Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 9.4s

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, 0.4× speedup

Math

\[\begin{array}{l} \\ \sin \left(u2 \cdot 6.28318530718\right) \cdot {\left({\left(\frac{u1 - 1}{u1}\right)}^{-2}\right)}^{0.25} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sin (* u2 6.28318530718)) (pow (pow (/ (- u1 1.0) u1) -2.0) 0.25)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf((u2 * 6.28318530718f)) * powf(powf(((u1 - 1.0f) / u1), -2.0f), 0.25f);
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sin((u2 * 6.28318530718e0)) * ((((u1 - 1.0e0) / u1) ** (-2.0e0)) ** 0.25e0)
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sin(Float32(u2 * Float32(6.28318530718))) * ((Float32(Float32(u1 - Float32(1.0)) / u1) ^ Float32(-2.0)) ^ Float32(0.25)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sin((u2 * single(6.28318530718))) * ((((u1 - single(1.0)) / u1) ^ single(-2.0)) ^ single(0.25));
end

Derivation

1. Initial program 98.4%

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

4. Applied rewrites 98.4%

   \[\leadsto \color{blue}{{\left({\left(\frac{u1 - 1}{u1}\right)}^{-2}\right)}^{0.25}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

5. Final simplification 98.4%

   \[\leadsto \sin \left(u2 \cdot 6.28318530718\right) \cdot {\left({\left(\frac{u1 - 1}{u1}\right)}^{-2}\right)}^{0.25} \]
6. Add Preprocessing

Alternative 2: 93.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.00139999995008111:\\ \;\;\;\;\left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(1 + u1\right) \cdot u1} \cdot \sin \left(u2 \cdot 6.28318530718\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.00139999995008111)
   (* (* u2 6.28318530718) (sqrt (/ u1 (- 1.0 u1))))
   (* (sqrt (* (+ 1.0 u1) u1)) (sin (* u2 6.28318530718)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.00139999995008111f) {
		tmp = (u2 * 6.28318530718f) * sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = sqrtf(((1.0f + u1) * u1)) * sinf((u2 * 6.28318530718f));
	}
	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 ((u2 * 6.28318530718e0) <= 0.00139999995008111e0) then
        tmp = (u2 * 6.28318530718e0) * sqrt((u1 / (1.0e0 - u1)))
    else
        tmp = sqrt(((1.0e0 + u1) * u1)) * sin((u2 * 6.28318530718e0))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.00139999995008111))
		tmp = Float32(Float32(u2 * Float32(6.28318530718)) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1))));
	else
		tmp = Float32(sqrt(Float32(Float32(Float32(1.0) + u1) * u1)) * sin(Float32(u2 * Float32(6.28318530718))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.00139999995008111))
		tmp = (u2 * single(6.28318530718)) * sqrt((u1 / (single(1.0) - u1)));
	else
		tmp = sqrt(((single(1.0) + u1) * u1)) * sin((u2 * single(6.28318530718)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00139999995

   1. Initial program 98.8%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   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 98.6

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

   5. Applied rewrites 98.6%

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

   if 0.00139999995 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.9%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. inv-pow N/A

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

      5. inv-pow N/A

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

      6. rem-square-sqrt N/A

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

      7. pow1/2 N/A

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

      8. pow1/2 N/A

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

      9. unpow-prod-down N/A

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

      10. inv-pow N/A

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

      11. inv-pow N/A

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

      12. pow-pow N/A

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

      13. pow-flip N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. pow1/2 N/A

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

      17. inv-pow N/A

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

      18. inv-pow N/A

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

      19. pow-pow N/A

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

      20. pow-flip N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. pow1/2 N/A

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

   4. Applied rewrites 97.9%

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

      1. lift-/.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-sqrt.f32 N/A

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

      4. lift-sqrt.f32 N/A

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

      5. rem-square-sqrt N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f32 N/A

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

      9. frac-2neg N/A

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

      10. metadata-eval N/A

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

      11. lift--.f32 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. remove-double-neg N/A

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

      16. sub-neg N/A

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

      17. lift--.f32 N/A

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

      18. lower-/.f32 97.8

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

   6. Applied rewrites 97.8%

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

   7. Taylor expanded in u1 around 0

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

      1. lower-+.f32 86.4

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

   9. Applied rewrites 86.4%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.00139999995008111:\\ \;\;\;\;\left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(1 + u1\right) \cdot u1} \cdot \sin \left(u2 \cdot 6.28318530718\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Final simplification 98.4%

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

Alternative 4: 89.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.041999999433755875:\\ \;\;\;\;\left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(u2 \cdot 6.28318530718\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.041999999433755875)
   (* (* u2 6.28318530718) (sqrt (/ u1 (- 1.0 u1))))
   (* (sqrt u1) (sin (* u2 6.28318530718)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.041999999433755875f) {
		tmp = (u2 * 6.28318530718f) * sqrtf((u1 / (1.0f - u1)));
	} else {
		tmp = sqrtf(u1) * sinf((u2 * 6.28318530718f));
	}
	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 ((u2 * 6.28318530718e0) <= 0.041999999433755875e0) then
        tmp = (u2 * 6.28318530718e0) * sqrt((u1 / (1.0e0 - u1)))
    else
        tmp = sqrt(u1) * sin((u2 * 6.28318530718e0))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.041999999433755875))
		tmp = Float32(Float32(u2 * Float32(6.28318530718)) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1))));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(u2 * Float32(6.28318530718))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.041999999433755875))
		tmp = (u2 * single(6.28318530718)) * sqrt((u1 / (single(1.0) - u1)));
	else
		tmp = sqrt(u1) * sin((u2 * single(6.28318530718)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.0419999994

   1. Initial program 98.7%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   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 93.5

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

   5. Applied rewrites 93.5%

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

   if 0.0419999994 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.4%

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

   3. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 75.5

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

   5. Applied rewrites 75.5%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.041999999433755875:\\ \;\;\;\;\left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(u2 \cdot 6.28318530718\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.0% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 81.8

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

5. Applied rewrites 81.8%

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

6. Final simplification 81.8%

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

Alternative 6: 72.7% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 81.8

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

5. Applied rewrites 81.8%

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

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 65.7

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

8. Applied rewrites 65.7%

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

Alternative 7: 72.6% accurate, 4.7× speedup

Math

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

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(sqrt(Float32(Float32(u1 - Float32(-1.0)) * u1)) * Float32(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 98.4%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 81.8

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

5. Applied rewrites 81.8%

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

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 65.7

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

8. Applied rewrites 65.7%

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

Alternative 8: 64.4% accurate, 6.4× speedup

Math

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

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(sqrt(u1) * Float32(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 98.4%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 81.8

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

5. Applied rewrites 81.8%

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

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 65.7

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

8. Applied rewrites 65.7%

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

Alternative 9: 19.6% accurate, 12.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 81.8

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

5. Applied rewrites 81.8%

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

   1. rem-square-sqrt N/A

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

   2. sqrt-unprod N/A

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

   3. lift--.f32 N/A

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

   4. flip-- N/A

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

   5. associate-*r/ N/A

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

   6. sqrt-div N/A

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

   7. lower-/.f32 N/A

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

7. Applied rewrites 74.6%

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

8. Taylor expanded in u1 around inf

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

10. Applied rewrites 19.9%

    \[\leadsto \color{blue}{1} \cdot \left(u2 \cdot 6.28318530718\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024254 
(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))))