Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 14.2s

Alternatives: 19

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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{u1} + -1\\ \frac{\sin \left(6.28318530718 \cdot u2\right)}{{\left(t\_0 \cdot t\_0\right)}^{0.25}} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (+ (/ 1.0 u1) -1.0)))
   (/ (sin (* 6.28318530718 u2)) (pow (* t_0 t_0) 0.25))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (1.0f / u1) + -1.0f;
	return sinf((6.28318530718f * u2)) / powf((t_0 * t_0), 0.25f);
}

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
    t_0 = (1.0e0 / u1) + (-1.0e0)
    code = sin((6.28318530718e0 * u2)) / ((t_0 * t_0) ** 0.25e0)
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(1.0) / u1) + Float32(-1.0))
	return Float32(sin(Float32(Float32(6.28318530718) * u2)) / (Float32(t_0 * t_0) ^ Float32(0.25)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	t_0 = (single(1.0) / u1) + single(-1.0);
	tmp = sin((single(6.28318530718) * u2)) / ((t_0 * t_0) ^ single(0.25));
end

Derivation

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 98.4%

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

4. Applied egg-rr 98.4%

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

6. Applied egg-rr 98.4%

   \[\leadsto \frac{\sin \left(6.28318530718 \cdot u2\right)}{\color{blue}{{\left(\left(\frac{1}{u1} + -1\right) \cdot \left(\frac{1}{u1} + -1\right)\right)}^{0.25}}} \]
7. Add Preprocessing

Alternative 2: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.7200000286102295:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot \left(-41.341702240407926 + \left(u2 \cdot u2\right) \cdot \left(81.6052492761019 + \left(u2 \cdot u2\right) \cdot -76.70585975309672\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.7200000286102295)
   (*
    (sqrt (/ u1 (- 1.0 u1)))
    (*
     u2
     (+
      6.28318530718
      (*
       (* u2 u2)
       (+
        -41.341702240407926
        (*
         (* u2 u2)
         (+ 81.6052492761019 (* (* u2 u2) -76.70585975309672))))))))
   (* (sin (* 6.28318530718 u2)) (sqrt (* u1 (+ 1.0 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.7200000286102295f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * (u2 * (6.28318530718f + ((u2 * u2) * (-41.341702240407926f + ((u2 * u2) * (81.6052492761019f + ((u2 * u2) * -76.70585975309672f)))))));
	} else {
		tmp = sinf((6.28318530718f * u2)) * sqrtf((u1 * (1.0f + u1)));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((6.28318530718e0 * u2) <= 0.7200000286102295e0) then
        tmp = sqrt((u1 / (1.0e0 - u1))) * (u2 * (6.28318530718e0 + ((u2 * u2) * ((-41.341702240407926e0) + ((u2 * u2) * (81.6052492761019e0 + ((u2 * u2) * (-76.70585975309672e0))))))))
    else
        tmp = sin((6.28318530718e0 * u2)) * sqrt((u1 * (1.0e0 + u1)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.7200000286102295))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(u2 * Float32(Float32(6.28318530718) + Float32(Float32(u2 * u2) * Float32(Float32(-41.341702240407926) + Float32(Float32(u2 * u2) * Float32(Float32(81.6052492761019) + Float32(Float32(u2 * u2) * Float32(-76.70585975309672)))))))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(Float32(u1 * Float32(Float32(1.0) + u1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.7200000286102295))
		tmp = sqrt((u1 / (single(1.0) - u1))) * (u2 * (single(6.28318530718) + ((u2 * u2) * (single(-41.341702240407926) + ((u2 * u2) * (single(81.6052492761019) + ((u2 * u2) * single(-76.70585975309672))))))));
	else
		tmp = sin((single(6.28318530718) * u2)) * sqrt((u1 * (single(1.0) + u1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.720000029

