Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 10.7s

Alternatives: 15

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, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right)\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (log1p (expm1 (* (sqrt (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return log1pf(expm1f((sqrtf((u1 / (1.0f - u1))) * sinf((6.28318530718f * u2)))));
}

Julia

function code(cosTheta_i, u1, u2)
	return log1p(expm1(Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * sin(Float32(Float32(6.28318530718) * u2)))))
end

Derivation

1. Initial program 98.4%

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

   1. log1p-expm1-u 98.4%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right)\right)\right)} \]

3. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right)\right)\right)} \]

4. Final simplification 98.4%

   \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right)\right)\right) \]

Alternative 2: 93.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.3499999940395355:\\ \;\;\;\;\frac{1}{\sqrt{-1 + \frac{1}{u1}} \cdot \left(\frac{0.15915494309188485}{u2} + u2 \cdot 1.0471975511966667\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.3499999940395355)
   (/
    1.0
    (*
     (sqrt (+ -1.0 (/ 1.0 u1)))
     (+ (/ 0.15915494309188485 u2) (* u2 1.0471975511966667))))
   (* (sin (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.3499999940395355f) {
		tmp = 1.0f / (sqrtf((-1.0f + (1.0f / u1))) * ((0.15915494309188485f / u2) + (u2 * 1.0471975511966667f)));
	} else {
		tmp = sinf((6.28318530718f * u2)) * sqrtf(u1);
	}
	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) <= 0.3499999940395355e0) then
        tmp = 1.0e0 / (sqrt(((-1.0e0) + (1.0e0 / u1))) * ((0.15915494309188485e0 / u2) + (u2 * 1.0471975511966667e0)))
    else
        tmp = sin((6.28318530718e0 * u2)) * sqrt(u1)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.3499999940395355))
		tmp = Float32(Float32(1.0) / Float32(sqrt(Float32(Float32(-1.0) + Float32(Float32(1.0) / u1))) * Float32(Float32(Float32(0.15915494309188485) / u2) + Float32(u2 * Float32(1.0471975511966667)))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.3499999940395355))
		tmp = single(1.0) / (sqrt((single(-1.0) + (single(1.0) / u1))) * ((single(0.15915494309188485) / u2) + (u2 * single(1.0471975511966667))));
	else
		tmp = sin((single(6.28318530718) * u2)) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.349999994

   1. Initial program 98.6%

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

      1. log1p-expm1-u 98.6%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(6.28318530718 \cdot u2\right)\right)\right)} \]

   3. Applied egg-rr 98.6%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(6.28318530718 \cdot u2\right)\right)\right)} \]
   4. Step-by-step derivation

      1. clear-num 98.5%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{1 - u1}{u1}}}} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(6.28318530718 \cdot u2\right)\right)\right) \]

      2. sqrt-div 98.4%

         \[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{1 - u1}{u1}}}} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(6.28318530718 \cdot u2\right)\right)\right) \]

      3. metadata-eval 98.4%

         \[\leadsto \frac{\color{blue}{1}}{\sqrt{\frac{1 - u1}{u1}}} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(6.28318530718 \cdot u2\right)\right)\right) \]

      4. log1p-expm1-u 98.4%

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

      5. add-sqr-sqrt 97.8%

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

      6. associate-*r* 97.7%

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

      7. metadata-eval 97.7%

         \[\leadsto \frac{1}{\sqrt{\frac{1 - u1}{u1}}} \cdot \sin \left(\left(\color{blue}{\sqrt{39.47841760436263}} \cdot \sqrt{u2}\right) \cdot \sqrt{u2}\right) \]

      8. sqrt-prod 98.3%

         \[\leadsto \frac{1}{\sqrt{\frac{1 - u1}{u1}}} \cdot \sin \left(\color{blue}{\sqrt{39.47841760436263 \cdot u2}} \cdot \sqrt{u2}\right) \]

      9. sqrt-prod 98.4%

         \[\leadsto \frac{1}{\sqrt{\frac{1 - u1}{u1}}} \cdot \sin \color{blue}{\left(\sqrt{\left(39.47841760436263 \cdot u2\right) \cdot u2}\right)} \]

      10. associate-*l/ 98.4%

          \[\leadsto \color{blue}{\frac{1 \cdot \sin \left(\sqrt{\left(39.47841760436263 \cdot u2\right) \cdot u2}\right)}{\sqrt{\frac{1 - u1}{u1}}}} \]

      11. associate-/l* 98.3%

          \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\frac{1 - u1}{u1}}}{\sin \left(\sqrt{\left(39.47841760436263 \cdot u2\right) \cdot u2}\right)}}} \]

      12. div-sub 98.2%

          \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\frac{1}{u1} - \frac{u1}{u1}}}}{\sin \left(\sqrt{\left(39.47841760436263 \cdot u2\right) \cdot u2}\right)}} \]

      13. *-inverses 98.2%

          \[\leadsto \frac{1}{\frac{\sqrt{\frac{1}{u1} - \color{blue}{1}}}{\sin \left(\sqrt{\left(39.47841760436263 \cdot u2\right) \cdot u2}\right)}} \]

