Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 13.6s

Alternatives: 20

Speedup: 1.0×

Specification

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

Alternative 2: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.800000011920929:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot \left(-41.341702240407926 + u2 \cdot \left(u2 \cdot \left(81.6052492761019 + \left(u2 \cdot u2\right) \cdot -76.70585975309672\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.800000011920929)
   (*
    (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 (+ u1 1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.800000011920929f) {
		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 * (u1 + 1.0f)));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((6.28318530718e0 * u2) <= 0.800000011920929e0) 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 * (u1 + 1.0e0)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.800000011920929))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(u2 * Float32(Float32(6.28318530718) + Float32(Float32(u2 * u2) * Float32(Float32(-41.341702240407926) + Float32(u2 * Float32(u2 * Float32(Float32(81.6052492761019) + Float32(Float32(u2 * u2) * Float32(-76.70585975309672))))))))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(Float32(u1 * Float32(u1 + Float32(1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.800000011920929))
		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 * (u1 + single(1.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.800000012

   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. unpow2 N/A

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

      11. associate-*l* N/A

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

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

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

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

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

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

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

      15. *-commutative N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f32 98.5%

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

   5. Simplified 98.5%

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

   if 0.800000012 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 96.8%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   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 86.2%

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

   5. Simplified 86.2%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.800000011920929:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot \left(-41.341702240407926 + u2 \cdot \left(u2 \cdot \left(81.6052492761019 + \left(u2 \cdot u2\right) \cdot -76.70585975309672\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in u2 around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

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

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

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

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

   17. unpow2 N/A

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

   18. *-lowering-*.f32 93.7%

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

5. Simplified 93.7%

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

Alternative 4: 84.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.009999999776482582:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot {\left(\frac{u1}{1 - u1}\right)}^{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{u1}\right)}^{-0.5} \cdot \left(u2 \cdot \left(6.28318530718 + u2 \cdot \left(u2 \cdot -41.341702240407926\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.009999999776482582)
   (* 6.28318530718 (* u2 (pow (/ u1 (- 1.0 u1)) 0.5)))
   (*
    (pow (/ 1.0 u1) -0.5)
    (* u2 (+ 6.28318530718 (* u2 (* u2 -41.341702240407926)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.009999999776482582f) {
		tmp = 6.28318530718f * (u2 * powf((u1 / (1.0f - u1)), 0.5f));
	} else {
		tmp = powf((1.0f / u1), -0.5f) * (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) <= 0.009999999776482582e0) then
        tmp = 6.28318530718e0 * (u2 * ((u1 / (1.0e0 - u1)) ** 0.5e0))
    else
        tmp = ((1.0e0 / u1) ** (-0.5e0)) * (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(0.009999999776482582))
		tmp = Float32(Float32(6.28318530718) * Float32(u2 * (Float32(u1 / Float32(Float32(1.0) - u1)) ^ Float32(0.5))));
	else
		tmp = Float32((Float32(Float32(1.0) / u1) ^ Float32(-0.5)) * Float32(u2 * Float32(Float32(6.28318530718) + Float32(u2 * Float32(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(0.009999999776482582))
		tmp = single(6.28318530718) * (u2 * ((u1 / (single(1.0) - u1)) ^ single(0.5)));
	else
		tmp = ((single(1.0) / u1) ^ single(-0.5)) * (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) < 0.00999999978

   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. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. pow1/2 N/A

