Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 12.7s

Alternatives: 14

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, 0.7× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return powf(powf((u1 / (1.0f - u1)), 2.0f), 0.25f) * 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 = (((u1 / (1.0e0 - u1)) ** 2.0e0) ** 0.25e0) * sin((6.28318530718e0 * u2))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. pow1/2 98.5%

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

   2. metadata-eval 98.5%

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

   3. metadata-eval 98.5%

      \[\leadsto {\left(\frac{u1}{1 - u1}\right)}^{\left(\color{blue}{0.5 \cdot 0.5} + 0.25\right)} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   4. metadata-eval 98.5%

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

   5. pow-prod-up 97.8%

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

   6. pow-prod-down 98.5%

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

   7. pow2 98.5%

      \[\leadsto {\color{blue}{\left({\left(\frac{u1}{1 - u1}\right)}^{2}\right)}}^{\left(0.5 \cdot 0.5\right)} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   8. metadata-eval 98.5%

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

3. Applied egg-rr 98.5%

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

4. Final simplification 98.5%

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

Alternative 2: 98.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. add-sqr-sqrt 97.6%

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

   2. pow1/2 97.6%

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

   3. pow1/2 97.6%

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

   4. pow-prod-down 98.5%

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

   5. swap-sqr 98.1%

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

   6. metadata-eval 98.5%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left({\left(\color{blue}{39.47841760436263} \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right) \]

3. Applied egg-rr 98.5%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{\left({\left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}^{0.5}\right)} \]
4. Step-by-step derivation

   1. unpow1/2 98.5%

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

5. Simplified 98.5%

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

6. Final simplification 98.5%

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

Alternative 3: 90.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.004900000058114529:\\ \;\;\;\;\sqrt{39.47841760436263 \cdot \frac{u1 \cdot \left(u2 \cdot u2\right)}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.004900000058114529)
   (sqrt (* 39.47841760436263 (/ (* u1 (* u2 u2)) (- 1.0 u1))))
   (* (sin (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.004900000058114529f) {
		tmp = sqrtf((39.47841760436263f * ((u1 * (u2 * u2)) / (1.0f - u1))));
	} 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) <= 0.004900000058114529e0) then
        tmp = sqrt((39.47841760436263e0 * ((u1 * (u2 * u2)) / (1.0e0 - u1))))
    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(0.004900000058114529))
		tmp = sqrt(Float32(Float32(39.47841760436263) * Float32(Float32(u1 * Float32(u2 * u2)) / Float32(Float32(1.0) - u1))));
	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(0.004900000058114529))
		tmp = sqrt((single(39.47841760436263) * ((u1 * (u2 * u2)) / (single(1.0) - u1))));
	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) < 0.00490000006

   1. Initial program 98.6%

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

   2. Taylor expanded in u2 around 0 97.7%

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

      1. add-sqr-sqrt 97.2%

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

      2. sqrt-unprod 97.7%

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

      3. *-commutative 97.7%

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

      4. *-commutative 97.7%

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

      5. swap-sqr 97.5%

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

      6. swap-sqr 97.6%

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

      7. add-sqr-sqrt 97.8%

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

      8. metadata-eval 98.2%

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

   4. Applied egg-rr 98.2%

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

      1. associate-*l/ 98.3%

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

   6. Simplified 98.3%

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

   if 0.00490000006 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 98.2%

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

   2. Taylor expanded in u1 around 0 78.5%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.004900000058114529:\\ \;\;\;\;\sqrt{39.47841760436263 \cdot \frac{u1 \cdot \left(u2 \cdot u2\right)}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 4: 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 98.5%

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

2. Final simplification 98.5%

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

Alternative 5: 82.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in u2 around 0 84.4%

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

   1. pow1/2 84.4%

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

   2. sqr-pow 84.3%

      \[\leadsto 6.28318530718 \cdot \left(\color{blue}{\left({\left(\frac{u1}{1 - u1}\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\frac{u1}{1 - u1}\right)}^{\left(\frac{0.5}{2}\right)}\right)} \cdot u2\right) \]