   1. Initial program 98.5%

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

   3. Taylor expanded in u2 around 0

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f32 98.5%

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

   5. Simplified 98.5%

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

   if 0.720000029 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.5%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\color{blue}{\left(u1 \cdot \left(1 + u1\right)\right)}\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]
   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 88.7%

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

   5. Simplified 88.7%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.7200000286102295:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot \left(-41.341702240407926 + \left(u2 \cdot u2\right) \cdot \left(81.6052492761019 + \left(u2 \cdot u2\right) \cdot -76.70585975309672\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
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 98.4%

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

4. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\frac{\sin \left(6.28318530718 \cdot u2\right)}{{\left(\frac{1}{u1} + -1\right)}^{0.5}}} \]
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. metadata-eval N/A

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

   3. sub-neg N/A

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

   4. *-inverses N/A

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

   5. div-sub N/A

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

   6. flip3-- N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right), \left(\sqrt{\frac{\frac{{1}^{3} - {u1}^{3}}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}}{u1}}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right), \left(\sqrt{\frac{\frac{{1}^{3} - {u1}^{3}}{1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}}{u1}}\right)\right) \]

   8. distribute-rgt-in N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right), \left(\sqrt{\frac{\frac{{1}^{3} - {u1}^{3}}{1 + u1 \cdot \left(u1 + 1\right)}}{u1}}\right)\right) \]

   9. associate-/l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right), \left(\sqrt{\frac{{1}^{3} - {u1}^{3}}{u1 \cdot \left(1 + u1 \cdot \left(u1 + 1\right)\right)}}\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right), \left(\sqrt{\frac{1 - {u1}^{3}}{u1 \cdot \left(1 + u1 \cdot \left(u1 + 1\right)\right)}}\right)\right) \]

   11. cube-unmult N/A

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

   12. sub-neg N/A

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

   13. metadata-eval N/A

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

   14. distribute-neg-in N/A

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

   15. associate-/l/ N/A

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

6. Applied egg-rr 98.4%

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

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Final simplification 98.4%

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

Alternative 5: 93.6% accurate, 1.6× speedup

Math

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

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(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 98.4%

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

3. Taylor expanded in u2 around 0

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

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

   11. unpow2 N/A

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

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

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

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

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

   14. *-commutative N/A

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

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

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

   16. unpow2 N/A

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

   17. *-lowering-*.f32 94.2%

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

5. Simplified 94.2%

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

Alternative 6: 91.3% 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 98.4%

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

3. Taylor expanded in u2 around 0

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

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

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

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

   14. *-lowering-*.f32 91.8%

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

5. Simplified 91.8%

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

Alternative 7: 84.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.013000000268220901:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot {\left(\frac{1}{u1} + -1\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot -41.341702240407926\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.013000000268220901)
   (* 6.28318530718 (* u2 (pow (+ (/ 1.0 u1) -1.0) -0.5)))
   (* (sqrt u1) (* u2 (+ 6.28318530718 (* (* u2 u2) -41.341702240407926))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.013000000268220901f) {
		tmp = 6.28318530718f * (u2 * powf(((1.0f / u1) + -1.0f), -0.5f));
	} else {
		tmp = sqrtf(u1) * (u2 * (6.28318530718f + ((u2 * u2) * -41.341702240407926f)));
	}
	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) <= 0.013000000268220901e0) then
        tmp = 6.28318530718e0 * (u2 * (((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0)))
    else
        tmp = sqrt(u1) * (u2 * (6.28318530718e0 + ((u2 * u2) * (-41.341702240407926e0))))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.013000000268220901))
		tmp = Float32(Float32(6.28318530718) * Float32(u2 * (Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)) ^ Float32(-0.5))));
	else
		tmp = Float32(sqrt(u1) * Float32(u2 * Float32(Float32(6.28318530718) + Float32(Float32(u2 * u2) * Float32(-41.341702240407926)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.013000000268220901))
		tmp = single(6.28318530718) * (u2 * (((single(1.0) / u1) + single(-1.0)) ^ single(-0.5)));
	else
		tmp = sqrt(u1) * (u2 * (single(6.28318530718) + ((u2 * u2) * single(-41.341702240407926))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.0130000003