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

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

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

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

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

      8. --lowering--.f32 96.7%

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

   7. Applied egg-rr 96.7%

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

   if 0.00999999978 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.9%

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

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

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

      2. pow1/2 N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

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

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

      7. div-sub N/A

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

      8. sub-neg N/A

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

      9. *-inverses N/A

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

      10. metadata-eval N/A

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

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

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

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

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

      13. metadata-eval N/A

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

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

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

      15. *-lowering-*.f32 97.8%

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

   4. Applied egg-rr 97.8%

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

   5. Taylor expanded in u2 around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f32 68.9%

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

   7. Simplified 68.9%

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

   8. Taylor expanded in u1 around 0

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

      1. /-lowering-/.f32 56.7%

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

   10. Simplified 56.7%

       \[\leadsto {\color{blue}{\left(\frac{1}{u1}\right)}}^{-0.5} \cdot \left(u2 \cdot \left(6.28318530718 + u2 \cdot \left(u2 \cdot -41.341702240407926\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.009999999776482582:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot {\left(\frac{u1}{1 - u1}\right)}^{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{u1}\right)}^{-0.5} \cdot \left(u2 \cdot \left(6.28318530718 + u2 \cdot \left(u2 \cdot -41.341702240407926\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.9% 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 98.2%

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

3. Taylor expanded in u2 around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

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

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

   10. unpow2 N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

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

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

   14. *-commutative N/A

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

   15. *-lowering-*.f32 91.8%

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

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 6: 84.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.009999999776482582:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot {\left(\frac{u1}{1 - u1}\right)}^{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot -41.341702240407926\right)\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.009999999776482582)
   (* 6.28318530718 (* u2 (pow (/ u1 (- 1.0 u1)) 0.5)))
   (* (* u2 (+ 6.28318530718 (* (* u2 u2) -41.341702240407926))) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.009999999776482582f) {
		tmp = 6.28318530718f * (u2 * powf((u1 / (1.0f - u1)), 0.5f));
	} else {
		tmp = (u2 * (6.28318530718f + ((u2 * u2) * -41.341702240407926f))) * sqrtf(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) <= 0.009999999776482582e0) then
        tmp = 6.28318530718e0 * (u2 * ((u1 / (1.0e0 - u1)) ** 0.5e0))
    else
        tmp = (u2 * (6.28318530718e0 + ((u2 * u2) * (-41.341702240407926e0)))) * sqrt(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(0.009999999776482582))
		tmp = Float32(Float32(6.28318530718) * Float32(u2 * (Float32(u1 / Float32(Float32(1.0) - u1)) ^ Float32(0.5))));
	else
		tmp = Float32(Float32(u2 * Float32(Float32(6.28318530718) + Float32(Float32(u2 * u2) * Float32(-41.341702240407926)))) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.009999999776482582))
		tmp = single(6.28318530718) * (u2 * ((u1 / (single(1.0) - u1)) ^ single(0.5)));
	else
		tmp = (u2 * (single(6.28318530718) + ((u2 * u2) * single(-41.341702240407926)))) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00999999978

   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. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. pow1/2 N/A

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

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

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

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

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

      8. --lowering--.f32 96.7%

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

   7. Applied egg-rr 96.7%

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

   if 0.00999999978 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.9%

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

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

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

      2. pow1/2 N/A

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

      3. clear-num N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

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

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

      7. div-sub N/A

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

      8. sub-neg N/A

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

      9. *-inverses N/A

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

      10. metadata-eval N/A

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

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

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

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

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

      13. metadata-eval N/A

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

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

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

      15. *-lowering-*.f32 97.8%

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

   4. Applied egg-rr 97.8%

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

   5. Taylor expanded in u2 around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f32 68.9%

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

   7. Simplified 68.9%

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

   8. Taylor expanded in u1 around 0

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

      1. *-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 56.7%

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

   10. Simplified 56.7%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.009999999776482582:\\ \;\;\;\;6.28318530718 \cdot \left(u2 \cdot {\left(\frac{u1}{1 - u1}\right)}^{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot -41.341702240407926\right)\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in u2 around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

   5. *-lowering-*.f32 89.2%

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

5. Simplified 89.2%

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

6. Final simplification 89.2%

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

Alternative 8: 89.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{u2 \cdot \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot {u2}^{2}\right) + \frac{314159265359}{50000000000} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]
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 89.1%