   3. clear-num 84.4%

      \[\leadsto 6.28318530718 \cdot \left(\left({\color{blue}{\left(\frac{1}{\frac{1 - u1}{u1}}\right)}}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\frac{u1}{1 - u1}\right)}^{\left(\frac{0.5}{2}\right)}\right) \cdot u2\right) \]

   4. inv-pow 84.4%

      \[\leadsto 6.28318530718 \cdot \left(\left({\color{blue}{\left({\left(\frac{1 - u1}{u1}\right)}^{-1}\right)}}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\frac{u1}{1 - u1}\right)}^{\left(\frac{0.5}{2}\right)}\right) \cdot u2\right) \]

   5. metadata-eval 84.4%

      \[\leadsto 6.28318530718 \cdot \left(\left({\left({\left(\frac{1 - u1}{u1}\right)}^{-1}\right)}^{\color{blue}{0.25}} \cdot {\left(\frac{u1}{1 - u1}\right)}^{\left(\frac{0.5}{2}\right)}\right) \cdot u2\right) \]

   6. metadata-eval 84.4%

      \[\leadsto 6.28318530718 \cdot \left(\left({\left({\left(\frac{1 - u1}{u1}\right)}^{-1}\right)}^{\color{blue}{\left(0.5 \cdot 0.5\right)}} \cdot {\left(\frac{u1}{1 - u1}\right)}^{\left(\frac{0.5}{2}\right)}\right) \cdot u2\right) \]

   7. pow-pow 84.4%

      \[\leadsto 6.28318530718 \cdot \left(\left(\color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{\left(-1 \cdot \left(0.5 \cdot 0.5\right)\right)}} \cdot {\left(\frac{u1}{1 - u1}\right)}^{\left(\frac{0.5}{2}\right)}\right) \cdot u2\right) \]

   8. metadata-eval 84.4%

      \[\leadsto 6.28318530718 \cdot \left(\left({\left(\frac{1 - u1}{u1}\right)}^{\left(-1 \cdot \color{blue}{0.25}\right)} \cdot {\left(\frac{u1}{1 - u1}\right)}^{\left(\frac{0.5}{2}\right)}\right) \cdot u2\right) \]

   9. metadata-eval 84.4%

      \[\leadsto 6.28318530718 \cdot \left(\left({\left(\frac{1 - u1}{u1}\right)}^{\color{blue}{-0.25}} \cdot {\left(\frac{u1}{1 - u1}\right)}^{\left(\frac{0.5}{2}\right)}\right) \cdot u2\right) \]

   10. clear-num 84.3%

       \[\leadsto 6.28318530718 \cdot \left(\left({\left(\frac{1 - u1}{u1}\right)}^{-0.25} \cdot {\color{blue}{\left(\frac{1}{\frac{1 - u1}{u1}}\right)}}^{\left(\frac{0.5}{2}\right)}\right) \cdot u2\right) \]

   11. inv-pow 84.3%

       \[\leadsto 6.28318530718 \cdot \left(\left({\left(\frac{1 - u1}{u1}\right)}^{-0.25} \cdot {\color{blue}{\left({\left(\frac{1 - u1}{u1}\right)}^{-1}\right)}}^{\left(\frac{0.5}{2}\right)}\right) \cdot u2\right) \]

   12. metadata-eval 84.3%

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

   13. metadata-eval 84.3%

       \[\leadsto 6.28318530718 \cdot \left(\left({\left(\frac{1 - u1}{u1}\right)}^{-0.25} \cdot {\left({\left(\frac{1 - u1}{u1}\right)}^{-1}\right)}^{\color{blue}{\left(0.5 \cdot 0.5\right)}}\right) \cdot u2\right) \]

   14. pow-pow 84.3%

       \[\leadsto 6.28318530718 \cdot \left(\left({\left(\frac{1 - u1}{u1}\right)}^{-0.25} \cdot \color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{\left(-1 \cdot \left(0.5 \cdot 0.5\right)\right)}}\right) \cdot u2\right) \]