   1. Initial program 98.4%

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

   3. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\frac{314159265359}{50000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)} \]
   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 96.6%

      \[\leadsto \color{blue}{\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]
   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. *-lowering-*.f32 N/A

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

   7. Applied egg-rr 96.7%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      7. /-lowering-/.f32 96.7%

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

   9. Applied egg-rr 96.7%

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

   if 0.0130000003 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   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 98.1%

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

   4. Applied egg-rr 98.1%

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

   6. Applied egg-rr 98.1%

      \[\leadsto \frac{\sin \left(6.28318530718 \cdot u2\right)}{\color{blue}{{\left(\left(\frac{1}{u1} + -1\right) \cdot \left(\frac{1}{u1} + -1\right)\right)}^{0.25}}} \]

   7. Taylor expanded in u2 around 0

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

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

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

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

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

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

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \left({u2}^{2}\right)\right)\right)\right), \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), \color{blue}{-1}\right)\right), \frac{1}{4}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \left(u2 \cdot u2\right)\right)\right)\right), \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right)\right), \frac{1}{4}\right)\right) \]

      5. *-lowering-*.f32 66.2%

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \mathsf{*.f32}\left(u2, u2\right)\right)\right)\right), \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(1, u1\right), -1\right)\right), \frac{1}{4}\right)\right) \]

   9. Simplified 66.2%

      \[\leadsto \frac{\color{blue}{u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)}}{{\left(\left(\frac{1}{u1} + -1\right) \cdot \left(\frac{1}{u1} + -1\right)\right)}^{0.25}} \]

   10. Taylor expanded in u1 around 0

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

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

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

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

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

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

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

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

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

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

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

       6. unpow2 N/A

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

       7. *-lowering-*.f32 57.9%

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

   12. Simplified 57.9%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.013000000268220901:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot {\left(\frac{1}{u1} + -1\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot -41.341702240407926\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.8% 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 98.4%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]
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 89.2%

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

6. Final simplification 89.2%

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

Alternative 9: 81.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\frac{314159265359}{50000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)} \]
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 81.5%

   \[\leadsto \color{blue}{\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]
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. *-lowering-*.f32 N/A

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

7. Applied egg-rr 81.6%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

   7. /-lowering-/.f32 81.6%

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

9. Applied egg-rr 81.6%

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

10. Final simplification 81.6%

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

Alternative 10: 81.1% 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 98.4%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
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 98.4%

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

4. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\frac{\sin \left(6.28318530718 \cdot u2\right)}{{\left(\frac{1}{u1} + -1\right)}^{0.5}}} \]
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. metadata-eval N/A

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

   3. sub-neg N/A

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

   4. *-inverses N/A

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

   5. div-sub N/A

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

   6. flip3-- N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right), \left(\sqrt{\frac{\frac{{1}^{3} - {u1}^{3}}{1 \cdot 1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}}{u1}}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right), \left(\sqrt{\frac{\frac{{1}^{3} - {u1}^{3}}{1 + \left(u1 \cdot u1 + 1 \cdot u1\right)}}{u1}}\right)\right) \]

   8. distribute-rgt-in N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right), \left(\sqrt{\frac{\frac{{1}^{3} - {u1}^{3}}{1 + u1 \cdot \left(u1 + 1\right)}}{u1}}\right)\right) \]

   9. associate-/l/ N/A

      \[\leadsto \mathsf{/.f32}\left(\mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right), \left(\sqrt{\frac{{1}^{3} - {u1}^{3}}{u1 \cdot \left(1 + u1 \cdot \left(u1 + 1\right)\right)}}\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(\mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right), \left(\sqrt{\frac{1 - {u1}^{3}}{u1 \cdot \left(1 + u1 \cdot \left(u1 + 1\right)\right)}}\right)\right) \]