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

6. Final simplification 89.1%

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

Alternative 9: 81.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

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

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

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

   5. *-rgt-identity N/A

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

   6. sub-neg N/A

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

   7. rgt-mult-inverse N/A

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

   8. mul-1-neg N/A

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

   9. distribute-neg-frac2 N/A

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

   10. mul-1-neg N/A

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

   11. *-rgt-identity N/A

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

   12. distribute-lft-in N/A

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

   13. +-commutative N/A

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

   14. sub-neg N/A

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

   15. associate-*r* N/A

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

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

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

   17. *-rgt-identity N/A

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

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

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

   19. associate-*r* N/A

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

   20. sub-neg N/A

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

   21. +-commutative N/A

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

5. Simplified 80.5%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

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

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

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

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

   5. pow1/2 N/A

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

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

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

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

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

   8. --lowering--.f32 80.5%

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

7. Applied egg-rr 80.5%

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

8. Final simplification 80.5%

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

Alternative 10: 81.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

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

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

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

   5. *-rgt-identity N/A

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

   6. sub-neg N/A

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

   7. rgt-mult-inverse N/A

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

   8. mul-1-neg N/A

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

   9. distribute-neg-frac2 N/A

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

   10. mul-1-neg N/A

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

   11. *-rgt-identity N/A

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

   12. distribute-lft-in N/A

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

   13. +-commutative N/A

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

   14. sub-neg N/A

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

   15. associate-*r* N/A

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

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

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

   17. *-rgt-identity N/A

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

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

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

   19. associate-*r* N/A

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

   20. sub-neg N/A

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

   21. +-commutative N/A

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

5. Simplified 80.5%

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

6. Final simplification 80.5%

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

Alternative 11: 73.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

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

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

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

   5. *-rgt-identity N/A

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

   6. sub-neg N/A

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

   7. rgt-mult-inverse N/A

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

   8. mul-1-neg N/A

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

   9. distribute-neg-frac2 N/A

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

   10. mul-1-neg N/A

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

   11. *-rgt-identity N/A

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

   12. distribute-lft-in N/A

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

   13. +-commutative N/A

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

   14. sub-neg N/A

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

   15. associate-*r* N/A

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

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

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

   17. *-rgt-identity N/A

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

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

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

   19. associate-*r* N/A

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

   20. sub-neg N/A

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

   21. +-commutative N/A

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

5. Simplified 80.5%

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

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. +-commutative N/A

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

   3. +-lowering-+.f32 71.5%

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

8. Simplified 71.5%

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

Alternative 12: 64.9% 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 98.2%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