   15. metadata-eval 84.3%

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

   16. metadata-eval 84.3%

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

4. Applied egg-rr 84.3%

   \[\leadsto 6.28318530718 \cdot \left(\color{blue}{\left({\left(\frac{1 - u1}{u1}\right)}^{-0.25} \cdot {\left(\frac{1 - u1}{u1}\right)}^{-0.25}\right)} \cdot u2\right) \]
5. Step-by-step derivation

   1. pow-sqr 84.5%

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

   2. metadata-eval 84.5%

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

6. Simplified 84.5%

   \[\leadsto 6.28318530718 \cdot \left(\color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{-0.5}} \cdot u2\right) \]
7. Step-by-step derivation

   1. add-sqr-sqrt 84.1%

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

   2. sqrt-unprod 84.5%

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

   3. swap-sqr 84.4%

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

   4. metadata-eval 84.6%

      \[\leadsto \sqrt{\color{blue}{39.47841760436263} \cdot \left(\left({\left(\frac{1 - u1}{u1}\right)}^{-0.5} \cdot u2\right) \cdot \left({\left(\frac{1 - u1}{u1}\right)}^{-0.5} \cdot u2\right)\right)} \]

   5. swap-sqr 84.7%

      \[\leadsto \sqrt{39.47841760436263 \cdot \color{blue}{\left(\left({\left(\frac{1 - u1}{u1}\right)}^{-0.5} \cdot {\left(\frac{1 - u1}{u1}\right)}^{-0.5}\right) \cdot \left(u2 \cdot u2\right)\right)}} \]

   6. pow-prod-up 84.8%

      \[\leadsto \sqrt{39.47841760436263 \cdot \left(\color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{\left(-0.5 + -0.5\right)}} \cdot \left(u2 \cdot u2\right)\right)} \]

   7. metadata-eval 84.8%

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

   8. inv-pow 84.8%

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

   9. clear-num 84.8%

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

8. Applied egg-rr 84.8%

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

   1. *-commutative 84.8%

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

   2. associate-*l/ 84.9%

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

   3. associate-*l/ 84.8%

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

10. Simplified 84.8%

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

11. Taylor expanded in u2 around 0 84.9%

    \[\leadsto \sqrt{\color{blue}{39.47841760436263 \cdot \frac{u1 \cdot {u2}^{2}}{1 - u1}}} \]
12. Step-by-step derivation

    1. associate-*r/ 84.8%

       \[\leadsto \sqrt{\color{blue}{\frac{39.47841760436263 \cdot \left(u1 \cdot {u2}^{2}\right)}{1 - u1}}} \]

    2. *-commutative 84.8%

       \[\leadsto \sqrt{\frac{\color{blue}{\left(u1 \cdot {u2}^{2}\right) \cdot 39.47841760436263}}{1 - u1}} \]

    3. unpow2 84.8%

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

    4. associate-*r* 84.8%

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

    5. associate-*r/ 84.8%

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

    6. associate-*l* 84.8%

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

    7. associate-/l* 84.9%

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

13. Simplified 84.9%

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

14. Final simplification 84.9%

    \[\leadsto \sqrt{u1 \cdot \frac{u2}{\frac{1 - u1}{u2 \cdot 39.47841760436263}}} \]

Alternative 6: 82.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in u2 around 0 84.4%

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

   1. add-sqr-sqrt 84.1%

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

   2. sqrt-unprod 84.4%

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

   3. *-commutative 84.4%

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

   4. *-commutative 84.4%

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

   5. swap-sqr 84.3%

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

   6. swap-sqr 84.4%

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

   7. add-sqr-sqrt 84.5%

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

   8. metadata-eval 84.8%

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

4. Applied egg-rr 84.8%

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

   1. associate-*l/ 84.9%

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

6. Simplified 84.9%

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

7. Final simplification 84.9%

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

Alternative 7: 81.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return 6.28318530718f * (u2 * powf(((1.0f / u1) + -1.0f), -0.5f));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = 6.28318530718e0 * (u2 * (((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0)))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (u2 * (((single(1.0) / u1) + single(-1.0)) ^ single(-0.5)));
end