   11. cube-unmult N/A

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

   12. sub-neg N/A

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

   13. metadata-eval N/A

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

   14. distribute-neg-in N/A

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

   15. associate-/l/ N/A

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

6. Applied egg-rr 98.4%

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

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 81.6%

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

9. Simplified 81.6%

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

Alternative 11: 81.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\frac{314159265359}{50000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)} \]
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 81.5%

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

Alternative 12: 72.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\frac{314159265359}{50000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)} \]
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 81.5%

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

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 72.1%

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

8. Simplified 72.1%

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

Alternative 13: 64.4% 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 98.4%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\frac{314159265359}{50000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)} \]
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 81.5%

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

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

Alternative 14: 21.1% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 6.28318530718 + \left(u2 \cdot u2\right) \cdot \left(-41.341702240407926 + u2 \cdot \left(u2 \cdot \left(81.6052492761019 + \left(u2 \cdot u2\right) \cdot -76.70585975309672\right)\right)\right)\\ u1 \cdot \left(u2 \cdot t\_0 + \frac{t\_0 \cdot \left(u2 \cdot 0.5\right)}{u1}\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0
         (+
          6.28318530718
          (*
           (* u2 u2)
           (+
            -41.341702240407926
            (*
             u2
             (* u2 (+ 81.6052492761019 (* (* u2 u2) -76.70585975309672)))))))))
   (* u1 (+ (* u2 t_0) (/ (* t_0 (* u2 0.5)) u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = 6.28318530718f + ((u2 * u2) * (-41.341702240407926f + (u2 * (u2 * (81.6052492761019f + ((u2 * u2) * -76.70585975309672f))))));
	return u1 * ((u2 * t_0) + ((t_0 * (u2 * 0.5f)) / 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
    real(4) :: t_0
    t_0 = 6.28318530718e0 + ((u2 * u2) * ((-41.341702240407926e0) + (u2 * (u2 * (81.6052492761019e0 + ((u2 * u2) * (-76.70585975309672e0)))))))
    code = u1 * ((u2 * t_0) + ((t_0 * (u2 * 0.5e0)) / u1))
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = 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))))))))
	return Float32(u1 * Float32(Float32(u2 * t_0) + Float32(Float32(t_0 * Float32(u2 * Float32(0.5))) / u1)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	t_0 = single(6.28318530718) + ((u2 * u2) * (single(-41.341702240407926) + (u2 * (u2 * (single(81.6052492761019) + ((u2 * u2) * single(-76.70585975309672)))))));
	tmp = u1 * ((u2 * t_0) + ((t_0 * (u2 * single(0.5))) / u1));
end

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in 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 86.4%

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

5. Simplified 86.4%

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

6. Taylor expanded in u2 around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

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

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

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

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

   11. unpow2 N/A

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

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

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

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

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

   14. *-commutative N/A

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

   15. unpow2 N/A

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

   16. associate-*l* N/A

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

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

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

   18. *-lowering-*.f32 83.0%

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

8. Simplified 83.0%

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

9. Taylor expanded in u1 around inf

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

11. Simplified 21.2%

    \[\leadsto \color{blue}{u1 \cdot \left(u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot \left(-41.341702240407926 + u2 \cdot \left(u2 \cdot \left(81.6052492761019 + \left(u2 \cdot u2\right) \cdot -76.70585975309672\right)\right)\right)\right) + \frac{\left(6.28318530718 + \left(u2 \cdot u2\right) \cdot \left(-41.341702240407926 + u2 \cdot \left(u2 \cdot \left(81.6052492761019 + \left(u2 \cdot u2\right) \cdot -76.70585975309672\right)\right)\right)\right) \cdot \left(u2 \cdot 0.5\right)}{u1}\right)} \]
12. Add Preprocessing

