Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.4%

Time: 12.6s

Alternatives: 21

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: 98.4% 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 98.1%

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

   1. add-sqr-sqrt 97.8%

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

   2. sqrt-unprod 98.1%

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

   3. swap-sqr 98.0%

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

   4. metadata-eval 98.4%

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

3. Applied egg-rr 98.4%

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

4. Final simplification 98.4%

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

Alternative 2: 90.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.014999999664723873:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1} \cdot \left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(u2 \cdot 6.28318530718\right)}{\sqrt{\frac{1}{u1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.014999999664723873)
   (sqrt (* (/ u1 (- 1.0 u1)) (* 39.47841760436263 (* u2 u2))))
   (/ (sin (* u2 6.28318530718)) (sqrt (/ 1.0 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.014999999664723873f) {
		tmp = sqrtf(((u1 / (1.0f - u1)) * (39.47841760436263f * (u2 * u2))));
	} else {
		tmp = sinf((u2 * 6.28318530718f)) / sqrtf((1.0f / 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) <= 0.014999999664723873e0) then
        tmp = sqrt(((u1 / (1.0e0 - u1)) * (39.47841760436263e0 * (u2 * u2))))
    else
        tmp = sin((u2 * 6.28318530718e0)) / sqrt((1.0e0 / 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(0.014999999664723873))
		tmp = sqrt(Float32(Float32(u1 / Float32(Float32(1.0) - u1)) * Float32(Float32(39.47841760436263) * Float32(u2 * u2))));
	else
		tmp = Float32(sin(Float32(u2 * Float32(6.28318530718))) / sqrt(Float32(Float32(1.0) / u1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.014999999664723873))
		tmp = sqrt(((u1 / (single(1.0) - u1)) * (single(39.47841760436263) * (u2 * u2))));
	else
		tmp = sin((u2 * single(6.28318530718))) / sqrt((single(1.0) / u1));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.0149999997

   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 96.6%

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

      1. associate-*r* 96.6%

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

   4. Simplified 96.6%

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

      1. add-sqr-sqrt 96.1%

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

      2. sqrt-unprod 96.6%

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

      3. *-commutative 96.6%

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

      4. *-commutative 96.6%

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

      5. swap-sqr 96.4%

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

      6. add-sqr-sqrt 96.4%

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

      7. pow2 96.4%

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

      8. *-commutative 96.4%

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

   6. Applied egg-rr 96.4%

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

      1. unpow2 96.4%

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

      2. swap-sqr 96.6%

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

      3. metadata-eval 97.1%

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

   8. Simplified 97.1%

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

   if 0.0149999997 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.3%

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

      1. add-sqr-sqrt 97.3%

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

      2. sqrt-unprod 97.3%

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

      3. swap-sqr 97.2%

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

      4. metadata-eval 97.7%

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

   3. Applied egg-rr 97.7%

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

      1. expm1-log1p-u 97.6%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right)\right)\right)} \]

      2. expm1-udef 94.3%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right)\right)} - 1\right)} \]

      3. *-commutative 94.3%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(e^{\mathsf{log1p}\left(\sin \left(\sqrt{\color{blue}{\left(u2 \cdot u2\right) \cdot 39.47841760436263}}\right)\right)} - 1\right) \]

      4. sqrt-prod 94.2%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(e^{\mathsf{log1p}\left(\sin \color{blue}{\left(\sqrt{u2 \cdot u2} \cdot \sqrt{39.47841760436263}\right)}\right)} - 1\right) \]

      5. sqrt-prod 93.8%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(e^{\mathsf{log1p}\left(\sin \left(\color{blue}{\left(\sqrt{u2} \cdot \sqrt{u2}\right)} \cdot \sqrt{39.47841760436263}\right)\right)} - 1\right) \]

      6. add-sqr-sqrt 94.2%

         \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(e^{\mathsf{log1p}\left(\sin \left(\color{blue}{u2} \cdot \sqrt{39.47841760436263}\right)\right)} - 1\right) \]

