

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

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

      16. /-lowering-/.f32 98.3%

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

   4. Applied egg-rr 98.3%

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

   5. Taylor expanded in u1 around 0

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

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

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

      2. /-lowering-/.f32 96.2%

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

   7. Simplified 96.2%

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

   8. Taylor expanded in u2 around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f32 87.1%

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

   10. Simplified 87.1%

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

   if 7.99999998e-6 < (/.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

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

      2. associate-*r* N/A

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

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

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

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

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

      5. *-rgt-identity N/A

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

      6. sub-neg N/A

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

      7. rgt-mult-inverse N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. mul-1-neg N/A

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

      11. *-rgt-identity N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. associate-*r* N/A

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

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

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

      17. *-rgt-identity N/A

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

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

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

      19. associate-*r* N/A

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

      20. sub-neg N/A

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

      21. +-commutative N/A

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

   5. Simplified 83.9%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

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

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

      5. pow1/2 N/A

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

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

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

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

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

      8. --lowering--.f32 84.0%

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

   7. Applied egg-rr 84.0%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{u1}{1 - u1} \leq 7.999999979801942 \cdot 10^{-6}:\\ \;\;\;\;\frac{u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot -41.341702240407926\right)}{\sqrt{\frac{1}{u1}}}\\ \mathbf{else}:\\ \;\;\;\;u2 \cdot \left(6.28318530718 \cdot {\left(\frac{u1}{1 - u1}\right)}^{0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return (u2 * (6.28318530718f + ((u2 * u2) * (-41.341702240407926f + ((u2 * u2) * 81.6052492761019f))))) / 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 = (u2 * (6.28318530718e0 + ((u2 * u2) * ((-41.341702240407926e0) + ((u2 * u2) * 81.6052492761019e0))))) / sqrt(((1.0e0 / u1) + (-1.0e0)))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (u2 * (single(6.28318530718) + ((u2 * u2) * (single(-41.341702240407926) + ((u2 * u2) * single(81.6052492761019)))))) / 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. *-commutative N/A

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

   2. clear-num N/A

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

   3. sqrt-div N/A

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

   4. metadata-eval N/A

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

   5. un-div-inv N/A

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

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

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

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

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

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

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

   9. pow1/2 N/A

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

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

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

   11. div-sub N/A

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

   12. sub-neg N/A

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

   13. *-inverses N/A

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

   14. metadata-eval N/A

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

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

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

   16. /-lowering-/.f32 98.3%

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

4. Applied egg-rr 98.3%

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

   1. unpow1/2 N/A

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

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

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

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

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

   4. /-lowering-/.f32 98.3%

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

6. Applied egg-rr 98.3%

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

7. Taylor expanded in u2 around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

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

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

   10. *-commutative N/A

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

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

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

   12. unpow2 N/A

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

   13. *-lowering-*.f32 91.0%

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

9. Simplified 91.0%

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

Alternative 8: 91.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 / (1.0e0 - u1))) * (u2 * (6.28318530718e0 + ((u2 * u2) * ((-41.341702240407926e0) + ((u2 * u2) * 81.6052492761019e0)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

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

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

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

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

   17. unpow2 N/A

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

   18. *-lowering-*.f32 93.3%

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

5. Simplified 93.3%

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

6. Taylor expanded in u2 around 0

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

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

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

   2. unpow2 N/A

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

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

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. +-commutative N/A

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

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

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

   8. *-commutative N/A

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

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

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

   10. unpow2 N/A

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

   11. *-lowering-*.f32 91.0%

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

8. Simplified 91.0%

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

Alternative 9: 91.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 / (1.0e0 - u1))) * (u2 * (6.28318530718e0 + ((u2 * u2) * ((-41.341702240407926e0) + (u2 * (u2 * 81.6052492761019e0))))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