Alternative 15: 21.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \left(u1 \cdot \left(-1 + \frac{-0.5}{u1}\right)\right) \cdot \left(u2 \cdot \left(\left(u2 \cdot u2\right) \cdot \left(\left(81.6052492761019 + u2 \cdot \left(u2 \cdot -76.70585975309672\right)\right) \cdot \left(0 - u2 \cdot u2\right) - -41.341702240407926\right) - 6.28318530718\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (* u1 (+ -1.0 (/ -0.5 u1)))
  (*
   u2
   (-
    (*
     (* u2 u2)
     (-
      (*
       (+ 81.6052492761019 (* u2 (* u2 -76.70585975309672)))
       (- 0.0 (* u2 u2)))
      -41.341702240407926))
    6.28318530718))))

C

float code(float cosTheta_i, float u1, float u2) {
	return (u1 * (-1.0f + (-0.5f / u1))) * (u2 * (((u2 * u2) * (((81.6052492761019f + (u2 * (u2 * -76.70585975309672f))) * (0.0f - (u2 * u2))) - -41.341702240407926f)) - 6.28318530718f));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = (u1 * ((-1.0e0) + ((-0.5e0) / u1))) * (u2 * (((u2 * u2) * (((81.6052492761019e0 + (u2 * (u2 * (-76.70585975309672e0)))) * (0.0e0 - (u2 * u2))) - (-41.341702240407926e0))) - 6.28318530718e0))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(u1 * Float32(Float32(-1.0) + Float32(Float32(-0.5) / u1))) * Float32(u2 * Float32(Float32(Float32(u2 * u2) * Float32(Float32(Float32(Float32(81.6052492761019) + Float32(u2 * Float32(u2 * Float32(-76.70585975309672)))) * Float32(Float32(0.0) - Float32(u2 * u2))) - Float32(-41.341702240407926))) - Float32(6.28318530718))))
end

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in 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 86.4%

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

5. Simplified 86.4%

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

6. Taylor expanded in u2 around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

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

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

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

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

   11. unpow2 N/A

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

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

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

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

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

   14. *-commutative N/A

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

   15. unpow2 N/A

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

   16. associate-*l* N/A

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

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

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

   18. *-lowering-*.f32 83.0%

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

8. Simplified 83.0%

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

9. Taylor expanded in u1 around -inf

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

    1. mul-1-neg N/A

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

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

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

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

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

    4. sub-neg N/A

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

    5. unpow2 N/A

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

    6. rem-square-sqrt N/A

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

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

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

    8. associate-*r/ N/A

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

    9. metadata-eval N/A

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

    10. distribute-neg-frac N/A

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

    11. metadata-eval N/A

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

    12. /-lowering-/.f32 21.2%

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

11. Simplified 21.2%

    \[\leadsto \color{blue}{\left(-u1 \cdot \left(-1 + \frac{-0.5}{u1}\right)\right)} \cdot \left(u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot \left(-41.341702240407926 + \left(u2 \cdot u2\right) \cdot \left(81.6052492761019 + u2 \cdot \left(u2 \cdot -76.70585975309672\right)\right)\right)\right)\right) \]

12. Final simplification 21.2%

    \[\leadsto \left(u1 \cdot \left(-1 + \frac{-0.5}{u1}\right)\right) \cdot \left(u2 \cdot \left(\left(u2 \cdot u2\right) \cdot \left(\left(81.6052492761019 + u2 \cdot \left(u2 \cdot -76.70585975309672\right)\right) \cdot \left(0 - u2 \cdot u2\right) - -41.341702240407926\right) - 6.28318530718\right)\right) \]
13. Add Preprocessing

Alternative 16: 21.1% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \left(u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot \left(-41.341702240407926 + \left(u2 \cdot u2\right) \cdot \left(81.6052492761019 + u2 \cdot \left(u2 \cdot -76.70585975309672\right)\right)\right)\right)\right) \cdot \left(u1 + 0.5\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (*
   u2
   (+
    6.28318530718
    (*
     (* u2 u2)
     (+
      -41.341702240407926
      (* (* u2 u2) (+ 81.6052492761019 (* u2 (* u2 -76.70585975309672))))))))
  (+ u1 0.5)))

