Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 10.6s

Alternatives: 9

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

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

   2. *-commutative N/A

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

   3. lift-sqrt.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. frac-2neg N/A

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

   6. metadata-eval N/A

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

   7. sqrt-div N/A

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

   8. metadata-eval N/A

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

   9. un-div-inv N/A

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

   10. lower-/.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f32 N/A

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

   14. lower-sqrt.f32 N/A

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

   15. lift-/.f32 N/A

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

   16. distribute-neg-frac N/A

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

   17. neg-sub0 N/A

       \[\leadsto \frac{\sin \left(u2 \cdot \frac{314159265359}{50000000000}\right)}{\sqrt{\frac{\color{blue}{0 - \left(u1 - 1\right)}}{u1}}} \]

   18. lift--.f32 N/A

       \[\leadsto \frac{\sin \left(u2 \cdot \frac{314159265359}{50000000000}\right)}{\sqrt{\frac{0 - \color{blue}{\left(u1 - 1\right)}}{u1}}} \]

   19. associate-+l- N/A

       \[\leadsto \frac{\sin \left(u2 \cdot \frac{314159265359}{50000000000}\right)}{\sqrt{\frac{\color{blue}{\left(0 - u1\right) + 1}}{u1}}} \]

   20. neg-sub0 N/A

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

   21. +-commutative N/A

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

   22. sub-neg N/A

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

   23. lift--.f32 N/A

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

   24. lower-/.f32 98.4

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

6. Applied rewrites 98.4%

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

7. Final simplification 98.4%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Final simplification 98.3%

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

Alternative 3: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.012000000104308128:\\ \;\;\;\;\frac{6.28318530718 \cdot u2}{\sqrt{\frac{1}{\frac{u1 \cdot u1}{\left(1 - u1\right) \cdot u1}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(6.28318530718 \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.012000000104308128)
   (/ (* 6.28318530718 u2) (sqrt (/ 1.0 (/ (* u1 u1) (* (- 1.0 u1) u1)))))
   (* (sqrt u1) (sin (* 6.28318530718 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.012000000104308128f) {
		tmp = (6.28318530718f * u2) / sqrtf((1.0f / ((u1 * u1) / ((1.0f - u1) * u1))));
	} else {
		tmp = sqrtf(u1) * sinf((6.28318530718f * u2));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((6.28318530718e0 * u2) <= 0.012000000104308128e0) then
        tmp = (6.28318530718e0 * u2) / sqrt((1.0e0 / ((u1 * u1) / ((1.0e0 - u1) * u1))))
    else
        tmp = sqrt(u1) * sin((6.28318530718e0 * u2))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.012000000104308128))
		tmp = Float32(Float32(Float32(6.28318530718) * u2) / sqrt(Float32(Float32(1.0) / Float32(Float32(u1 * u1) / Float32(Float32(Float32(1.0) - u1) * u1)))));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(Float32(6.28318530718) * u2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.012000000104308128))
		tmp = (single(6.28318530718) * u2) / sqrt((single(1.0) / ((u1 * u1) / ((single(1.0) - u1) * u1))));
	else
		tmp = sqrt(u1) * sin((single(6.28318530718) * u2));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.0120000001

   1. Initial program 98.6%

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

   4. Applied rewrites 98.6%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f32 N/A

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

      4. lift-/.f32 N/A

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

      5. frac-2neg N/A

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

      6. metadata-eval N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. un-div-inv N/A

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

      10. lower-/.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f32 N/A

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

      14. lower-sqrt.f32 N/A

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

      15. lift-/.f32 N/A

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

      16. distribute-neg-frac N/A

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

      17. neg-sub0 N/A

          \[\leadsto \frac{\sin \left(u2 \cdot \frac{314159265359}{50000000000}\right)}{\sqrt{\frac{\color{blue}{0 - \left(u1 - 1\right)}}{u1}}} \]

      18. lift--.f32 N/A

          \[\leadsto \frac{\sin \left(u2 \cdot \frac{314159265359}{50000000000}\right)}{\sqrt{\frac{0 - \color{blue}{\left(u1 - 1\right)}}{u1}}} \]

      19. associate-+l- N/A

          \[\leadsto \frac{\sin \left(u2 \cdot \frac{314159265359}{50000000000}\right)}{\sqrt{\frac{\color{blue}{\left(0 - u1\right) + 1}}{u1}}} \]

      20. neg-sub0 N/A

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

      21. +-commutative N/A

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

      22. sub-neg N/A

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

      23. lift--.f32 N/A

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

      24. lower-/.f32 98.7

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

   6. Applied rewrites 98.7%

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

      1. lift-/.f32 N/A

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

      2. lift--.f32 N/A

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

      3. div-sub N/A

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

      4. frac-sub N/A

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

      5. lift-*.f32 N/A

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

      6. clear-num N/A

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

      7. lower-/.f32 N/A

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

      8. lower-/.f32 N/A

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

      9. distribute-rgt-out-- N/A

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

      10. lift--.f32 N/A

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

      11. lower-*.f32 98.7

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

   8. Applied rewrites 98.7%

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

   9. Taylor expanded in u2 around 0

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

       1. *-commutative N/A

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

       2. lower-*.f32 96.7

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

   11. Applied rewrites 96.7%

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

   if 0.0120000001 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.5%

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

   3. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 74.5

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

   5. Applied rewrites 74.5%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.012000000104308128:\\ \;\;\;\;\frac{6.28318530718 \cdot u2}{\sqrt{\frac{1}{\frac{u1 \cdot u1}{\left(1 - u1\right) \cdot u1}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(6.28318530718 \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