      14. sub-neg 98.2%

          \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\frac{1}{u1} + \left(-1\right)}}}{\sin \left(\sqrt{\left(39.47841760436263 \cdot u2\right) \cdot u2}\right)}} \]

      15. metadata-eval 98.2%

          \[\leadsto \frac{1}{\frac{\sqrt{\frac{1}{u1} + \color{blue}{-1}}}{\sin \left(\sqrt{\left(39.47841760436263 \cdot u2\right) \cdot u2}\right)}} \]

      16. sqrt-prod 98.1%

          \[\leadsto \frac{1}{\frac{\sqrt{\frac{1}{u1} + -1}}{\sin \color{blue}{\left(\sqrt{39.47841760436263 \cdot u2} \cdot \sqrt{u2}\right)}}} \]

   5. Applied egg-rr 98.3%

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

   6. Taylor expanded in u2 around 0 95.9%

      \[\leadsto \frac{1}{\color{blue}{0.15915494309188485 \cdot \left(\frac{1}{u2} \cdot \sqrt{\frac{1}{u1} - 1}\right) + 1.0471975511966667 \cdot \left(u2 \cdot \sqrt{\frac{1}{u1} - 1}\right)}} \]
   7. Step-by-step derivation

      1. associate-*r* 95.9%

         \[\leadsto \frac{1}{\color{blue}{\left(0.15915494309188485 \cdot \frac{1}{u2}\right) \cdot \sqrt{\frac{1}{u1} - 1}} + 1.0471975511966667 \cdot \left(u2 \cdot \sqrt{\frac{1}{u1} - 1}\right)} \]

      2. sub-neg 95.9%

         \[\leadsto \frac{1}{\left(0.15915494309188485 \cdot \frac{1}{u2}\right) \cdot \sqrt{\color{blue}{\frac{1}{u1} + \left(-1\right)}} + 1.0471975511966667 \cdot \left(u2 \cdot \sqrt{\frac{1}{u1} - 1}\right)} \]

      3. metadata-eval 95.9%

         \[\leadsto \frac{1}{\left(0.15915494309188485 \cdot \frac{1}{u2}\right) \cdot \sqrt{\frac{1}{u1} + \color{blue}{-1}} + 1.0471975511966667 \cdot \left(u2 \cdot \sqrt{\frac{1}{u1} - 1}\right)} \]

      4. +-commutative 95.9%

         \[\leadsto \frac{1}{\left(0.15915494309188485 \cdot \frac{1}{u2}\right) \cdot \sqrt{\color{blue}{-1 + \frac{1}{u1}}} + 1.0471975511966667 \cdot \left(u2 \cdot \sqrt{\frac{1}{u1} - 1}\right)} \]

      5. associate-*r* 95.9%

         \[\leadsto \frac{1}{\left(0.15915494309188485 \cdot \frac{1}{u2}\right) \cdot \sqrt{-1 + \frac{1}{u1}} + \color{blue}{\left(1.0471975511966667 \cdot u2\right) \cdot \sqrt{\frac{1}{u1} - 1}}} \]

      6. sub-neg 95.9%

         \[\leadsto \frac{1}{\left(0.15915494309188485 \cdot \frac{1}{u2}\right) \cdot \sqrt{-1 + \frac{1}{u1}} + \left(1.0471975511966667 \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{1}{u1} + \left(-1\right)}}} \]

      7. metadata-eval 95.9%

         \[\leadsto \frac{1}{\left(0.15915494309188485 \cdot \frac{1}{u2}\right) \cdot \sqrt{-1 + \frac{1}{u1}} + \left(1.0471975511966667 \cdot u2\right) \cdot \sqrt{\frac{1}{u1} + \color{blue}{-1}}} \]

      8. +-commutative 95.9%

         \[\leadsto \frac{1}{\left(0.15915494309188485 \cdot \frac{1}{u2}\right) \cdot \sqrt{-1 + \frac{1}{u1}} + \left(1.0471975511966667 \cdot u2\right) \cdot \sqrt{\color{blue}{-1 + \frac{1}{u1}}}} \]

      9. distribute-rgt-out 96.0%

         \[\leadsto \frac{1}{\color{blue}{\sqrt{-1 + \frac{1}{u1}} \cdot \left(0.15915494309188485 \cdot \frac{1}{u2} + 1.0471975511966667 \cdot u2\right)}} \]

      10. associate-*r/ 96.1%

          \[\leadsto \frac{1}{\sqrt{-1 + \frac{1}{u1}} \cdot \left(\color{blue}{\frac{0.15915494309188485 \cdot 1}{u2}} + 1.0471975511966667 \cdot u2\right)} \]

      11. metadata-eval 96.1%

          \[\leadsto \frac{1}{\sqrt{-1 + \frac{1}{u1}} \cdot \left(\frac{\color{blue}{0.15915494309188485}}{u2} + 1.0471975511966667 \cdot u2\right)} \]

   8. Simplified 96.1%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{-1 + \frac{1}{u1}} \cdot \left(\frac{0.15915494309188485}{u2} + 1.0471975511966667 \cdot u2\right)}} \]

   if 0.349999994 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.0%

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

   2. Taylor expanded in u1 around 0 70.1%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.3499999940395355:\\ \;\;\;\;\frac{1}{\sqrt{-1 + \frac{1}{u1}} \cdot \left(\frac{0.15915494309188485}{u2} + u2 \cdot 1.0471975511966667\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]

