Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 13.4s

Alternatives: 13

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

4. Applied egg-rr 98.2%

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

6. Applied egg-rr 98.3%

   \[\leadsto \color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{-0.5} \cdot \sin \left(6.28318530718 \cdot u2\right)} \]
7. 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 98.2%

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

3. Final simplification 98.2%

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

Alternative 3: 93.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (((single(1.0) - u1) / u1) ^ single(-0.5)) * (u2 * (single(6.28318530718) + ((u2 * u2) * (((u2 * u2) * (single(81.6052492761019) + (u2 * (u2 * single(-76.70585975309672))))) + 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 98.2%

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

6. Applied egg-rr 98.3%

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

7. Taylor expanded in u2 around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

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

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

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

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

   10. unpow2 N/A

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

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

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

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

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

   13. unpow2 N/A

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

   14. associate-*r* N/A

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

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

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

   16. *-lowering-*.f32 94.5%

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

9. Simplified 94.5%

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

10. Final simplification 94.5%

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

Alternative 4: 93.9% 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 94.5%

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

5. Simplified 94.5%

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

Alternative 5: 91.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 * (powf(((1.0f - u1) / u1), -0.5f) * (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 = u2 * ((((1.0e0 - u1) / u1) ** (-0.5e0)) * (6.28318530718e0 + (u2 * (u2 * ((-41.341702240407926e0) + (u2 * (u2 * 81.6052492761019e0)))))))
end function

Julia

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

MATLAB

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

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

5. Simplified 92.2%

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

7. Applied egg-rr 92.1%

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

8. Final simplification 92.1%

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

Alternative 6: 91.8% 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 92.0%

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

5. Simplified 92.0%

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

Alternative 7: 89.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return powf(((1.0f - u1) / u1), -0.5f) * (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 = (((1.0e0 - u1) / u1) ** (-0.5e0)) * (u2 * (6.28318530718e0 + ((u2 * u2) * (-41.341702240407926e0))))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (((single(1.0) - u1) / u1) ^ single(-0.5)) * (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 98.2%

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

6. Applied egg-rr 98.3%

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

7. Taylor expanded in u2 around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

   5. *-lowering-*.f32 88.9%

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

9. Simplified 88.9%

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

10. Final simplification 88.9%

    \[\leadsto {\left(\frac{1 - u1}{u1}\right)}^{-0.5} \cdot \left(u2 \cdot \left(6.28318530718 + \left(u2 \cdot u2\right) \cdot -41.341702240407926\right)\right) \]
11. 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 88.8%

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

6. Final simplification 88.8%

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

Alternative 9: 81.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u2 * ((((single(1.0) - u1) / u1) ^ single(-0.5)) * 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 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 80.2%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

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

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

7. Applied egg-rr 80.3%

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

8. Final simplification 80.3%

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

Alternative 10: 81.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

5. Simplified 92.2%

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

6. Taylor expanded in u2 around 0

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

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

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

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

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

   3. sub-neg N/A

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

   4. mul-1-neg N/A

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

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

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

   6. mul-1-neg N/A

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

   7. sub-neg N/A

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

   8. --lowering--.f32 80.2%

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

8. Simplified 80.2%

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

Alternative 11: 64.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 6.28318530718 \cdot \left(u2 \cdot {\left(u1 \cdot u1\right)}^{0.25}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* 6.28318530718 (* u2 (pow (* u1 u1) 0.25))))

C

float code(float cosTheta_i, float u1, float u2) {
	return 6.28318530718f * (u2 * powf((u1 * u1), 0.25f));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = 6.28318530718e0 * (u2 * ((u1 * u1) ** 0.25e0))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(6.28318530718) * Float32(u2 * (Float32(u1 * u1) ^ Float32(0.25))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (u2 * ((u1 * u1) ^ single(0.25)));
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 80.2%

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

6. Taylor expanded in u1 around 0

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

   1. sqrt-lowering-sqrt.f32 64.2%

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

8. Simplified 64.2%

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

9. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\frac{314159265359}{50000000000} \cdot \left(\sqrt{u1} \cdot u2\right)} \]
10. 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 64.2%

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

11. Simplified 64.2%

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

    1. pow1/2 N/A

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

    2. sqr-pow N/A

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

    3. pow-prod-down N/A

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

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

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

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

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

    6. metadata-eval 64.2%

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

13. Applied egg-rr 64.2%

    \[\leadsto 6.28318530718 \cdot \left(\color{blue}{{\left(u1 \cdot u1\right)}^{0.25}} \cdot u2\right) \]

14. Final simplification 64.2%

    \[\leadsto 6.28318530718 \cdot \left(u2 \cdot {\left(u1 \cdot u1\right)}^{0.25}\right) \]
15. Add Preprocessing

Alternative 12: 64.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u2 * (single(6.28318530718) * (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 80.2%

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

6. Taylor expanded in u1 around 0

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

   1. sqrt-lowering-sqrt.f32 64.2%

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

8. Simplified 64.2%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

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

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

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

   5. pow1/2 N/A

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

   6. pow-lowering-pow.f32 64.2%

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

10. Applied egg-rr 64.2%

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

11. Final simplification 64.2%

    \[\leadsto u2 \cdot \left(6.28318530718 \cdot {u1}^{0.5}\right) \]
12. Add Preprocessing

Alternative 13: 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 80.2%

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

6. Taylor expanded in u1 around 0

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

   1. sqrt-lowering-sqrt.f32 64.2%

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

8. Simplified 64.2%

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

9. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\frac{314159265359}{50000000000} \cdot \left(\sqrt{u1} \cdot u2\right)} \]
10. 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 64.2%

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

11. Simplified 64.2%

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

12. Final simplification 64.2%

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

Reproduce

herbie shell --seed 2024152 
(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))))