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

   2. *-commutative N/A

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

   3. lift-sqrt.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. frac-2neg N/A

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

   6. metadata-eval N/A

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

   7. sqrt-div N/A

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

   8. metadata-eval N/A

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

   9. un-div-inv N/A

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

   10. lower-/.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f32 N/A

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

   14. lower-sqrt.f32 N/A

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

   15. lift-/.f32 N/A

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

   16. distribute-neg-frac N/A

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

   17. neg-sub0 N/A

       \[\leadsto \frac{\sin \left(u2 \cdot \frac{314159265359}{50000000000}\right)}{\sqrt{\frac{\color{blue}{0 - \left(u1 - 1\right)}}{u1}}} \]

   18. lift--.f32 N/A

       \[\leadsto \frac{\sin \left(u2 \cdot \frac{314159265359}{50000000000}\right)}{\sqrt{\frac{0 - \color{blue}{\left(u1 - 1\right)}}{u1}}} \]

   19. associate-+l- N/A

       \[\leadsto \frac{\sin \left(u2 \cdot \frac{314159265359}{50000000000}\right)}{\sqrt{\frac{\color{blue}{\left(0 - u1\right) + 1}}{u1}}} \]

   20. neg-sub0 N/A

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

   21. +-commutative N/A

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

   22. sub-neg N/A

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

   23. lift--.f32 N/A

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

   24. lower-/.f32 98.4

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

6. Applied rewrites 98.4%

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

   1. lift-/.f32 N/A

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

   2. lift--.f32 N/A

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

   3. div-sub N/A

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

   4. frac-sub N/A

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

   5. lift-*.f32 N/A

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

   6. clear-num N/A

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

   7. lower-/.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. distribute-rgt-out-- N/A

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

   10. lift--.f32 N/A

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

   11. lower-*.f32 98.3

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

8. Applied rewrites 98.3%

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

9. Taylor expanded in u2 around 0

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

    1. *-commutative N/A

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

    2. lower-*.f32 82.4

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

11. Applied rewrites 82.4%

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

12. Final simplification 82.4%

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

Alternative 5: 81.3% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 82.4

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

5. Applied rewrites 82.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-sqrt.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. clear-num N/A

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

   6. sqrt-div N/A

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

   7. lift--.f32 N/A

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

   8. sub-neg N/A

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

   9. +-commutative N/A

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

   10. neg-sub0 N/A

       \[\leadsto \left(u2 \cdot \frac{314159265359}{50000000000}\right) \cdot \frac{\sqrt{1}}{\sqrt{\frac{\color{blue}{\left(0 - u1\right)} + 1}{u1}}} \]

   11. associate-+l- N/A

       \[\leadsto \left(u2 \cdot \frac{314159265359}{50000000000}\right) \cdot \frac{\sqrt{1}}{\sqrt{\frac{\color{blue}{0 - \left(u1 - 1\right)}}{u1}}} \]

   12. lift--.f32 N/A

       \[\leadsto \left(u2 \cdot \frac{314159265359}{50000000000}\right) \cdot \frac{\sqrt{1}}{\sqrt{\frac{0 - \color{blue}{\left(u1 - 1\right)}}{u1}}} \]

   13. neg-sub0 N/A

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

   14. distribute-neg-frac N/A

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

   15. lift-/.f32 N/A

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

   16. metadata-eval N/A

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

   17. un-div-inv N/A

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

   18. lower-/.f32 N/A

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

7. Applied rewrites 82.4%

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

8. Final simplification 82.4%

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

Alternative 6: 81.3% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 82.4

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

5. Applied rewrites 82.4%

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

6. Final simplification 82.4%

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

Alternative 7: 73.0% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(((u1 * u1) + 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

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

   1. *-commutative N/A

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

   2. lower-*.f32 82.4

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

5. Applied rewrites 82.4%

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

6. Taylor expanded in u1 around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f32 65.2

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

8. Applied rewrites 65.0%

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

10. Applied rewrites 74.0%

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

11. Final simplification 74.0%

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

Alternative 8: 73.0% accurate, 4.7× speedup

Math

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

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(Float32(u1 - 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

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

   1. *-commutative N/A

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

   2. lower-*.f32 82.4

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

5. Applied rewrites 82.4%

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

6. Taylor expanded in u1 around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f32 65.2

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

8. Applied rewrites 65.0%

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

10. Applied rewrites 74.0%

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

11. Final simplification 74.0%

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

Alternative 9: 64.8% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 82.4

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

5. Applied rewrites 82.4%

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

6. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 65.2

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

8. Applied rewrites 65.2%

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

9. Final simplification 65.2%

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

Reproduce

herbie shell --seed 2024243 
(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))))