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

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

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

   5. *-rgt-identity N/A

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

   6. sub-neg N/A

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

   7. rgt-mult-inverse N/A

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

   8. mul-1-neg N/A

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

   9. distribute-neg-frac2 N/A

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

   10. mul-1-neg N/A

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

   11. *-rgt-identity N/A

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

   12. distribute-lft-in N/A

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

   13. +-commutative N/A

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

   14. sub-neg N/A

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

   15. associate-*r* N/A

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

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

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

   17. *-rgt-identity N/A

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

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

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

   19. associate-*r* N/A

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

   20. sub-neg N/A

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

   21. +-commutative N/A

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

5. Simplified 80.5%

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

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. *-lowering-*.f32 N/A

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

   3. sqrt-lowering-sqrt.f32 63.7%

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

8. Simplified 63.7%

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

9. Final simplification 63.7%

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

Alternative 13: 21.0% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{0.5}{u1}\\ u2 \cdot \left(t\_0 \cdot \left(u1 \cdot 6.28318530718\right) + \left(u2 \cdot u2\right) \cdot \left(t\_0 \cdot \left(u1 \cdot -41.341702240407926\right) + \left(u2 \cdot u2\right) \cdot \left(\left(\left(u2 \cdot u2\right) \cdot t\_0\right) \cdot \left(u1 \cdot -76.70585975309672\right) + t\_0 \cdot \left(u1 \cdot 81.6052492761019\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (+ 1.0 (/ 0.5 u1))))
   (*
    u2
    (+
     (* t_0 (* u1 6.28318530718))
     (*
      (* u2 u2)
      (+
       (* t_0 (* u1 -41.341702240407926))
       (*
        (* u2 u2)
        (+
         (* (* (* u2 u2) t_0) (* u1 -76.70585975309672))
         (* t_0 (* u1 81.6052492761019))))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = 1.0f + (0.5f / u1);
	return u2 * ((t_0 * (u1 * 6.28318530718f)) + ((u2 * u2) * ((t_0 * (u1 * -41.341702240407926f)) + ((u2 * u2) * ((((u2 * u2) * t_0) * (u1 * -76.70585975309672f)) + (t_0 * (u1 * 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
    real(4) :: t_0
    t_0 = 1.0e0 + (0.5e0 / u1)
    code = u2 * ((t_0 * (u1 * 6.28318530718e0)) + ((u2 * u2) * ((t_0 * (u1 * (-41.341702240407926e0))) + ((u2 * u2) * ((((u2 * u2) * t_0) * (u1 * (-76.70585975309672e0))) + (t_0 * (u1 * 81.6052492761019e0)))))))
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(1.0) + Float32(Float32(0.5) / u1))
	return Float32(u2 * Float32(Float32(t_0 * Float32(u1 * Float32(6.28318530718))) + Float32(Float32(u2 * u2) * Float32(Float32(t_0 * Float32(u1 * Float32(-41.341702240407926))) + Float32(Float32(u2 * u2) * Float32(Float32(Float32(Float32(u2 * u2) * t_0) * Float32(u1 * Float32(-76.70585975309672))) + Float32(t_0 * Float32(u1 * Float32(81.6052492761019)))))))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	t_0 = single(1.0) + (single(0.5) / u1);
	tmp = u2 * ((t_0 * (u1 * single(6.28318530718))) + ((u2 * u2) * ((t_0 * (u1 * single(-41.341702240407926))) + ((u2 * u2) * ((((u2 * u2) * t_0) * (u1 * single(-76.70585975309672))) + (t_0 * (u1 * single(81.6052492761019))))))));
end

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in 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 85.2%

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

5. Simplified 85.2%

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

6. Taylor expanded in u1 around -inf

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

   1. mul-1-neg N/A

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

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

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

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

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

   4. neg-sub0 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \left(0 - \left({\left(\sqrt{-1}\right)}^{2} - \frac{1}{2} \cdot \frac{1}{u1}\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \color{blue}{u2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \left({\left(\sqrt{-1}\right)}^{2} - \frac{1}{2} \cdot \frac{1}{u1}\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \color{blue}{u2}\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \left({\left(\sqrt{-1}\right)}^{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \left(\sqrt{-1} \cdot \sqrt{-1} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   8. rem-square-sqrt N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \left(-1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   10. associate-*r/ N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   12. distribute-neg-frac N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{u1}\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \left(\frac{\frac{-1}{2}}{u1}\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   14. /-lowering-/.f32 21.2%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \mathsf{/.f32}\left(\frac{-1}{2}, u1\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

8. Simplified 21.2%

   \[\leadsto \color{blue}{\left(u1 \cdot \left(0 - \left(-1 + \frac{-0.5}{u1}\right)\right)\right)} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

9. Taylor expanded in u2 around 0

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

11. Simplified 21.0%

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

Alternative 14: 21.0% accurate, 6.7× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return (u1 * (-1.0f + (-0.5f / u1))) * (u2 * ((u2 * (u2 * (((u2 * u2) * ((0.0f - ((u2 * u2) * -76.70585975309672f)) - 81.6052492761019f)) - -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 * (((u2 * u2) * ((0.0e0 - ((u2 * u2) * (-76.70585975309672e0))) - 81.6052492761019e0)) - (-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(u2 * Float32(u2 * Float32(Float32(Float32(u2 * u2) * Float32(Float32(Float32(0.0) - Float32(Float32(u2 * u2) * Float32(-76.70585975309672))) - Float32(81.6052492761019))) - 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 * (((u2 * u2) * ((single(0.0) - ((u2 * u2) * single(-76.70585975309672))) - single(81.6052492761019))) - single(-41.341702240407926)))) - single(6.28318530718)));
end