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 98.4%

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

2. Final simplification 98.4%

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

Alternative 4: 87.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{-1 + \frac{1}{u1}} \cdot \left(\frac{0.15915494309188485}{u2} + u2 \cdot 1.0471975511966667\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (/
  1.0
  (*
   (sqrt (+ -1.0 (/ 1.0 u1)))
   (+ (/ 0.15915494309188485 u2) (* u2 1.0471975511966667)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return 1.0f / (sqrtf((-1.0f + (1.0f / u1))) * ((0.15915494309188485f / u2) + (u2 * 1.0471975511966667f)));
}

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 / (sqrt(((-1.0e0) + (1.0e0 / u1))) * ((0.15915494309188485e0 / u2) + (u2 * 1.0471975511966667e0)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(1.0) / Float32(sqrt(Float32(Float32(-1.0) + Float32(Float32(1.0) / u1))) * Float32(Float32(Float32(0.15915494309188485) / u2) + Float32(u2 * Float32(1.0471975511966667)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(1.0) / (sqrt((single(-1.0) + (single(1.0) / u1))) * ((single(0.15915494309188485) / u2) + (u2 * single(1.0471975511966667))));
end

Derivation

1. Initial program 98.4%

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

   1. log1p-expm1-u 98.4%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(6.28318530718 \cdot u2\right)\right)\right)} \]

3. Applied egg-rr 98.4%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(6.28318530718 \cdot u2\right)\right)\right)} \]
4. Step-by-step derivation

   1. clear-num 98.4%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{1 - u1}{u1}}}} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(6.28318530718 \cdot u2\right)\right)\right) \]

   2. sqrt-div 98.3%

      \[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{1 - u1}{u1}}}} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(6.28318530718 \cdot u2\right)\right)\right) \]

   3. metadata-eval 98.3%

      \[\leadsto \frac{\color{blue}{1}}{\sqrt{\frac{1 - u1}{u1}}} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(6.28318530718 \cdot u2\right)\right)\right) \]

   4. log1p-expm1-u 98.3%

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

   5. add-sqr-sqrt 97.6%

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

   6. associate-*r* 97.5%

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

   7. metadata-eval 97.5%

      \[\leadsto \frac{1}{\sqrt{\frac{1 - u1}{u1}}} \cdot \sin \left(\left(\color{blue}{\sqrt{39.47841760436263}} \cdot \sqrt{u2}\right) \cdot \sqrt{u2}\right) \]

   8. sqrt-prod 98.2%

      \[\leadsto \frac{1}{\sqrt{\frac{1 - u1}{u1}}} \cdot \sin \left(\color{blue}{\sqrt{39.47841760436263 \cdot u2}} \cdot \sqrt{u2}\right) \]

   9. sqrt-prod 98.3%

      \[\leadsto \frac{1}{\sqrt{\frac{1 - u1}{u1}}} \cdot \sin \color{blue}{\left(\sqrt{\left(39.47841760436263 \cdot u2\right) \cdot u2}\right)} \]

   10. associate-*l/ 98.3%

       \[\leadsto \color{blue}{\frac{1 \cdot \sin \left(\sqrt{\left(39.47841760436263 \cdot u2\right) \cdot u2}\right)}{\sqrt{\frac{1 - u1}{u1}}}} \]

   11. associate-/l* 98.2%

       \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\frac{1 - u1}{u1}}}{\sin \left(\sqrt{\left(39.47841760436263 \cdot u2\right) \cdot u2}\right)}}} \]

   12. div-sub 98.1%

       \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\frac{1}{u1} - \frac{u1}{u1}}}}{\sin \left(\sqrt{\left(39.47841760436263 \cdot u2\right) \cdot u2}\right)}} \]

   13. *-inverses 98.1%

       \[\leadsto \frac{1}{\frac{\sqrt{\frac{1}{u1} - \color{blue}{1}}}{\sin \left(\sqrt{\left(39.47841760436263 \cdot u2\right) \cdot u2}\right)}} \]

   14. sub-neg 98.1%

       \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\frac{1}{u1} + \left(-1\right)}}}{\sin \left(\sqrt{\left(39.47841760436263 \cdot u2\right) \cdot u2}\right)}} \]

   15. metadata-eval 98.1%

       \[\leadsto \frac{1}{\frac{\sqrt{\frac{1}{u1} + \color{blue}{-1}}}{\sin \left(\sqrt{\left(39.47841760436263 \cdot u2\right) \cdot u2}\right)}} \]

   16. sqrt-prod 98.0%

       \[\leadsto \frac{1}{\frac{\sqrt{\frac{1}{u1} + -1}}{\sin \color{blue}{\left(\sqrt{39.47841760436263 \cdot u2} \cdot \sqrt{u2}\right)}}} \]

5. Applied egg-rr 98.2%

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

6. Taylor expanded in u2 around 0 89.3%

   \[\leadsto \frac{1}{\color{blue}{0.15915494309188485 \cdot \left(\frac{1}{u2} \cdot \sqrt{\frac{1}{u1} - 1}\right) + 1.0471975511966667 \cdot \left(u2 \cdot \sqrt{\frac{1}{u1} - 1}\right)}} \]
7. Step-by-step derivation