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

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

   10. unpow2 N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

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

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

   14. *-commutative N/A

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

   15. *-lowering-*.f32 91.0%

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

5. Simplified 91.0%

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

Alternative 10: 84.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{u1}{1 - u1}\\ \mathbf{if}\;t\_0 \leq 7.999999979801942 \cdot 10^{-6}:\\ \;\;\;\;\left(u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot -41.341702240407926\right)\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;u2 \cdot \left(6.28318530718 \cdot {t\_0}^{0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (/ u1 (- 1.0 u1))))
   (if (<= t_0 7.999999979801942e-6)
     (* (* u2 (+ 6.28318530718 (* (* u2 u2) -41.341702240407926))) (sqrt u1))
     (* u2 (* 6.28318530718 (pow t_0 0.5))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u1 / (1.0f - u1);
	float tmp;
	if (t_0 <= 7.999999979801942e-6f) {
		tmp = (u2 * (6.28318530718f + ((u2 * u2) * -41.341702240407926f))) * sqrtf(u1);
	} else {
		tmp = u2 * (6.28318530718f * powf(t_0, 0.5f));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: t_0
    real(4) :: tmp
    t_0 = u1 / (1.0e0 - u1)
    if (t_0 <= 7.999999979801942e-6) then
        tmp = (u2 * (6.28318530718e0 + ((u2 * u2) * (-41.341702240407926e0)))) * sqrt(u1)
    else
        tmp = u2 * (6.28318530718e0 * (t_0 ** 0.5e0))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u1 / Float32(Float32(1.0) - u1))
	tmp = Float32(0.0)
	if (t_0 <= Float32(7.999999979801942e-6))
		tmp = Float32(Float32(u2 * Float32(Float32(6.28318530718) + Float32(Float32(u2 * u2) * Float32(-41.341702240407926)))) * sqrt(u1));
	else
		tmp = Float32(u2 * Float32(Float32(6.28318530718) * (t_0 ^ Float32(0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = u1 / (single(1.0) - u1);
	tmp = single(0.0);
	if (t_0 <= single(7.999999979801942e-6))
		tmp = (u2 * (single(6.28318530718) + ((u2 * u2) * single(-41.341702240407926)))) * sqrt(u1);
	else
		tmp = u2 * (single(6.28318530718) * (t_0 ^ single(0.5)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 u1 (-.f32 #s(literal 1 binary32) u1)) < 7.99999998e-6

   1. Initial program 98.4%

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in u2 around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f32 88.7%

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

   7. Simplified 88.7%

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

   8. Taylor expanded in u1 around 0

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

   10. Simplified 87.1%

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

   if 7.99999998e-6 < (/.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

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

      2. associate-*r* N/A

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

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

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

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

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

      5. *-rgt-identity N/A

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

      6. sub-neg N/A

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

      7. rgt-mult-inverse N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. mul-1-neg N/A

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

      11. *-rgt-identity N/A

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

      12. distribute-lft-in N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. associate-*r* N/A

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

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

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

      17. *-rgt-identity N/A

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

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

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

      19. associate-*r* N/A

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

      20. sub-neg N/A

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

      21. +-commutative N/A

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

   5. Simplified 83.9%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

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

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

      5. pow1/2 N/A

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

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

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

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

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

      8. --lowering--.f32 84.0%

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

   7. Applied egg-rr 84.0%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{u1}{1 - u1} \leq 7.999999979801942 \cdot 10^{-6}:\\ \;\;\;\;\left(u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot -41.341702240407926\right)\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;u2 \cdot \left(6.28318530718 \cdot {\left(\frac{u1}{1 - u1}\right)}^{0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 89.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return (u2 * (6.28318530718f + ((u2 * u2) * -41.341702240407926f))) / 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 = (u2 * (6.28318530718e0 + ((u2 * u2) * (-41.341702240407926e0)))) / sqrt(((1.0e0 / u1) + (-1.0e0)))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (u2 * (single(6.28318530718) + ((u2 * u2) * single(-41.341702240407926)))) / 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. *-commutative N/A

