Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.2%

Time: 10.2s

Alternatives: 10

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: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{\frac{-1}{u1} + -1}{1 - {u1}^{-2}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (/ (+ (/ -1.0 u1) -1.0) (- 1.0 (pow u1 -2.0))))
  (sin (* 6.28318530718 u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((((-1.0f / u1) + -1.0f) / (1.0f - powf(u1, -2.0f)))) * 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(((((-1.0e0) / u1) + (-1.0e0)) / (1.0e0 - (u1 ** (-2.0e0))))) * sin((6.28318530718e0 * u2))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(Float32(Float32(Float32(-1.0) / u1) + Float32(-1.0)) / Float32(Float32(1.0) - (u1 ^ Float32(-2.0))))) * sin(Float32(Float32(6.28318530718) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((((single(-1.0) / u1) + single(-1.0)) / (single(1.0) - (u1 ^ single(-2.0))))) * sin((single(6.28318530718) * u2));
end

Derivation

1. Initial program 98.3%

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

4. Applied rewrites 98.3%

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

   1. lift-/.f32 N/A

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

   2. lift--.f32 N/A

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

   3. div-sub N/A

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

   4. *-inverses N/A

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

   5. flip-- N/A

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

   6. clear-num N/A

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

   7. lower-/.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. lower-+.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. metadata-eval N/A

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

   12. lower--.f32 N/A

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

   13. inv-pow N/A

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

   14. inv-pow N/A

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

   15. pow-prod-up N/A

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

   16. metadata-eval N/A

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

   17. lower-pow.f32 98.4

       \[\leadsto \sqrt{\frac{-1}{\frac{1}{\frac{1 + \frac{1}{u1}}{1 - \color{blue}{{u1}^{-2}}}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

6. Applied rewrites 98.4%

   \[\leadsto \sqrt{\frac{-1}{\color{blue}{\frac{1}{\frac{1 + \frac{1}{u1}}{1 - {u1}^{-2}}}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
7. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/r/ N/A

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

   4. metadata-eval N/A

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

   5. neg-mul-1 N/A

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

   6. lift-/.f32 N/A

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

   7. distribute-neg-frac N/A

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

   8. lower-/.f32 N/A

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

   9. lift-+.f32 N/A

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

   10. +-commutative N/A

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

   11. distribute-neg-in N/A

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

   12. metadata-eval N/A

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

   13. lower-+.f32 N/A

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

   14. lift-/.f32 N/A

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

   15. distribute-neg-frac N/A

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

   16. metadata-eval N/A

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

   17. lower-/.f32 98.4

       \[\leadsto \sqrt{\frac{\color{blue}{\frac{-1}{u1}} + -1}{1 - {u1}^{-2}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

8. Applied rewrites 98.4%

   \[\leadsto \sqrt{\color{blue}{\frac{\frac{-1}{u1} + -1}{1 - {u1}^{-2}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
9. Add Preprocessing

Alternative 2: 98.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

4. Applied rewrites 98.3%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

Alternative 4: 93.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.25600001215934753:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-76.70585975309672, u2 \cdot u2, 81.6052492761019\right) \cdot u2, u2, -41.341702240407926\right) \cdot u2\right) \cdot u2 + 6.28318530718\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(6.28318530718 \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.25600001215934753)
   (*
    (sqrt (/ u1 (- 1.0 u1)))
    (*
     (+
      (*
       (*
        (fma
         (* (fma -76.70585975309672 (* u2 u2) 81.6052492761019) u2)
         u2
         -41.341702240407926)
        u2)
       u2)
      6.28318530718)
     u2))
   (* (sqrt u1) (sin (* 6.28318530718 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.25600001215934753f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * ((((fmaf((fmaf(-76.70585975309672f, (u2 * u2), 81.6052492761019f) * u2), u2, -41.341702240407926f) * u2) * u2) + 6.28318530718f) * u2);
	} else {
		tmp = sqrtf(u1) * sinf((6.28318530718f * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.25600001215934753))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(Float32(Float32(fma(Float32(fma(Float32(-76.70585975309672), Float32(u2 * u2), Float32(81.6052492761019)) * u2), u2, Float32(-41.341702240407926)) * u2) * u2) + Float32(6.28318530718)) * u2));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(Float32(6.28318530718) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.256000012

   1. Initial program 98.4%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f32 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f32 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f32 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f32 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f32 91.7

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

   5. Applied rewrites 91.7%

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

   7. Applied rewrites 91.7%

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

   9. Applied rewrites 97.4%

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

   if 0.256000012 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.9%

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

   3. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 74.1

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

   5. Applied rewrites 74.1%

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

Alternative 5: 88.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. sub-neg N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f32 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f32 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f32 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f32 83.8

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

5. Applied rewrites 83.8%

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

7. Applied rewrites 83.8%

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

9. Applied rewrites 90.3%

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

Alternative 6: 80.8% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

5. Applied rewrites 83.8%

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

7. Applied rewrites 84.0%

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

Alternative 7: 80.8% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

5. Applied rewrites 83.8%

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

7. Applied rewrites 83.9%

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

Alternative 8: 80.8% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

5. Applied rewrites 83.8%

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

Alternative 9: 64.3% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

5. Applied rewrites 83.8%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 65.3%

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

Alternative 10: 64.3% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

5. Applied rewrites 83.8%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 65.3%

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

10. Applied rewrites 65.3%

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

Reproduce

herbie shell --seed 2024307 
(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))))