Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 9.5s

Alternatives: 9

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

   1. add-sqr-sqrt 98.9%

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

   2. pow1/2 98.9%

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

   3. pow1/2 98.9%

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

   4. pow-prod-down 99.0%

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

   5. swap-sqr 98.9%

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

   6. metadata-eval 99.1%

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

3. Applied egg-rr 99.1%

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

   1. unpow1/2 99.1%

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

   2. unpow2 99.1%

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

   3. *-commutative 99.1%

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

   4. unpow2 99.1%

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

5. Simplified 99.1%

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

6. Final simplification 99.1%

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

Alternative 2: 99.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

   1. add-sqr-sqrt 98.9%

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

   2. pow1/2 98.9%

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

   3. pow1/2 98.9%

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

   4. pow-prod-down 99.0%

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

   5. swap-sqr 98.9%

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

   6. metadata-eval 99.1%

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

3. Applied egg-rr 99.1%

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

   1. unpow1/2 99.1%

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

   2. associate-*r* 99.1%

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

   3. *-commutative 99.1%

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

   4. *-commutative 99.1%

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

5. Simplified 99.1%

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

6. Final simplification 99.1%

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

Alternative 3: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \cos \left(u2 \cdot {39.47841760436263}^{0.5}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (cos (* u2 (pow 39.47841760436263 0.5)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * cosf((u2 * powf(39.47841760436263f, 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 = sqrt((u1 / (1.0e0 - u1))) * cos((u2 * (39.47841760436263e0 ** 0.5e0)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * cos(Float32(u2 * (Float32(39.47841760436263) ^ Float32(0.5)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * cos((u2 * (single(39.47841760436263) ^ single(0.5))));
end

Derivation

1. Initial program 99.0%

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

   1. add-sqr-sqrt 98.9%

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

   2. pow1/2 98.9%

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

   3. pow1/2 98.9%

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

   4. pow-prod-down 99.0%

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

   5. swap-sqr 98.9%

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

   6. metadata-eval 99.1%

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

3. Applied egg-rr 99.1%

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

   1. unpow1/2 99.1%

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

   2. unpow2 99.1%

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

   3. *-commutative 99.1%

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

   4. unpow2 99.1%

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

5. Simplified 99.1%

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

   1. pow1/2 99.1%

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

   2. *-commutative 99.1%

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

   3. unpow-prod-down 99.1%

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

   4. pow1/2 99.1%

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

   5. sqrt-prod 99.0%

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

   6. add-sqr-sqrt 99.1%

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

7. Applied egg-rr 99.1%

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

8. Final simplification 99.1%

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

Alternative 4: 94.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.10000000149011612:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.10000000149011612)
   (* (sqrt (/ u1 (- 1.0 u1))) (+ 1.0 (* (* u2 u2) -19.739208802181317)))
   (* (cos (* u2 6.28318530718)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.10000000149011612f) {
		tmp = sqrtf((u1 / (1.0f - u1))) * (1.0f + ((u2 * u2) * -19.739208802181317f));
	} else {
		tmp = cosf((u2 * 6.28318530718f)) * 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 ((u2 * 6.28318530718e0) <= 0.10000000149011612e0) then
        tmp = sqrt((u1 / (1.0e0 - u1))) * (1.0e0 + ((u2 * u2) * (-19.739208802181317e0)))
    else
        tmp = cos((u2 * 6.28318530718e0)) * sqrt(u1)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.10000000149011612))
		tmp = Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(1.0) + Float32(Float32(u2 * u2) * Float32(-19.739208802181317))));
	else
		tmp = Float32(cos(Float32(u2 * Float32(6.28318530718))) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.10000000149011612))
		tmp = sqrt((u1 / (single(1.0) - u1))) * (single(1.0) + ((u2 * u2) * single(-19.739208802181317)));
	else
		tmp = cos((u2 * single(6.28318530718))) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.100000001

   1. Initial program 99.4%

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

      1. add-sqr-sqrt 99.4%

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

      2. pow1/2 99.4%

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

      3. pow1/2 99.4%

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

      4. pow-prod-down 99.4%

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

      5. clear-num 99.3%

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

      6. clear-num 99.3%

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

      7. inv-pow 99.3%

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

      8. inv-pow 99.3%

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

      9. pow-prod-up 99.3%

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

      10. div-sub 99.2%

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

      11. inv-pow 99.2%

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

      12. pow1 99.2%

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

      13. pow1 99.2%

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

      14. pow-div 99.2%

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

      15. metadata-eval 99.2%

          \[\leadsto \sqrt{{\left({\left({u1}^{-1} - {u1}^{\color{blue}{0}}\right)}^{\left(-1 + -1\right)}\right)}^{0.5}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