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

   2. clear-num N/A

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

   3. sqrt-div N/A

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

   4. metadata-eval N/A

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

   5. un-div-inv N/A

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

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

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

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

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

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

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

   9. pow1/2 N/A

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

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

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

   11. div-sub N/A

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

   12. sub-neg N/A

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

   13. *-inverses N/A

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

   14. metadata-eval N/A

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

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

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

   16. /-lowering-/.f32 98.3%

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

4. Applied egg-rr 98.3%

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

   1. unpow1/2 N/A

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

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

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

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

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

   4. /-lowering-/.f32 98.3%

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

6. Applied egg-rr 98.3%

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

7. Taylor expanded in u2 around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

   5. *-lowering-*.f32 89.5%

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

9. Simplified 89.5%

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

10. Final simplification 89.5%

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

Alternative 12: 89.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

   5. *-lowering-*.f32 89.5%

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

5. Simplified 89.5%

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

   1. *-commutative N/A

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

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

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

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

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f32 89.5%

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

7. Applied egg-rr 89.5%

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

8. Final simplification 89.5%

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

Alternative 13: 89.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

   5. *-lowering-*.f32 89.5%

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

5. Simplified 89.5%

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

6. Final simplification 89.5%

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

Alternative 14: 89.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0

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

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

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

   5. +-commutative N/A

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

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

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

5. Simplified 89.5%

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

6. Final simplification 89.5%

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

Alternative 15: 81.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

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

   2. associate-*r* N/A

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

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

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

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

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

   5. *-rgt-identity N/A

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

   6. sub-neg N/A

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

   7. rgt-mult-inverse N/A

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

   8. mul-1-neg N/A

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

   9. distribute-neg-frac2 N/A

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

   10. mul-1-neg N/A

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

   11. *-rgt-identity N/A

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

   12. distribute-lft-in N/A

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

   13. +-commutative N/A

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

   14. sub-neg N/A

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

   15. associate-*r* N/A

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

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

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

   17. *-rgt-identity N/A

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

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

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

   19. associate-*r* N/A

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

   20. sub-neg N/A

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

   21. +-commutative N/A

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

5. Simplified 81.8%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

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

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

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

   5. pow1/2 N/A

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

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

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

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

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

   8. --lowering--.f32 81.9%

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

7. Applied egg-rr 81.9%

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

8. Final simplification 81.9%

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

Alternative 16: 81.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{6.28318530718 \cdot u2}{\sqrt{\frac{1}{u1} + -1}} \end{array} \]

FPCore

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

C

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

Julia

function code(cosTheta_i, u1, u2)
	return Float32(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 = (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. *-commutative N/A

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

   2. clear-num N/A

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

   3. sqrt-div N/A

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

   4. metadata-eval N/A

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

   5. un-div-inv N/A

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

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

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

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

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

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

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

   9. pow1/2 N/A

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

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

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

   11. div-sub N/A

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

   12. sub-neg N/A

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

   13. *-inverses N/A

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

   14. metadata-eval N/A

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

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

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

   16. /-lowering-/.f32 98.3%

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

4. Applied egg-rr 98.3%

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

   1. unpow1/2 N/A

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

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

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

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

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

   4. /-lowering-/.f32 98.3%

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

6. Applied egg-rr 98.3%

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

7. Taylor expanded in u2 around 0

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

   1. *-lowering-*.f32 81.9%

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

9. Simplified 81.9%

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

Alternative 17: 81.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

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

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

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

   5. *-rgt-identity N/A

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

   6. sub-neg N/A

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

   7. rgt-mult-inverse N/A

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

   8. mul-1-neg N/A

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

   9. distribute-neg-frac2 N/A

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

   10. mul-1-neg N/A

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

   11. *-rgt-identity N/A

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

   12. distribute-lft-in N/A

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

   13. +-commutative N/A

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

   14. sub-neg N/A

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

   15. associate-*r* N/A

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

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

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

   17. *-rgt-identity N/A

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

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

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

   19. associate-*r* N/A

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

   20. sub-neg N/A

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

   21. +-commutative N/A

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

5. Simplified 81.8%

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

6. Final simplification 81.8%

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

Alternative 18: 73.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (single(6.28318530718) * u2) * sqrt((u1 * (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. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

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

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

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

   5. *-rgt-identity N/A

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

   6. sub-neg N/A

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

   7. rgt-mult-inverse N/A

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

   8. mul-1-neg N/A

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

   9. distribute-neg-frac2 N/A

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

   10. mul-1-neg N/A

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

   11. *-rgt-identity N/A

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

   12. distribute-lft-in N/A

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

   13. +-commutative N/A

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

   14. sub-neg N/A

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

   15. associate-*r* N/A

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

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

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

   17. *-rgt-identity N/A

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

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

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

   19. associate-*r* N/A

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

   20. sub-neg N/A

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

   21. +-commutative N/A

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

5. Simplified 81.8%

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

6. Taylor expanded in u1 around 0

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

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

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

   2. +-lowering-+.f32 73.5%

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

8. Simplified 73.5%

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

9. Final simplification 73.5%

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

Alternative 19: 64.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{6.28318530718 \cdot u2}{\sqrt{\frac{1}{u1}}} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (single(6.28318530718) * u2) / sqrt((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. Step-by-step derivation

   1. *-commutative N/A

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

   2. clear-num N/A

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

   3. sqrt-div N/A

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

   4. metadata-eval N/A

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

   5. un-div-inv N/A

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

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

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

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

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

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

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

   9. pow1/2 N/A

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

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

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

   11. div-sub N/A

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

   12. sub-neg N/A

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

   13. *-inverses N/A

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

   14. metadata-eval N/A

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

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

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

   16. /-lowering-/.f32 98.3%

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

4. Applied egg-rr 98.3%

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

5. Taylor expanded in u1 around 0

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

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

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

   2. /-lowering-/.f32 74.0%

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

7. Simplified 74.0%

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

8. Taylor expanded in u2 around 0

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

   1. *-lowering-*.f32 64.0%

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

10. Simplified 64.0%

    \[\leadsto \frac{\color{blue}{6.28318530718 \cdot u2}}{\sqrt{\frac{1}{u1}}} \]
11. Add Preprocessing