   1. associate-*r* 89.4%

      \[\leadsto \frac{1}{\color{blue}{\left(0.15915494309188485 \cdot \frac{1}{u2}\right) \cdot \sqrt{\frac{1}{u1} - 1}} + 1.0471975511966667 \cdot \left(u2 \cdot \sqrt{\frac{1}{u1} - 1}\right)} \]

   2. sub-neg 89.4%

      \[\leadsto \frac{1}{\left(0.15915494309188485 \cdot \frac{1}{u2}\right) \cdot \sqrt{\color{blue}{\frac{1}{u1} + \left(-1\right)}} + 1.0471975511966667 \cdot \left(u2 \cdot \sqrt{\frac{1}{u1} - 1}\right)} \]

   3. metadata-eval 89.4%

      \[\leadsto \frac{1}{\left(0.15915494309188485 \cdot \frac{1}{u2}\right) \cdot \sqrt{\frac{1}{u1} + \color{blue}{-1}} + 1.0471975511966667 \cdot \left(u2 \cdot \sqrt{\frac{1}{u1} - 1}\right)} \]

   4. +-commutative 89.4%

      \[\leadsto \frac{1}{\left(0.15915494309188485 \cdot \frac{1}{u2}\right) \cdot \sqrt{\color{blue}{-1 + \frac{1}{u1}}} + 1.0471975511966667 \cdot \left(u2 \cdot \sqrt{\frac{1}{u1} - 1}\right)} \]

   5. associate-*r* 89.4%

      \[\leadsto \frac{1}{\left(0.15915494309188485 \cdot \frac{1}{u2}\right) \cdot \sqrt{-1 + \frac{1}{u1}} + \color{blue}{\left(1.0471975511966667 \cdot u2\right) \cdot \sqrt{\frac{1}{u1} - 1}}} \]

   6. sub-neg 89.4%

      \[\leadsto \frac{1}{\left(0.15915494309188485 \cdot \frac{1}{u2}\right) \cdot \sqrt{-1 + \frac{1}{u1}} + \left(1.0471975511966667 \cdot u2\right) \cdot \sqrt{\color{blue}{\frac{1}{u1} + \left(-1\right)}}} \]

   7. metadata-eval 89.4%

      \[\leadsto \frac{1}{\left(0.15915494309188485 \cdot \frac{1}{u2}\right) \cdot \sqrt{-1 + \frac{1}{u1}} + \left(1.0471975511966667 \cdot u2\right) \cdot \sqrt{\frac{1}{u1} + \color{blue}{-1}}} \]

   8. +-commutative 89.4%

      \[\leadsto \frac{1}{\left(0.15915494309188485 \cdot \frac{1}{u2}\right) \cdot \sqrt{-1 + \frac{1}{u1}} + \left(1.0471975511966667 \cdot u2\right) \cdot \sqrt{\color{blue}{-1 + \frac{1}{u1}}}} \]

   9. distribute-rgt-out 89.4%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{-1 + \frac{1}{u1}} \cdot \left(0.15915494309188485 \cdot \frac{1}{u2} + 1.0471975511966667 \cdot u2\right)}} \]

   10. associate-*r/ 89.5%

       \[\leadsto \frac{1}{\sqrt{-1 + \frac{1}{u1}} \cdot \left(\color{blue}{\frac{0.15915494309188485 \cdot 1}{u2}} + 1.0471975511966667 \cdot u2\right)} \]

   11. metadata-eval 89.5%

       \[\leadsto \frac{1}{\sqrt{-1 + \frac{1}{u1}} \cdot \left(\frac{\color{blue}{0.15915494309188485}}{u2} + 1.0471975511966667 \cdot u2\right)} \]

8. Simplified 89.5%

   \[\leadsto \frac{1}{\color{blue}{\sqrt{-1 + \frac{1}{u1}} \cdot \left(\frac{0.15915494309188485}{u2} + 1.0471975511966667 \cdot u2\right)}} \]

9. Final simplification 89.5%

   \[\leadsto \frac{1}{\sqrt{-1 + \frac{1}{u1}} \cdot \left(\frac{0.15915494309188485}{u2} + u2 \cdot 1.0471975511966667\right)} \]

Alternative 5: 81.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1} \cdot \left(u2 \cdot \left(u2 \cdot 39.47841760436263\right)\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (sqrt (* (/ u1 (- 1.0 u1)) (* u2 (* u2 39.47841760436263)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(((u1 / (1.0f - u1)) * (u2 * (u2 * 39.47841760436263f))));
}

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 * (u2 * 39.47841760436263e0))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(Float32(u1 / Float32(Float32(1.0) - u1)) * Float32(u2 * Float32(u2 * Float32(39.47841760436263)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(((u1 / (single(1.0) - u1)) * (u2 * (u2 * single(39.47841760436263)))));
end