      16. metadata-eval 99.2%

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

      17. metadata-eval 99.2%

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

   3. Applied egg-rr 99.2%

      \[\leadsto \sqrt{\color{blue}{{\left({\left({u1}^{-1} - 1\right)}^{-2}\right)}^{0.5}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
   4. Step-by-step derivation

      1. exp-to-pow 96.4%

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

      2. *-commutative 96.4%

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

      3. log-pow 96.4%

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

      4. associate-*r* 96.4%

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

      5. metadata-eval 96.4%

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

      6. *-commutative 96.4%

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

      7. exp-to-pow 99.2%

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

      8. unpow-1 99.2%

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

      9. unpow-1 99.2%

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

      10. sub-neg 99.2%

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

      11. metadata-eval 99.2%

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

      12. +-commutative 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in u2 around 0 98.8%

      \[\leadsto \color{blue}{-19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}\right) + \sqrt{\frac{1}{\frac{1}{u1} - 1}}} \]
   7. Step-by-step derivation

      1. associate-*r* 98.8%

         \[\leadsto \color{blue}{\left(-19.739208802181317 \cdot {u2}^{2}\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} + \sqrt{\frac{1}{\frac{1}{u1} - 1}} \]

      2. distribute-lft1-in 98.8%

         \[\leadsto \color{blue}{\left(-19.739208802181317 \cdot {u2}^{2} + 1\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} \]

      3. unpow2 98.8%

         \[\leadsto \left(-19.739208802181317 \cdot \color{blue}{\left(u2 \cdot u2\right)} + 1\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}} \]

      4. *-inverses 98.8%

         \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - \color{blue}{\frac{u1}{u1}}}} \]

      5. div-sub 98.8%

         \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{1}{\color{blue}{\frac{1 - u1}{u1}}}} \]

      6. *-lft-identity 98.8%

         \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{1}{\color{blue}{1 \cdot \frac{1 - u1}{u1}}}} \]

      7. *-lft-identity 98.8%

         \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{1}{\color{blue}{\frac{1 - u1}{u1}}}} \]

      8. associate-/l* 99.0%

         \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\color{blue}{\frac{1 \cdot u1}{1 - u1}}} \]

      9. *-lft-identity 99.0%

         \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{\color{blue}{u1}}{1 - u1}} \]

   8. Simplified 99.0%

      \[\leadsto \color{blue}{\left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

   if 0.100000001 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.1%

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

   2. Taylor expanded in u1 around 0 74.3%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.10000000149011612:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 5: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Final simplification 99.0%

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

Alternative 6: 88.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (+ 1.0 (* (* u2 u2) -19.739208802181317))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * (1.0f + ((u2 * u2) * -19.739208802181317f));
}

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))) * (1.0e0 + ((u2 * u2) * (-19.739208802181317e0)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * Float32(Float32(1.0) + Float32(Float32(u2 * u2) * Float32(-19.739208802181317))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * (single(1.0) + ((u2 * u2) * single(-19.739208802181317)));
end

Derivation

1. Initial program 99.0%

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

   1. add-sqr-sqrt 99.0%

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

   2. pow1/2 99.0%

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

   3. pow1/2 99.0%

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

   4. pow-prod-down 99.0%

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

   5. clear-num 99.0%

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

   6. clear-num 99.0%

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

   7. inv-pow 99.0%

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

   8. inv-pow 99.0%

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

   9. pow-prod-up 99.0%

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

   10. div-sub 98.8%

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

   11. inv-pow 98.8%

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

   12. pow1 98.8%

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

   13. pow1 98.8%

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

   14. pow-div 98.8%

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

   15. metadata-eval 98.8%

       \[\leadsto \sqrt{{\left({\left({u1}^{-1} - {u1}^{\color{blue}{0}}\right)}^{\left(-1 + -1\right)}\right)}^{0.5}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]

   16. metadata-eval 98.8%

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

   17. metadata-eval 98.8%

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

3. Applied egg-rr 98.8%

   \[\leadsto \sqrt{\color{blue}{{\left({\left({u1}^{-1} - 1\right)}^{-2}\right)}^{0.5}}} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
4. Step-by-step derivation