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in u2 around 0 84.4%

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

   1. pow1/2 84.4%

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

   2. sqr-pow 84.3%

      \[\leadsto 6.28318530718 \cdot \left(\color{blue}{\left({\left(\frac{u1}{1 - u1}\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\frac{u1}{1 - u1}\right)}^{\left(\frac{0.5}{2}\right)}\right)} \cdot u2\right) \]

   3. clear-num 84.4%

      \[\leadsto 6.28318530718 \cdot \left(\left({\color{blue}{\left(\frac{1}{\frac{1 - u1}{u1}}\right)}}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\frac{u1}{1 - u1}\right)}^{\left(\frac{0.5}{2}\right)}\right) \cdot u2\right) \]

   4. inv-pow 84.4%

      \[\leadsto 6.28318530718 \cdot \left(\left({\color{blue}{\left({\left(\frac{1 - u1}{u1}\right)}^{-1}\right)}}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\frac{u1}{1 - u1}\right)}^{\left(\frac{0.5}{2}\right)}\right) \cdot u2\right) \]

   5. metadata-eval 84.4%

      \[\leadsto 6.28318530718 \cdot \left(\left({\left({\left(\frac{1 - u1}{u1}\right)}^{-1}\right)}^{\color{blue}{0.25}} \cdot {\left(\frac{u1}{1 - u1}\right)}^{\left(\frac{0.5}{2}\right)}\right) \cdot u2\right) \]

   6. metadata-eval 84.4%

      \[\leadsto 6.28318530718 \cdot \left(\left({\left({\left(\frac{1 - u1}{u1}\right)}^{-1}\right)}^{\color{blue}{\left(0.5 \cdot 0.5\right)}} \cdot {\left(\frac{u1}{1 - u1}\right)}^{\left(\frac{0.5}{2}\right)}\right) \cdot u2\right) \]

   7. pow-pow 84.4%

      \[\leadsto 6.28318530718 \cdot \left(\left(\color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{\left(-1 \cdot \left(0.5 \cdot 0.5\right)\right)}} \cdot {\left(\frac{u1}{1 - u1}\right)}^{\left(\frac{0.5}{2}\right)}\right) \cdot u2\right) \]

   8. metadata-eval 84.4%

      \[\leadsto 6.28318530718 \cdot \left(\left({\left(\frac{1 - u1}{u1}\right)}^{\left(-1 \cdot \color{blue}{0.25}\right)} \cdot {\left(\frac{u1}{1 - u1}\right)}^{\left(\frac{0.5}{2}\right)}\right) \cdot u2\right) \]

   9. metadata-eval 84.4%

      \[\leadsto 6.28318530718 \cdot \left(\left({\left(\frac{1 - u1}{u1}\right)}^{\color{blue}{-0.25}} \cdot {\left(\frac{u1}{1 - u1}\right)}^{\left(\frac{0.5}{2}\right)}\right) \cdot u2\right) \]

   10. clear-num 84.3%

       \[\leadsto 6.28318530718 \cdot \left(\left({\left(\frac{1 - u1}{u1}\right)}^{-0.25} \cdot {\color{blue}{\left(\frac{1}{\frac{1 - u1}{u1}}\right)}}^{\left(\frac{0.5}{2}\right)}\right) \cdot u2\right) \]

   11. inv-pow 84.3%

       \[\leadsto 6.28318530718 \cdot \left(\left({\left(\frac{1 - u1}{u1}\right)}^{-0.25} \cdot {\color{blue}{\left({\left(\frac{1 - u1}{u1}\right)}^{-1}\right)}}^{\left(\frac{0.5}{2}\right)}\right) \cdot u2\right) \]