Alternative 20: 64.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

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

   2. associate-*r* N/A

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

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

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

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

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

   5. *-rgt-identity N/A

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

   6. sub-neg N/A

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

   7. rgt-mult-inverse N/A

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

   8. mul-1-neg N/A

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

   9. distribute-neg-frac2 N/A

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

   10. mul-1-neg N/A

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

   11. *-rgt-identity N/A

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

   12. distribute-lft-in N/A

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

   13. +-commutative N/A

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

   14. sub-neg N/A

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

   15. associate-*r* N/A

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

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

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

   17. *-rgt-identity N/A

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

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

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

   19. associate-*r* N/A

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

   20. sub-neg N/A

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

   21. +-commutative N/A

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

5. Simplified 81.8%

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

6. Taylor expanded in u1 around 0

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

   1. sqrt-lowering-sqrt.f32 63.9%

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

8. Simplified 63.9%

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

Alternative 21: 64.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (u2 * sqrt(u1));
end

Derivation

1. Initial program 98.3%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
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

      \[\leadsto \frac{314159265359}{50000000000} \cdot \left(u2 \cdot \color{blue}{\sqrt{\frac{u1}{1 - u1}}}\right) \]

   2. associate-*r* N/A

      \[\leadsto \left(\frac{314159265359}{50000000000} \cdot u2\right) \cdot \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(\frac{314159265359}{50000000000} \cdot u2\right), \color{blue}{\left(\sqrt{\frac{u1}{1 - u1}}\right)}\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\color{blue}{\frac{u1}{1 - u1}}}\right)\right) \]

   5. *-rgt-identity N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{1 - u1}}\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{1 + \left(\mathsf{neg}\left(u1\right)\right)}}\right)\right) \]

   7. rgt-mult-inverse N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \frac{1}{-1 \cdot u1} + \left(\mathsf{neg}\left(u1\right)\right)}}\right)\right) \]

   8. mul-1-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \frac{1}{\mathsf{neg}\left(u1\right)} + \left(\mathsf{neg}\left(u1\right)\right)}}\right)\right) \]

   9. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \left(\mathsf{neg}\left(u1\right)\right)}}\right)\right) \]

   10. mul-1-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + -1 \cdot u1}}\right)\right) \]

   11. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + \left(-1 \cdot u1\right) \cdot 1}}\right)\right) \]

   12. distribute-lft-in N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)}}\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)}}\right)\right) \]

   14. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{\left(-1 \cdot u1\right) \cdot \left(1 - \frac{1}{u1}\right)}}\right)\right) \]

   15. associate-*r* N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \left(\sqrt{\frac{u1 \cdot 1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}}\right)\right) \]

   16. sqrt-lowering-sqrt.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \mathsf{sqrt.f32}\left(\left(\frac{u1 \cdot 1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}\right)\right)\right) \]

   17. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \mathsf{sqrt.f32}\left(\left(\frac{u1}{-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)}\right)\right)\right) \]

   18. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(-1 \cdot \left(u1 \cdot \left(1 - \frac{1}{u1}\right)\right)\right)\right)\right)\right) \]

   19. associate-*r* N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\left(-1 \cdot u1\right) \cdot \left(1 - \frac{1}{u1}\right)\right)\right)\right)\right) \]

   20. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\left(-1 \cdot u1\right) \cdot \left(1 + \left(\mathsf{neg}\left(\frac{1}{u1}\right)\right)\right)\right)\right)\right)\right) \]

   21. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right), \mathsf{sqrt.f32}\left(\mathsf{/.f32}\left(u1, \left(\left(-1 \cdot u1\right) \cdot \left(\left(\mathsf{neg}\left(\frac{1}{u1}\right)\right) + 1\right)\right)\right)\right)\right) \]

5. Simplified 81.8%

   \[\leadsto \color{blue}{\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

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. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \color{blue}{\left(\sqrt{u1} \cdot u2\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \left(u2 \cdot \color{blue}{\sqrt{u1}}\right)\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(u2, \color{blue}{\left(\sqrt{u1}\right)}\right)\right) \]

   4. sqrt-lowering-sqrt.f32 63.9%

      \[\leadsto \mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(u2, \mathsf{sqrt.f32}\left(u1\right)\right)\right) \]

8. Simplified 63.9%

   \[\leadsto \color{blue}{6.28318530718 \cdot \left(u2 \cdot \sqrt{u1}\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024145 
(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))))