Derivation

1. Initial program 98.4%

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

2. Taylor expanded in u2 around 0 83.4%

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

   1. add-sqr-sqrt 83.0%

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

   2. sqrt-unprod 83.4%

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

   3. associate-*r* 83.4%

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

   4. associate-*r* 83.5%

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

   5. swap-sqr 83.5%

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

   6. swap-sqr 83.3%

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

   7. metadata-eval 83.5%

      \[\leadsto \sqrt{\left(\color{blue}{39.47841760436263} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   8. associate-*l* 83.5%

      \[\leadsto \sqrt{\color{blue}{\left(\left(39.47841760436263 \cdot u2\right) \cdot u2\right)} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   9. *-commutative 83.5%

      \[\leadsto \sqrt{\color{blue}{\left(u2 \cdot \left(39.47841760436263 \cdot u2\right)\right)} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   10. *-commutative 83.5%

       \[\leadsto \sqrt{\left(u2 \cdot \color{blue}{\left(u2 \cdot 39.47841760436263\right)}\right) \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   11. add-sqr-sqrt 83.8%

       \[\leadsto \sqrt{\left(u2 \cdot \left(u2 \cdot 39.47841760436263\right)\right) \cdot \color{blue}{\frac{u1}{1 - u1}}} \]

4. Applied egg-rr 83.8%

   \[\leadsto \color{blue}{\sqrt{\left(u2 \cdot \left(u2 \cdot 39.47841760436263\right)\right) \cdot \frac{u1}{1 - u1}}} \]
5. Step-by-step derivation

   1. *-commutative 83.8%

      \[\leadsto \sqrt{\color{blue}{\frac{u1}{1 - u1} \cdot \left(u2 \cdot \left(u2 \cdot 39.47841760436263\right)\right)}} \]

6. Simplified 83.8%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1} \cdot \left(u2 \cdot \left(u2 \cdot 39.47841760436263\right)\right)}} \]

7. Final simplification 83.8%

   \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \left(u2 \cdot \left(u2 \cdot 39.47841760436263\right)\right)} \]

Alternative 6: 81.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1} \cdot \left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (sqrt (* (/ u1 (- 1.0 u1)) (* 39.47841760436263 (* u2 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(((u1 / (1.0f - u1)) * (39.47841760436263f * (u2 * 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)) * (39.47841760436263e0 * (u2 * u2))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(Float32(u1 / Float32(Float32(1.0) - u1)) * Float32(Float32(39.47841760436263) * Float32(u2 * u2))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(((u1 / (single(1.0) - u1)) * (single(39.47841760436263) * (u2 * u2))));
end

Derivation

1. Initial program 98.4%

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

2. Taylor expanded in u2 around 0 83.4%

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

   1. add-sqr-sqrt 83.0%

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

   2. sqrt-unprod 83.4%

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

   3. associate-*r* 83.4%

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

   4. associate-*r* 83.5%

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

   5. swap-sqr 83.5%

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

   6. swap-sqr 83.3%

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

   7. metadata-eval 83.5%

      \[\leadsto \sqrt{\left(\color{blue}{39.47841760436263} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   8. associate-*l* 83.5%

      \[\leadsto \sqrt{\color{blue}{\left(\left(39.47841760436263 \cdot u2\right) \cdot u2\right)} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   9. *-commutative 83.5%

      \[\leadsto \sqrt{\color{blue}{\left(u2 \cdot \left(39.47841760436263 \cdot u2\right)\right)} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   10. *-commutative 83.5%

       \[\leadsto \sqrt{\left(u2 \cdot \color{blue}{\left(u2 \cdot 39.47841760436263\right)}\right) \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   11. add-sqr-sqrt 83.8%

       \[\leadsto \sqrt{\left(u2 \cdot \left(u2 \cdot 39.47841760436263\right)\right) \cdot \color{blue}{\frac{u1}{1 - u1}}} \]

4. Applied egg-rr 83.8%

   \[\leadsto \color{blue}{\sqrt{\left(u2 \cdot \left(u2 \cdot 39.47841760436263\right)\right) \cdot \frac{u1}{1 - u1}}} \]
5. Step-by-step derivation

   1. *-commutative 83.8%

      \[\leadsto \sqrt{\color{blue}{\frac{u1}{1 - u1} \cdot \left(u2 \cdot \left(u2 \cdot 39.47841760436263\right)\right)}} \]

   2. associate-*r* 83.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \color{blue}{\left(\left(u2 \cdot u2\right) \cdot 39.47841760436263\right)}} \]

6. Simplified 83.8%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1} \cdot \left(\left(u2 \cdot u2\right) \cdot 39.47841760436263\right)}} \]

7. Final simplification 83.8%

   \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)} \]

Alternative 7: 81.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1 \cdot \left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}{1 - u1}} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (sqrt (/ (* u1 (* 39.47841760436263 (* u2 u2))) (- 1.0 u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(((u1 * (39.47841760436263f * (u2 * u2))) / (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 = sqrt(((u1 * (39.47841760436263e0 * (u2 * u2))) / (1.0e0 - u1)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(Float32(u1 * Float32(Float32(39.47841760436263) * Float32(u2 * u2))) / Float32(Float32(1.0) - u1)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(((u1 * (single(39.47841760436263) * (u2 * u2))) / (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. Taylor expanded in u2 around 0 83.4%