   12. metadata-eval 84.3%

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

   13. metadata-eval 84.3%

       \[\leadsto 6.28318530718 \cdot \left(\left({\left(\frac{1 - u1}{u1}\right)}^{-0.25} \cdot {\left({\left(\frac{1 - u1}{u1}\right)}^{-1}\right)}^{\color{blue}{\left(0.5 \cdot 0.5\right)}}\right) \cdot u2\right) \]

   14. pow-pow 84.3%

       \[\leadsto 6.28318530718 \cdot \left(\left({\left(\frac{1 - u1}{u1}\right)}^{-0.25} \cdot \color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{\left(-1 \cdot \left(0.5 \cdot 0.5\right)\right)}}\right) \cdot u2\right) \]

   15. metadata-eval 84.3%

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

   16. metadata-eval 84.3%

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

4. Applied egg-rr 84.3%

   \[\leadsto 6.28318530718 \cdot \left(\color{blue}{\left({\left(\frac{1 - u1}{u1}\right)}^{-0.25} \cdot {\left(\frac{1 - u1}{u1}\right)}^{-0.25}\right)} \cdot u2\right) \]
5. Step-by-step derivation

   1. pow-sqr 84.5%

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

   2. metadata-eval 84.5%

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

   3. div-sub 84.5%

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

   4. *-inverses 84.5%

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

   5. sub-neg 84.5%

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

   6. metadata-eval 84.5%

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

6. Simplified 84.5%

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

7. Final simplification 84.5%

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

Alternative 8: 81.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return 6.28318530718f * (u2 * powf(((1.0f - u1) / u1), -0.5f));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = 6.28318530718e0 * (u2 * (((1.0e0 - u1) / u1) ** (-0.5e0)))
end function

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (u2 * (((single(1.0) - u1) / u1) ^ single(-0.5)));
end

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in u2 around 0 84.4%

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

   1. pow1/2 84.4%

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

   2. sqr-pow 84.3%

      \[\leadsto 6.28318530718 \cdot \left(\color{blue}{\left({\left(\frac{u1}{1 - u1}\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\frac{u1}{1 - u1}\right)}^{\left(\frac{0.5}{2}\right)}\right)} \cdot u2\right) \]

   3. clear-num 84.4%

      \[\leadsto 6.28318530718 \cdot \left(\left({\color{blue}{\left(\frac{1}{\frac{1 - u1}{u1}}\right)}}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\frac{u1}{1 - u1}\right)}^{\left(\frac{0.5}{2}\right)}\right) \cdot u2\right) \]

   4. inv-pow 84.4%

      \[\leadsto 6.28318530718 \cdot \left(\left({\color{blue}{\left({\left(\frac{1 - u1}{u1}\right)}^{-1}\right)}}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\frac{u1}{1 - u1}\right)}^{\left(\frac{0.5}{2}\right)}\right) \cdot u2\right) \]

   5. metadata-eval 84.4%

      \[\leadsto 6.28318530718 \cdot \left(\left({\left({\left(\frac{1 - u1}{u1}\right)}^{-1}\right)}^{\color{blue}{0.25}} \cdot {\left(\frac{u1}{1 - u1}\right)}^{\left(\frac{0.5}{2}\right)}\right) \cdot u2\right) \]

   6. metadata-eval 84.4%

      \[\leadsto 6.28318530718 \cdot \left(\left({\left({\left(\frac{1 - u1}{u1}\right)}^{-1}\right)}^{\color{blue}{\left(0.5 \cdot 0.5\right)}} \cdot {\left(\frac{u1}{1 - u1}\right)}^{\left(\frac{0.5}{2}\right)}\right) \cdot u2\right) \]

   7. pow-pow 84.4%

      \[\leadsto 6.28318530718 \cdot \left(\left(\color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{\left(-1 \cdot \left(0.5 \cdot 0.5\right)\right)}} \cdot {\left(\frac{u1}{1 - u1}\right)}^{\left(\frac{0.5}{2}\right)}\right) \cdot u2\right) \]