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in 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 85.2%

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

5. Simplified 85.2%

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

6. Taylor expanded in u1 around -inf

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

   1. mul-1-neg N/A

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

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

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

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

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

   4. neg-sub0 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \left(0 - \left({\left(\sqrt{-1}\right)}^{2} - \frac{1}{2} \cdot \frac{1}{u1}\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \color{blue}{u2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \left({\left(\sqrt{-1}\right)}^{2} - \frac{1}{2} \cdot \frac{1}{u1}\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \color{blue}{u2}\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \left({\left(\sqrt{-1}\right)}^{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \left(\sqrt{-1} \cdot \sqrt{-1} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   8. rem-square-sqrt N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \left(-1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   10. associate-*r/ N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   12. distribute-neg-frac N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{u1}\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \left(\frac{\frac{-1}{2}}{u1}\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   14. /-lowering-/.f32 21.2%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \mathsf{/.f32}\left(\frac{-1}{2}, u1\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

8. Simplified 21.2%

   \[\leadsto \color{blue}{\left(u1 \cdot \left(0 - \left(-1 + \frac{-0.5}{u1}\right)\right)\right)} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

9. Taylor expanded in u2 around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \mathsf{/.f32}\left(\frac{-1}{2}, u1\right)\right)\right)\right), \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)}\right) \]
10. Step-by-step derivation

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \mathsf{/.f32}\left(\frac{-1}{2}, u1\right)\right)\right)\right), \mathsf{*.f32}\left(u2, \color{blue}{\left(\frac{314159265359}{50000000000} + {u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \mathsf{/.f32}\left(\frac{-1}{2}, u1\right)\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \color{blue}{\left({u2}^{2} \cdot \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)}\right)\right)\right) \]

    3. unpow2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \mathsf{/.f32}\left(\frac{-1}{2}, u1\right)\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \left(\left(u2 \cdot u2\right) \cdot \left(\color{blue}{{u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)} - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)\right)\right) \]

    4. associate-*l* N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \mathsf{/.f32}\left(\frac{-1}{2}, u1\right)\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \left(u2 \cdot \color{blue}{\left(u2 \cdot \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \mathsf{/.f32}\left(\frac{-1}{2}, u1\right)\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(u2, \color{blue}{\left(u2 \cdot \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \mathsf{/.f32}\left(\frac{-1}{2}, u1\right)\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \color{blue}{\left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) - \frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)}\right)\right)\right)\right)\right) \]

    7. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \mathsf{/.f32}\left(\frac{-1}{2}, u1\right)\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)}\right)\right)\right)\right)\right)\right) \]

    8. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \mathsf{/.f32}\left(\frac{-1}{2}, u1\right)\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right) + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}\right)\right)\right)\right)\right)\right) \]

    9. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \mathsf{/.f32}\left(\frac{-1}{2}, u1\right)\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} + \color{blue}{{u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \mathsf{/.f32}\left(\frac{-1}{2}, u1\right)\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \color{blue}{\left({u2}^{2} \cdot \left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \mathsf{/.f32}\left(\frac{-1}{2}, u1\right)\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \mathsf{*.f32}\left(\left({u2}^{2}\right), \color{blue}{\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

    12. unpow2 N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \mathsf{/.f32}\left(\frac{-1}{2}, u1\right)\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \mathsf{*.f32}\left(\left(u2 \cdot u2\right), \left(\color{blue}{\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \mathsf{/.f32}\left(\frac{-1}{2}, u1\right)\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \left(\color{blue}{\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}} + \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \mathsf{/.f32}\left(\frac{-1}{2}, u1\right)\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \color{blue}{\left(\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000} \cdot {u2}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

    15. *-commutative N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \mathsf{/.f32}\left(\frac{-1}{2}, u1\right)\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \left({u2}^{2} \cdot \color{blue}{\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \mathsf{/.f32}\left(\frac{-1}{2}, u1\right)\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \mathsf{*.f32}\left(\left({u2}^{2}\right), \color{blue}{\frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