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

   1. add-sqr-sqrt 83.0%

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

   2. sqrt-unprod 83.4%

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

   3. associate-*r* 83.4%

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

   4. associate-*r* 83.5%

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

   5. swap-sqr 83.5%

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

   6. swap-sqr 83.3%

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

   7. metadata-eval 83.5%

      \[\leadsto \sqrt{\left(\color{blue}{39.47841760436263} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   8. associate-*l* 83.5%

      \[\leadsto \sqrt{\color{blue}{\left(\left(39.47841760436263 \cdot u2\right) \cdot u2\right)} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   9. *-commutative 83.5%

      \[\leadsto \sqrt{\color{blue}{\left(u2 \cdot \left(39.47841760436263 \cdot u2\right)\right)} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   10. *-commutative 83.5%

       \[\leadsto \sqrt{\left(u2 \cdot \color{blue}{\left(u2 \cdot 39.47841760436263\right)}\right) \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   11. add-sqr-sqrt 83.8%

       \[\leadsto \sqrt{\left(u2 \cdot \left(u2 \cdot 39.47841760436263\right)\right) \cdot \color{blue}{\frac{u1}{1 - u1}}} \]

4. Applied egg-rr 83.8%

   \[\leadsto \color{blue}{\sqrt{\left(u2 \cdot \left(u2 \cdot 39.47841760436263\right)\right) \cdot \frac{u1}{1 - u1}}} \]
5. Step-by-step derivation

   1. *-commutative 83.8%

      \[\leadsto \sqrt{\color{blue}{\frac{u1}{1 - u1} \cdot \left(u2 \cdot \left(u2 \cdot 39.47841760436263\right)\right)}} \]

6. Simplified 83.8%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1} \cdot \left(u2 \cdot \left(u2 \cdot 39.47841760436263\right)\right)}} \]
7. Step-by-step derivation

   1. associate-*l/ 83.8%

      \[\leadsto \sqrt{\color{blue}{\frac{u1 \cdot \left(u2 \cdot \left(u2 \cdot 39.47841760436263\right)\right)}{1 - u1}}} \]

   2. *-commutative 83.8%

      \[\leadsto \sqrt{\frac{u1 \cdot \color{blue}{\left(\left(u2 \cdot 39.47841760436263\right) \cdot u2\right)}}{1 - u1}} \]

   3. *-commutative 83.8%

      \[\leadsto \sqrt{\frac{u1 \cdot \left(\color{blue}{\left(39.47841760436263 \cdot u2\right)} \cdot u2\right)}{1 - u1}} \]

   4. associate-*l* 83.9%

      \[\leadsto \sqrt{\frac{u1 \cdot \color{blue}{\left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}}{1 - u1}} \]

8. Applied egg-rr 83.9%

   \[\leadsto \sqrt{\color{blue}{\frac{u1 \cdot \left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}{1 - u1}}} \]

9. Final simplification 83.9%

   \[\leadsto \sqrt{\frac{u1 \cdot \left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}{1 - u1}} \]

Alternative 8: 81.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

2. Taylor expanded in u2 around 0 83.4%

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

3. Final simplification 83.4%

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

Alternative 9: 81.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

2. Taylor expanded in u2 around 0 83.4%

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

   1. *-commutative 83.4%

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

   2. associate-*l* 83.5%

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

4. Simplified 83.5%

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

5. Final simplification 83.5%

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

Alternative 10: 64.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{39.47841760436263 \cdot \left(u1 \cdot \left(u2 \cdot u2\right)\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (sqrt (* 39.47841760436263 (* u1 (* u2 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((39.47841760436263f * (u1 * (u2 * 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((39.47841760436263e0 * (u1 * (u2 * u2))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(Float32(39.47841760436263) * Float32(u1 * Float32(u2 * u2))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((single(39.47841760436263) * (u1 * (u2 * u2))));
end

Derivation

1. Initial program 98.4%

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

2. Taylor expanded in u2 around 0 83.4%

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

3. Taylor expanded in u1 around 0 67.7%

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

   1. add-sqr-sqrt 67.5%

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

   2. sqrt-unprod 67.7%

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

   3. swap-sqr 67.7%

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

   4. metadata-eval 67.5%

      \[\leadsto \sqrt{\color{blue}{39.47841760436263} \cdot \left(\left(u2 \cdot \sqrt{u1}\right) \cdot \left(u2 \cdot \sqrt{u1}\right)\right)} \]

   5. *-commutative 67.5%

      \[\leadsto \sqrt{39.47841760436263 \cdot \left(\color{blue}{\left(\sqrt{u1} \cdot u2\right)} \cdot \left(u2 \cdot \sqrt{u1}\right)\right)} \]

   6. *-commutative 67.5%

      \[\leadsto \sqrt{39.47841760436263 \cdot \left(\left(\sqrt{u1} \cdot u2\right) \cdot \color{blue}{\left(\sqrt{u1} \cdot u2\right)}\right)} \]

   7. swap-sqr 67.6%

      \[\leadsto \sqrt{39.47841760436263 \cdot \color{blue}{\left(\left(\sqrt{u1} \cdot \sqrt{u1}\right) \cdot \left(u2 \cdot u2\right)\right)}} \]