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)))))))) * (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 + ((u2 * u2) * ((-41.341702240407926e0) + ((u2 * u2) * (81.6052492761019e0 + (u2 * (u2 * (-76.70585975309672e0))))))))) * (u1 + 0.5e0)
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))))))))) * Float32(u1 + Float32(0.5)))
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))))))))) * (u1 + single(0.5));
end

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in 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 86.4%

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

5. Simplified 86.4%

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

6. Taylor expanded in u2 around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

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

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

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

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

   11. unpow2 N/A

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

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

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

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

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

   14. *-commutative N/A

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

   15. unpow2 N/A

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

   16. associate-*l* N/A

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

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

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

   18. *-lowering-*.f32 83.0%

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

8. Simplified 83.0%

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

9. Taylor expanded in u1 around inf

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

    1. distribute-rgt-in N/A

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

    2. *-lft-identity N/A

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

    3. associate-*l* N/A

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

    4. lft-mult-inverse N/A

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

    5. metadata-eval N/A

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

    6. +-lowering-+.f32 21.2%

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

11. Simplified 21.2%

    \[\leadsto \color{blue}{\left(u1 + 0.5\right)} \cdot \left(u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot \left(-41.341702240407926 + \left(u2 \cdot u2\right) \cdot \left(81.6052492761019 + u2 \cdot \left(u2 \cdot -76.70585975309672\right)\right)\right)\right)\right) \]

12. Final simplification 21.2%

    \[\leadsto \left(u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot \left(-41.341702240407926 + \left(u2 \cdot u2\right) \cdot \left(81.6052492761019 + u2 \cdot \left(u2 \cdot -76.70585975309672\right)\right)\right)\right)\right) \cdot \left(u1 + 0.5\right) \]
13. Add Preprocessing

Alternative 17: 20.1% accurate, 10.0× speedup

Math

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

FPCore

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

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)))))));
}

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))))))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return 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 = u2 * (single(6.28318530718) + ((u2 * u2) * (single(-41.341702240407926) + (u2 * (u2 * (single(81.6052492761019) + ((u2 * u2) * single(-76.70585975309672))))))));
end

Derivation

1. Initial program 98.4%

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

4. Applied egg-rr 85.4%

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

5. Taylor expanded in u1 around inf

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

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

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

   2. *-lowering-*.f32 20.2%

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

7. Simplified 20.2%

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

8. Taylor expanded in u2 around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

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

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

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

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

   17. unpow2 N/A

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

   18. *-lowering-*.f32 20.2%

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

10. Simplified 20.2%

    \[\leadsto \color{blue}{u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot \left(-41.341702240407926 + u2 \cdot \left(u2 \cdot \left(81.6052492761019 + \left(u2 \cdot u2\right) \cdot -76.70585975309672\right)\right)\right)\right)} \]
11. Add Preprocessing

Alternative 18: 19.8% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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 85.4%

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

5. Taylor expanded in u1 around inf

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

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

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

   2. *-lowering-*.f32 20.2%

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

7. Simplified 20.2%

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

8. Taylor expanded in u2 around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

   5. *-lowering-*.f32 20.0%

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

10. Simplified 20.0%

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

11. Final simplification 20.0%

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

Alternative 19: 19.5% accurate, 69.7× speedup

Math

\[\begin{array}{l} \\ 6.28318530718 \cdot u2 \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

4. Applied egg-rr 85.4%

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

5. Taylor expanded in u1 around inf

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

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

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

   2. *-lowering-*.f32 20.2%

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

7. Simplified 20.2%

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

8. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\frac{314159265359}{50000000000} \cdot u2} \]
9. Step-by-step derivation

   1. *-lowering-*.f32 19.7%

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

10. Simplified 19.7%

    \[\leadsto \color{blue}{6.28318530718 \cdot u2} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024288 
(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))))