    17. unpow2 N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \mathsf{/.f32}\left(\frac{-1}{2}, u1\right)\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \mathsf{*.f32}\left(\left(u2 \cdot u2\right), \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

    18. *-lowering-*.f32 21.0%

        \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \mathsf{/.f32}\left(\frac{-1}{2}, u1\right)\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(u2, \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \mathsf{+.f32}\left(\frac{3060196847853821555298148281676017575122444629042460390799}{37500000000000000000000000000000000000000000000000000000}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(u2, u2\right), \frac{-302029322777818351566783844332719832329455959975176141755859165754785028165295919}{3937500000000000000000000000000000000000000000000000000000000000000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

11. Simplified 21.0%

    \[\leadsto \left(u1 \cdot \left(0 - \left(-1 + \frac{-0.5}{u1}\right)\right)\right) \cdot \color{blue}{\left(u2 \cdot \left(6.28318530718 + u2 \cdot \left(u2 \cdot \left(-41.341702240407926 + \left(u2 \cdot u2\right) \cdot \left(81.6052492761019 + \left(u2 \cdot u2\right) \cdot -76.70585975309672\right)\right)\right)\right)\right)} \]

12. Final simplification 21.0%

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

Alternative 15: 20.7% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{0.5}{u1}\\ u2 \cdot \left(t\_0 \cdot \left(u1 \cdot 6.28318530718\right) + t\_0 \cdot \left(-41.341702240407926 \cdot \left(u1 \cdot \left(u2 \cdot u2\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (+ 1.0 (/ 0.5 u1))))
   (*
    u2
    (+
     (* t_0 (* u1 6.28318530718))
     (* t_0 (* -41.341702240407926 (* u1 (* u2 u2))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = 1.0f + (0.5f / u1);
	return u2 * ((t_0 * (u1 * 6.28318530718f)) + (t_0 * (-41.341702240407926f * (u1 * (u2 * 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
    real(4) :: t_0
    t_0 = 1.0e0 + (0.5e0 / u1)
    code = u2 * ((t_0 * (u1 * 6.28318530718e0)) + (t_0 * ((-41.341702240407926e0) * (u1 * (u2 * u2)))))
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(1.0) + Float32(Float32(0.5) / u1))
	return Float32(u2 * Float32(Float32(t_0 * Float32(u1 * Float32(6.28318530718))) + Float32(t_0 * Float32(Float32(-41.341702240407926) * Float32(u1 * Float32(u2 * u2))))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	t_0 = single(1.0) + (single(0.5) / u1);
	tmp = u2 * ((t_0 * (u1 * single(6.28318530718))) + (t_0 * (single(-41.341702240407926) * (u1 * (u2 * u2)))));
end

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in 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 85.2%

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

5. Simplified 85.2%

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

6. Taylor expanded in u1 around -inf

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

   1. mul-1-neg N/A

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

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

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

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

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

   4. neg-sub0 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \left(0 - \left({\left(\sqrt{-1}\right)}^{2} - \frac{1}{2} \cdot \frac{1}{u1}\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \color{blue}{u2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \left({\left(\sqrt{-1}\right)}^{2} - \frac{1}{2} \cdot \frac{1}{u1}\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \color{blue}{u2}\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \left({\left(\sqrt{-1}\right)}^{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \left(\sqrt{-1} \cdot \sqrt{-1} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   8. rem-square-sqrt N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \left(-1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   10. associate-*r/ N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   12. distribute-neg-frac N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{u1}\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \left(\frac{\frac{-1}{2}}{u1}\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   14. /-lowering-/.f32 21.2%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \mathsf{/.f32}\left(\frac{-1}{2}, u1\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

8. Simplified 21.2%

   \[\leadsto \color{blue}{\left(u1 \cdot \left(0 - \left(-1 + \frac{-0.5}{u1}\right)\right)\right)} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