   8. metadata-eval 84.4%

      \[\leadsto 6.28318530718 \cdot \left(\left({\left(\frac{1 - u1}{u1}\right)}^{\left(-1 \cdot \color{blue}{0.25}\right)} \cdot {\left(\frac{u1}{1 - u1}\right)}^{\left(\frac{0.5}{2}\right)}\right) \cdot u2\right) \]

   9. metadata-eval 84.4%

      \[\leadsto 6.28318530718 \cdot \left(\left({\left(\frac{1 - u1}{u1}\right)}^{\color{blue}{-0.25}} \cdot {\left(\frac{u1}{1 - u1}\right)}^{\left(\frac{0.5}{2}\right)}\right) \cdot u2\right) \]

   10. clear-num 84.3%

       \[\leadsto 6.28318530718 \cdot \left(\left({\left(\frac{1 - u1}{u1}\right)}^{-0.25} \cdot {\color{blue}{\left(\frac{1}{\frac{1 - u1}{u1}}\right)}}^{\left(\frac{0.5}{2}\right)}\right) \cdot u2\right) \]

   11. inv-pow 84.3%

       \[\leadsto 6.28318530718 \cdot \left(\left({\left(\frac{1 - u1}{u1}\right)}^{-0.25} \cdot {\color{blue}{\left({\left(\frac{1 - u1}{u1}\right)}^{-1}\right)}}^{\left(\frac{0.5}{2}\right)}\right) \cdot u2\right) \]

   12. metadata-eval 84.3%

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

   13. metadata-eval 84.3%

       \[\leadsto 6.28318530718 \cdot \left(\left({\left(\frac{1 - u1}{u1}\right)}^{-0.25} \cdot {\left({\left(\frac{1 - u1}{u1}\right)}^{-1}\right)}^{\color{blue}{\left(0.5 \cdot 0.5\right)}}\right) \cdot u2\right) \]

   14. pow-pow 84.3%

       \[\leadsto 6.28318530718 \cdot \left(\left({\left(\frac{1 - u1}{u1}\right)}^{-0.25} \cdot \color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{\left(-1 \cdot \left(0.5 \cdot 0.5\right)\right)}}\right) \cdot u2\right) \]

   15. metadata-eval 84.3%

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

   16. metadata-eval 84.3%

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

4. Applied egg-rr 84.3%

   \[\leadsto 6.28318530718 \cdot \left(\color{blue}{\left({\left(\frac{1 - u1}{u1}\right)}^{-0.25} \cdot {\left(\frac{1 - u1}{u1}\right)}^{-0.25}\right)} \cdot u2\right) \]
5. Step-by-step derivation

   1. pow-sqr 84.5%

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

   2. metadata-eval 84.5%

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

6. Simplified 84.5%

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

7. Final simplification 84.5%

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

Alternative 9: 81.8% accurate, 1.9× speedup

Math

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

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(6.28318530718) * Float32(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 98.5%

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

2. Taylor expanded in u2 around 0 84.4%

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

3. Final simplification 84.4%

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

Alternative 10: 81.7% accurate, 1.9× 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 98.5%

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

2. Taylor expanded in u2 around 0 84.4%

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

   1. *-commutative 84.4%

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

   2. associate-*r* 84.4%

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

4. Simplified 84.4%

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

5. Final simplification 84.4%

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

Alternative 11: 64.8% 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 98.5%

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

2. Taylor expanded in u2 around 0 84.4%

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

3. Taylor expanded in u1 around 0 65.8%

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

4. Final simplification 65.8%

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

Alternative 12: 20.6% accurate, 29.9× speedup

Math

\[\begin{array}{l} \\ u2 \cdot \left(u1 \cdot 6.28318530718 + 3.14159265359\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* u2 (+ (* u1 6.28318530718) 3.14159265359)))

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 * ((u1 * 6.28318530718f) + 3.14159265359f);
}

