Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 9.4s

Alternatives: 7

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

4. Applied rewrites 98.1%

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

6. Applied rewrites 98.2%

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

7. Final simplification 98.2%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

3. Final simplification 98.1%

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

Alternative 3: 94.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.36000001430511475:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(u2 \cdot u2, -76.70585975309672, 81.6052492761019\right), u2 \cdot u2, -41.341702240407926\right) \cdot u2\right) \cdot u2 + 6.28318530718\right) \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(6.28318530718 \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.36000001430511475)
   (*
    (*
     (+
      (*
       (*
        (fma
         (fma (* u2 u2) -76.70585975309672 81.6052492761019)
         (* u2 u2)
         -41.341702240407926)
        u2)
       u2)
      6.28318530718)
     u2)
    (sqrt (/ u1 (- 1.0 u1))))
   (* (sqrt u1) (sin (* 6.28318530718 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.36000001430511475f) {
		tmp = ((((fmaf(fmaf((u2 * u2), -76.70585975309672f, 81.6052492761019f), (u2 * u2), -41.341702240407926f) * u2) * u2) + 6.28318530718f) * u2) * sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = sqrtf(u1) * sinf((6.28318530718f * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.36000001430511475))
		tmp = Float32(Float32(Float32(Float32(Float32(fma(fma(Float32(u2 * u2), Float32(-76.70585975309672), Float32(81.6052492761019)), Float32(u2 * u2), Float32(-41.341702240407926)) * u2) * u2) + Float32(6.28318530718)) * u2) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1))));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(Float32(6.28318530718) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.360000014

   1. Initial program 98.4%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 88.1

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

   5. Applied rewrites 88.1%

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

   6. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f32 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f32 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f32 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f32 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f32 88.1

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

   8. Applied rewrites 88.1%

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

   10. Applied rewrites 96.4%

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

   if 0.360000014 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 96.4%

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

   3. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 76.0

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

   5. Applied rewrites 76.0%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.36000001430511475:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(u2 \cdot u2, -76.70585975309672, 81.6052492761019\right), u2 \cdot u2, -41.341702240407926\right) \cdot u2\right) \cdot u2 + 6.28318530718\right) \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(6.28318530718 \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 78.6

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

5. Applied rewrites 78.6%

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

6. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. sub-neg N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f32 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f32 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f32 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f32 78.6

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

8. Applied rewrites 77.3%

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

10. Applied rewrites 86.7%

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

11. Final simplification 86.7%

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

Alternative 5: 81.1% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 78.6

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

5. Applied rewrites 78.6%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-sqrt.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. sqrt-div N/A

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

   6. pow1/2 N/A

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

   7. metadata-eval N/A

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

   8. pow-flip N/A

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

   9. lift-pow.f32 N/A

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

   10. lift-sqrt.f32 N/A

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

   11. associate-/r* N/A

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

   12. associate-/l/ N/A

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

   13. lift-sqrt.f32 N/A

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

   14. pow1/2 N/A

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

   15. pow-flip N/A

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

   16. metadata-eval N/A

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

   17. lift-pow.f32 N/A

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

   18. clear-num N/A

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

   19. un-div-inv N/A

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

   20. lower-/.f32 N/A

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

7. Applied rewrites 78.7%

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

8. Final simplification 78.7%

   \[\leadsto \frac{6.28318530718 \cdot u2}{\sqrt{\frac{1 - u1}{u1}}} \]
9. Add Preprocessing

Alternative 6: 81.1% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 78.6

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

5. Applied rewrites 78.6%

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

6. Final simplification 78.6%

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

Alternative 7: 64.2% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 78.6

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

5. Applied rewrites 78.6%

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

6. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 62.3

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

8. Applied rewrites 62.3%

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

9. Final simplification 62.3%

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

Reproduce

herbie shell --seed 2024276 
(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))))