9. Taylor expanded in u2 around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. associate-*r* N/A

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

    4. associate-*l* N/A

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

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

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

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

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

11. Simplified 20.9%

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

12. Final simplification 20.9%

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

Alternative 16: 20.7% accurate, 11.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = (u1 * ((-1.0e0) + ((-0.5e0) / u1))) * (u2 * (-6.28318530718e0 - ((u2 * u2) * (-41.341702240407926e0))))
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(6.28318530718)) - Float32(Float32(u2 * u2) * Float32(-41.341702240407926)))))
end

MATLAB

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

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in 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 85.2%

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

5. Simplified 85.2%

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

6. Taylor expanded in u1 around -inf

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

   1. mul-1-neg N/A

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

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

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

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

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

   4. neg-sub0 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \left(0 - \left({\left(\sqrt{-1}\right)}^{2} - \frac{1}{2} \cdot \frac{1}{u1}\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \color{blue}{u2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \left({\left(\sqrt{-1}\right)}^{2} - \frac{1}{2} \cdot \frac{1}{u1}\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \color{blue}{u2}\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \left({\left(\sqrt{-1}\right)}^{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \left(\sqrt{-1} \cdot \sqrt{-1} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   8. rem-square-sqrt N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \left(-1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   10. associate-*r/ N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   12. distribute-neg-frac N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{u1}\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \left(\frac{\frac{-1}{2}}{u1}\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   14. /-lowering-/.f32 21.2%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \mathsf{/.f32}\left(\frac{-1}{2}, u1\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

8. Simplified 21.2%

   \[\leadsto \color{blue}{\left(u1 \cdot \left(0 - \left(-1 + \frac{-0.5}{u1}\right)\right)\right)} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

9. Taylor expanded in u2 around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \mathsf{/.f32}\left(\frac{-1}{2}, u1\right)\right)\right)\right), \color{blue}{\left(u2 \cdot \left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)\right)}\right) \]
10. Step-by-step derivation

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \mathsf{/.f32}\left(\frac{-1}{2}, u1\right)\right)\right)\right), \mathsf{*.f32}\left(u2, \color{blue}{\left(\frac{314159265359}{50000000000} + \frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \mathsf{/.f32}\left(\frac{-1}{2}, u1\right)\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \color{blue}{\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000} \cdot {u2}^{2}\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \mathsf{/.f32}\left(\frac{-1}{2}, u1\right)\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \color{blue}{\left({u2}^{2}\right)}\right)\right)\right)\right) \]

    4. unpow2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \mathsf{/.f32}\left(\frac{-1}{2}, u1\right)\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \left(u2 \cdot \color{blue}{u2}\right)\right)\right)\right)\right) \]

    5. *-lowering-*.f32 20.9%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \mathsf{/.f32}\left(\frac{-1}{2}, u1\right)\right)\right)\right), \mathsf{*.f32}\left(u2, \mathsf{+.f32}\left(\frac{314159265359}{50000000000}, \mathsf{*.f32}\left(\frac{-31006276680305942139213528068663279}{750000000000000000000000000000000}, \mathsf{*.f32}\left(u2, \color{blue}{u2}\right)\right)\right)\right)\right) \]

11. Simplified 20.9%

    \[\leadsto \left(u1 \cdot \left(0 - \left(-1 + \frac{-0.5}{u1}\right)\right)\right) \cdot \color{blue}{\left(u2 \cdot \left(6.28318530718 + -41.341702240407926 \cdot \left(u2 \cdot u2\right)\right)\right)} \]

12. Final simplification 20.9%

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

Alternative 17: 20.5% accurate, 17.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in 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 85.2%