   8. add-sqr-sqrt 67.8%

      \[\leadsto \sqrt{39.47841760436263 \cdot \left(\color{blue}{u1} \cdot \left(u2 \cdot u2\right)\right)} \]

5. Applied egg-rr 67.8%

   \[\leadsto \color{blue}{\sqrt{39.47841760436263 \cdot \left(u1 \cdot \left(u2 \cdot u2\right)\right)}} \]

6. Final simplification 67.8%

   \[\leadsto \sqrt{39.47841760436263 \cdot \left(u1 \cdot \left(u2 \cdot u2\right)\right)} \]

Alternative 11: 64.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1 \cdot \left(u2 \cdot \left(u2 \cdot 39.47841760436263\right)\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (sqrt (* u1 (* u2 (* u2 39.47841760436263)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 * (u2 * (u2 * 39.47841760436263f))));
}

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 * (u2 * 39.47841760436263e0))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(u1 * Float32(u2 * Float32(u2 * Float32(39.47841760436263)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 * (u2 * (u2 * single(39.47841760436263)))));
end

Derivation

1. Initial program 98.4%

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

2. Taylor expanded in u2 around 0 83.4%

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

   1. add-sqr-sqrt 83.0%

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

   2. sqrt-unprod 83.4%

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

   3. associate-*r* 83.4%

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

   4. associate-*r* 83.5%

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

   5. swap-sqr 83.5%

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

   6. swap-sqr 83.3%

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

   7. metadata-eval 83.5%

      \[\leadsto \sqrt{\left(\color{blue}{39.47841760436263} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   8. associate-*l* 83.5%

      \[\leadsto \sqrt{\color{blue}{\left(\left(39.47841760436263 \cdot u2\right) \cdot u2\right)} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   9. *-commutative 83.5%

      \[\leadsto \sqrt{\color{blue}{\left(u2 \cdot \left(39.47841760436263 \cdot u2\right)\right)} \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   10. *-commutative 83.5%

       \[\leadsto \sqrt{\left(u2 \cdot \color{blue}{\left(u2 \cdot 39.47841760436263\right)}\right) \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

   11. add-sqr-sqrt 83.8%

       \[\leadsto \sqrt{\left(u2 \cdot \left(u2 \cdot 39.47841760436263\right)\right) \cdot \color{blue}{\frac{u1}{1 - u1}}} \]

4. Applied egg-rr 83.8%

   \[\leadsto \color{blue}{\sqrt{\left(u2 \cdot \left(u2 \cdot 39.47841760436263\right)\right) \cdot \frac{u1}{1 - u1}}} \]
5. Step-by-step derivation

   1. *-commutative 83.8%

      \[\leadsto \sqrt{\color{blue}{\frac{u1}{1 - u1} \cdot \left(u2 \cdot \left(u2 \cdot 39.47841760436263\right)\right)}} \]

6. Simplified 83.8%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1} \cdot \left(u2 \cdot \left(u2 \cdot 39.47841760436263\right)\right)}} \]

7. Taylor expanded in u1 around 0 67.8%

   \[\leadsto \sqrt{\color{blue}{u1} \cdot \left(u2 \cdot \left(u2 \cdot 39.47841760436263\right)\right)} \]

8. Final simplification 67.8%

   \[\leadsto \sqrt{u1 \cdot \left(u2 \cdot \left(u2 \cdot 39.47841760436263\right)\right)} \]

Alternative 12: 64.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1 \cdot \left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (sqrt (* u1 (* 39.47841760436263 (* u2 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 * (39.47841760436263f * (u2 * 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 * (39.47841760436263e0 * (u2 * u2))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(u1 * Float32(Float32(39.47841760436263) * Float32(u2 * u2))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 * (single(39.47841760436263) * (u2 * u2))));
end

Derivation

1. Initial program 98.4%

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

2. Taylor expanded in u2 around 0 83.4%

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

3. Taylor expanded in u1 around 0 67.7%

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

   1. add-sqr-sqrt 67.5%

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

   2. sqrt-unprod 67.7%

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

   3. swap-sqr 67.7%

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

   4. metadata-eval 67.5%

      \[\leadsto \sqrt{\color{blue}{39.47841760436263} \cdot \left(\left(u2 \cdot \sqrt{u1}\right) \cdot \left(u2 \cdot \sqrt{u1}\right)\right)} \]

   5. *-commutative 67.5%

      \[\leadsto \sqrt{39.47841760436263 \cdot \left(\color{blue}{\left(\sqrt{u1} \cdot u2\right)} \cdot \left(u2 \cdot \sqrt{u1}\right)\right)} \]

   6. *-commutative 67.5%

      \[\leadsto \sqrt{39.47841760436263 \cdot \left(\left(\sqrt{u1} \cdot u2\right) \cdot \color{blue}{\left(\sqrt{u1} \cdot u2\right)}\right)} \]

   7. swap-sqr 67.6%

      \[\leadsto \sqrt{39.47841760436263 \cdot \color{blue}{\left(\left(\sqrt{u1} \cdot \sqrt{u1}\right) \cdot \left(u2 \cdot u2\right)\right)}} \]