   1. exp-to-pow 96.1%

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

   2. *-commutative 96.1%

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

   3. log-pow 96.1%

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

   4. associate-*r* 96.1%

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

   5. metadata-eval 96.1%

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

   6. *-commutative 96.1%

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

   7. exp-to-pow 98.9%

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

   8. unpow-1 98.9%

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

   9. unpow-1 98.9%

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

   10. sub-neg 98.9%

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

   11. metadata-eval 98.9%

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

   12. +-commutative 98.9%

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

5. Simplified 98.9%

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

6. Taylor expanded in u2 around 0 89.7%

   \[\leadsto \color{blue}{-19.739208802181317 \cdot \left({u2}^{2} \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}\right) + \sqrt{\frac{1}{\frac{1}{u1} - 1}}} \]
7. Step-by-step derivation

   1. associate-*r* 89.7%

      \[\leadsto \color{blue}{\left(-19.739208802181317 \cdot {u2}^{2}\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} + \sqrt{\frac{1}{\frac{1}{u1} - 1}} \]

   2. distribute-lft1-in 89.6%

      \[\leadsto \color{blue}{\left(-19.739208802181317 \cdot {u2}^{2} + 1\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}}} \]

   3. unpow2 89.6%

      \[\leadsto \left(-19.739208802181317 \cdot \color{blue}{\left(u2 \cdot u2\right)} + 1\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - 1}} \]

   4. *-inverses 89.6%

      \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} - \color{blue}{\frac{u1}{u1}}}} \]

   5. div-sub 89.7%

      \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{1}{\color{blue}{\frac{1 - u1}{u1}}}} \]

   6. *-lft-identity 89.7%

      \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{1}{\color{blue}{1 \cdot \frac{1 - u1}{u1}}}} \]

   7. *-lft-identity 89.7%

      \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{1}{\color{blue}{\frac{1 - u1}{u1}}}} \]

   8. associate-/l* 89.8%

      \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\color{blue}{\frac{1 \cdot u1}{1 - u1}}} \]

   9. *-lft-identity 89.8%

      \[\leadsto \left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{\color{blue}{u1}}{1 - u1}} \]

8. Simplified 89.8%

   \[\leadsto \color{blue}{\left(-19.739208802181317 \cdot \left(u2 \cdot u2\right) + 1\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \]

9. Final simplification 89.8%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(1 + \left(u2 \cdot u2\right) \cdot -19.739208802181317\right) \]

Alternative 7: 80.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in u2 around 0 82.3%

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

3. Final simplification 82.3%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \]

Alternative 8: 63.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return 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 = sqrt(u1)
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(u1)
end

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in u2 around 0 82.3%

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

3. Taylor expanded in u1 around 0 64.8%

   \[\leadsto \sqrt{\color{blue}{u1}} \]

4. Final simplification 64.8%

   \[\leadsto \sqrt{u1} \]

Alternative 9: 19.1% accurate, 209.0× speedup

Math

\[\begin{array}{l} \\ u1 \end{array} \]

FPCore

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

C

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

Julia

function code(cosTheta_i, u1, u2)
	return u1
end

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in u1 around 0 86.0%

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

   1. unpow2 86.0%

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

   2. fma-udef 86.0%

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

4. Simplified 86.0%

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

5. Taylor expanded in u2 around 0 73.0%

   \[\leadsto \color{blue}{\sqrt{{u1}^{2} + u1}} \]
6. Step-by-step derivation

   1. unpow2 73.0%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot u1} + u1} \]

   2. unpow1 73.0%

      \[\leadsto \sqrt{u1 \cdot u1 + \color{blue}{{u1}^{1}}} \]

   3. sqr-pow 72.8%

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

   4. hypot-def 72.8%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(u1, {u1}^{\left(\frac{1}{2}\right)}\right)} \]

   5. metadata-eval 72.8%

      \[\leadsto \mathsf{hypot}\left(u1, {u1}^{\color{blue}{0.5}}\right) \]

   6. unpow1/2 72.8%

      \[\leadsto \mathsf{hypot}\left(u1, \color{blue}{\sqrt{u1}}\right) \]

7. Simplified 72.8%

   \[\leadsto \color{blue}{\mathsf{hypot}\left(u1, \sqrt{u1}\right)} \]

8. Taylor expanded in u1 around inf 19.0%

   \[\leadsto \color{blue}{u1} \]

9. Final simplification 19.0%

   \[\leadsto u1 \]

Reproduce

herbie shell --seed 2023230 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_x"
  :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))) (cos (* 6.28318530718 u2))))