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

5. Simplified 85.2%

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

6. Taylor expanded in u1 around -inf

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

   1. mul-1-neg N/A

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

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

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

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

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

   4. neg-sub0 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \left(0 - \left({\left(\sqrt{-1}\right)}^{2} - \frac{1}{2} \cdot \frac{1}{u1}\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \color{blue}{u2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \left({\left(\sqrt{-1}\right)}^{2} - \frac{1}{2} \cdot \frac{1}{u1}\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \color{blue}{u2}\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \left({\left(\sqrt{-1}\right)}^{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \left(\sqrt{-1} \cdot \sqrt{-1} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   8. rem-square-sqrt N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \left(-1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   10. associate-*r/ N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   12. distribute-neg-frac N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{u1}\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \left(\frac{\frac{-1}{2}}{u1}\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   14. /-lowering-/.f32 21.2%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \mathsf{/.f32}\left(\frac{-1}{2}, u1\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

8. Simplified 21.2%

   \[\leadsto \color{blue}{\left(u1 \cdot \left(0 - \left(-1 + \frac{-0.5}{u1}\right)\right)\right)} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

9. Taylor expanded in u2 around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \mathsf{/.f32}\left(\frac{-1}{2}, u1\right)\right)\right)\right), \color{blue}{\left(\frac{314159265359}{50000000000} \cdot u2\right)}\right) \]
10. Step-by-step derivation

    1. *-lowering-*.f32 20.5%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \mathsf{/.f32}\left(\frac{-1}{2}, u1\right)\right)\right)\right), \mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \color{blue}{u2}\right)\right) \]

11. Simplified 20.5%

    \[\leadsto \left(u1 \cdot \left(0 - \left(-1 + \frac{-0.5}{u1}\right)\right)\right) \cdot \color{blue}{\left(6.28318530718 \cdot u2\right)} \]

12. Final simplification 20.5%

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

Alternative 18: 20.5% accurate, 19.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return (1.0f + (0.5f / u1)) * (u2 * (u1 * 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 = (1.0e0 + (0.5e0 / u1)) * (u2 * (u1 * 6.28318530718e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

3. Taylor expanded in 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 85.2%

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

5. Simplified 85.2%

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

6. Taylor expanded in u1 around -inf

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

   1. mul-1-neg N/A

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

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

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

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

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

   4. neg-sub0 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \left(0 - \left({\left(\sqrt{-1}\right)}^{2} - \frac{1}{2} \cdot \frac{1}{u1}\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \color{blue}{u2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \left({\left(\sqrt{-1}\right)}^{2} - \frac{1}{2} \cdot \frac{1}{u1}\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, \color{blue}{u2}\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \left({\left(\sqrt{-1}\right)}^{2} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \left(\sqrt{-1} \cdot \sqrt{-1} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   8. rem-square-sqrt N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \left(-1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   10. associate-*r/ N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{u1}\right)\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   12. distribute-neg-frac N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{u1}\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \left(\frac{\frac{-1}{2}}{u1}\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

   14. /-lowering-/.f32 21.2%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(u1, \mathsf{\_.f32}\left(0, \mathsf{+.f32}\left(-1, \mathsf{/.f32}\left(\frac{-1}{2}, u1\right)\right)\right)\right), \mathsf{sin.f32}\left(\mathsf{*.f32}\left(\frac{314159265359}{50000000000}, u2\right)\right)\right) \]

8. Simplified 21.2%

   \[\leadsto \color{blue}{\left(u1 \cdot \left(0 - \left(-1 + \frac{-0.5}{u1}\right)\right)\right)} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

9. Taylor expanded in u2 around 0

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

    1. associate-*r* N/A

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

    2. associate-*r* N/A

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

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

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

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

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

    5. *-commutative N/A

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

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

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

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

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

    8. associate-*r/ N/A

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

    9. metadata-eval N/A

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

    10. /-lowering-/.f32 20.5%

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

11. Simplified 20.5%

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

12. Final simplification 20.5%

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

Alternative 19: 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 98.2%

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

4. Applied egg-rr 90.6%

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

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 19.9%

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

10. Simplified 19.9%

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

11. Final simplification 19.9%

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

Alternative 20: 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 98.2%

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

4. Applied egg-rr 90.6%

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

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 19.5%

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

10. Simplified 19.5%

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

Reproduce

herbie shell --seed 2024163 
(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))))