   8. add-sqr-sqrt 67.8%

      \[\leadsto \sqrt{39.47841760436263 \cdot \left(\color{blue}{u1} \cdot \left(u2 \cdot u2\right)\right)} \]

5. Applied egg-rr 67.8%

   \[\leadsto \color{blue}{\sqrt{39.47841760436263 \cdot \left(u1 \cdot \left(u2 \cdot u2\right)\right)}} \]
6. Step-by-step derivation

   1. *-commutative 67.8%

      \[\leadsto \sqrt{\color{blue}{\left(u1 \cdot \left(u2 \cdot u2\right)\right) \cdot 39.47841760436263}} \]

   2. unpow2 67.8%

      \[\leadsto \sqrt{\left(u1 \cdot \color{blue}{{u2}^{2}}\right) \cdot 39.47841760436263} \]

   3. associate-*l* 67.8%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left({u2}^{2} \cdot 39.47841760436263\right)}} \]

   4. unpow2 67.8%

      \[\leadsto \sqrt{u1 \cdot \left(\color{blue}{\left(u2 \cdot u2\right)} \cdot 39.47841760436263\right)} \]

7. Simplified 67.8%

   \[\leadsto \color{blue}{\sqrt{u1 \cdot \left(\left(u2 \cdot u2\right) \cdot 39.47841760436263\right)}} \]

8. Final simplification 67.8%

   \[\leadsto \sqrt{u1 \cdot \left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)} \]

Alternative 13: 64.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(u2 \cdot u2\right) \cdot \left(u1 \cdot 39.47841760436263\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (sqrt (* (* u2 u2) (* u1 39.47841760436263))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(((u2 * u2) * (u1 * 39.47841760436263f)));
}

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(((u2 * u2) * (u1 * 39.47841760436263e0)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(Float32(u2 * u2) * Float32(u1 * Float32(39.47841760436263))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(((u2 * u2) * (u1 * single(39.47841760436263))));
end

Derivation

1. Initial program 98.4%

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

2. Taylor expanded in u2 around 0 83.4%

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

3. Taylor expanded in u1 around 0 67.7%

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

   1. add-sqr-sqrt 67.5%

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

   2. sqrt-unprod 67.7%

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

   3. swap-sqr 67.7%

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

   4. metadata-eval 67.5%

      \[\leadsto \sqrt{\color{blue}{39.47841760436263} \cdot \left(\left(u2 \cdot \sqrt{u1}\right) \cdot \left(u2 \cdot \sqrt{u1}\right)\right)} \]

   5. *-commutative 67.5%

      \[\leadsto \sqrt{39.47841760436263 \cdot \left(\color{blue}{\left(\sqrt{u1} \cdot u2\right)} \cdot \left(u2 \cdot \sqrt{u1}\right)\right)} \]

   6. *-commutative 67.5%

      \[\leadsto \sqrt{39.47841760436263 \cdot \left(\left(\sqrt{u1} \cdot u2\right) \cdot \color{blue}{\left(\sqrt{u1} \cdot u2\right)}\right)} \]

   7. swap-sqr 67.6%

      \[\leadsto \sqrt{39.47841760436263 \cdot \color{blue}{\left(\left(\sqrt{u1} \cdot \sqrt{u1}\right) \cdot \left(u2 \cdot u2\right)\right)}} \]

   8. add-sqr-sqrt 67.8%

      \[\leadsto \sqrt{39.47841760436263 \cdot \left(\color{blue}{u1} \cdot \left(u2 \cdot u2\right)\right)} \]

5. Applied egg-rr 67.8%

   \[\leadsto \color{blue}{\sqrt{39.47841760436263 \cdot \left(u1 \cdot \left(u2 \cdot u2\right)\right)}} \]
6. Step-by-step derivation

   1. unpow2 67.8%

      \[\leadsto \sqrt{39.47841760436263 \cdot \left(u1 \cdot \color{blue}{{u2}^{2}}\right)} \]

   2. associate-*r* 67.8%

      \[\leadsto \sqrt{\color{blue}{\left(39.47841760436263 \cdot u1\right) \cdot {u2}^{2}}} \]

   3. unpow2 67.8%

      \[\leadsto \sqrt{\left(39.47841760436263 \cdot u1\right) \cdot \color{blue}{\left(u2 \cdot u2\right)}} \]

7. Simplified 67.8%

   \[\leadsto \color{blue}{\sqrt{\left(39.47841760436263 \cdot u1\right) \cdot \left(u2 \cdot u2\right)}} \]

8. Final simplification 67.8%

   \[\leadsto \sqrt{\left(u2 \cdot u2\right) \cdot \left(u1 \cdot 39.47841760436263\right)} \]

Alternative 14: 64.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

2. Taylor expanded in u2 around 0 83.4%

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

3. Taylor expanded in u1 around 0 67.7%

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

4. Final simplification 67.7%

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

Alternative 15: 64.7% accurate, 2.0× 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 98.4%

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

2. Taylor expanded in u1 around 0 74.9%

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

3. Taylor expanded in u2 around 0 67.7%

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

4. Final simplification 67.7%

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

Reproduce

herbie shell --seed 2023250 
(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))))