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 * ((u1 * 6.28318530718e0) + 3.14159265359e0)
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u2 * Float32(Float32(u1 * Float32(6.28318530718)) + Float32(3.14159265359)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u2 * ((u1 * single(6.28318530718)) + single(3.14159265359));
end

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in u1 around 0 86.8%

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

   1. +-commutative 86.8%

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

   2. unpow2 86.8%

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

   3. fma-udef 86.8%

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

4. Simplified 86.8%

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

5. Taylor expanded in u2 around 0 74.9%

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

   1. associate-*r* 74.8%

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

   2. +-commutative 74.8%

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

   3. unpow2 74.8%

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

   4. fma-udef 74.8%

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

7. Simplified 74.8%

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

8. Taylor expanded in u1 around inf 20.4%

   \[\leadsto \color{blue}{3.14159265359 \cdot u2 + 6.28318530718 \cdot \left(u1 \cdot u2\right)} \]
9. Step-by-step derivation

   1. +-commutative 20.4%

      \[\leadsto \color{blue}{6.28318530718 \cdot \left(u1 \cdot u2\right) + 3.14159265359 \cdot u2} \]

   2. associate-*r* 20.4%

      \[\leadsto \color{blue}{\left(6.28318530718 \cdot u1\right) \cdot u2} + 3.14159265359 \cdot u2 \]

   3. distribute-rgt-out 20.4%

      \[\leadsto \color{blue}{u2 \cdot \left(6.28318530718 \cdot u1 + 3.14159265359\right)} \]

10. Simplified 20.4%

    \[\leadsto \color{blue}{u2 \cdot \left(6.28318530718 \cdot u1 + 3.14159265359\right)} \]

11. Final simplification 20.4%

    \[\leadsto u2 \cdot \left(u1 \cdot 6.28318530718 + 3.14159265359\right) \]

Alternative 13: 20.6% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (single(6.28318530718) * u2) * (u1 + single(0.5));
end

Derivation

1. Initial program 98.5%

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

2. Taylor expanded in u1 around 0 86.8%

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

   1. +-commutative 86.8%

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

   2. unpow2 86.8%

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

   3. fma-udef 86.8%

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

4. Simplified 86.8%

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

5. Taylor expanded in u2 around 0 74.9%

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

   1. associate-*r* 74.8%

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

   2. +-commutative 74.8%

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

   3. unpow2 74.8%

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

   4. fma-udef 74.8%

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

7. Simplified 74.8%

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

8. Taylor expanded in u1 around inf 20.4%

   \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \color{blue}{\left(0.5 + u1\right)} \]
9. Step-by-step derivation

   1. +-commutative 20.4%

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

10. Simplified 20.4%

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

11. Final simplification 20.4%

    \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \left(u1 + 0.5\right) \]

Alternative 14: 19.6% accurate, 69.7× speedup

Math

\[\begin{array}{l} \\ 6.28318530718 \cdot u2 \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. pow1/2 98.5%

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

   2. metadata-eval 98.5%

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

   3. metadata-eval 98.5%

      \[\leadsto {\left(\frac{u1}{1 - u1}\right)}^{\left(\color{blue}{0.5 \cdot 0.5} + 0.25\right)} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   4. metadata-eval 98.5%

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

   5. pow-prod-up 97.8%

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

   6. pow-prod-down 98.5%

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

   7. pow2 98.5%

      \[\leadsto {\color{blue}{\left({\left(\frac{u1}{1 - u1}\right)}^{2}\right)}}^{\left(0.5 \cdot 0.5\right)} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   8. metadata-eval 98.5%

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

3. Applied egg-rr 98.5%

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

4. Taylor expanded in u1 around -inf 20.1%

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

5. Taylor expanded in u2 around 0 19.4%

   \[\leadsto \color{blue}{6.28318530718 \cdot u2} \]

6. Final simplification 19.4%

   \[\leadsto 6.28318530718 \cdot u2 \]

Reproduce

herbie shell --seed 2023292 
(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))))