      7. metadata-eval 94.2%

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

   5. Applied egg-rr 94.2%

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

      1. expm1-def 97.2%

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

      2. expm1-log1p-u 97.3%

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

      3. *-commutative 97.3%

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

      4. clear-num 97.3%

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

      5. sqrt-div 97.2%

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

      6. metadata-eval 97.2%

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

      7. sqrt-undiv 97.1%

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

      8. *-commutative 97.1%

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

      9. div-inv 97.2%

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

      10. *-commutative 97.2%

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

      11. sqrt-undiv 97.6%

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

   7. Applied egg-rr 97.6%

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

   8. Taylor expanded in u1 around 0 77.0%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.014999999664723873:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1} \cdot \left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(u2 \cdot 6.28318530718\right)}{\sqrt{\frac{1}{u1}}}\\ \end{array} \]

Alternative 3: 90.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.009999999776482582:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1} \cdot \left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.009999999776482582)
   (sqrt (* (/ u1 (- 1.0 u1)) (* 39.47841760436263 (* u2 u2))))
   (* (sin (* u2 6.28318530718)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.009999999776482582f) {
		tmp = sqrtf(((u1 / (1.0f - u1)) * (39.47841760436263f * (u2 * u2))));
	} else {
		tmp = sinf((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) <= 0.009999999776482582e0) then
        tmp = sqrt(((u1 / (1.0e0 - u1)) * (39.47841760436263e0 * (u2 * u2))))
    else
        tmp = sin((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(0.009999999776482582))
		tmp = sqrt(Float32(Float32(u1 / Float32(Float32(1.0) - u1)) * Float32(Float32(39.47841760436263) * Float32(u2 * u2))));
	else
		tmp = Float32(sin(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(0.009999999776482582))
		tmp = sqrt(((u1 / (single(1.0) - u1)) * (single(39.47841760436263) * (u2 * u2))));
	else
		tmp = sin((u2 * single(6.28318530718))) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.00999999978

   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 96.9%

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

      1. associate-*r* 96.9%

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

   4. Simplified 96.9%

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

      1. add-sqr-sqrt 96.3%

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

      2. sqrt-unprod 96.9%

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

      3. *-commutative 96.9%

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

      4. *-commutative 96.9%

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

      5. swap-sqr 96.7%

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

      6. add-sqr-sqrt 96.7%

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

      7. pow2 96.7%

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

      8. *-commutative 96.7%

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

   6. Applied egg-rr 96.7%

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

      1. unpow2 96.7%

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

      2. swap-sqr 96.9%

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

      3. metadata-eval 97.3%

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

   8. Simplified 97.3%

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

   if 0.00999999978 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.4%

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

   2. Taylor expanded in u1 around 0 77.0%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.009999999776482582:\\ \;\;\;\;\sqrt{\frac{u1}{1 - u1} \cdot \left(39.47841760436263 \cdot \left(u2 \cdot u2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

2. Final simplification 98.1%

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

Alternative 5: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. add-sqr-sqrt 97.8%

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

   2. sqrt-unprod 98.1%

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

   3. swap-sqr 98.0%

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

   4. metadata-eval 98.4%

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

3. Applied egg-rr 98.4%

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

   1. expm1-log1p-u 98.4%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right)\right)\right)} \]

   2. expm1-udef 61.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right)\right)} - 1\right)} \]

   3. *-commutative 61.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(e^{\mathsf{log1p}\left(\sin \left(\sqrt{\color{blue}{\left(u2 \cdot u2\right) \cdot 39.47841760436263}}\right)\right)} - 1\right) \]

   4. sqrt-prod 61.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(e^{\mathsf{log1p}\left(\sin \color{blue}{\left(\sqrt{u2 \cdot u2} \cdot \sqrt{39.47841760436263}\right)}\right)} - 1\right) \]

   5. sqrt-prod 61.7%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(e^{\mathsf{log1p}\left(\sin \left(\color{blue}{\left(\sqrt{u2} \cdot \sqrt{u2}\right)} \cdot \sqrt{39.47841760436263}\right)\right)} - 1\right) \]

   6. add-sqr-sqrt 61.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(e^{\mathsf{log1p}\left(\sin \left(\color{blue}{u2} \cdot \sqrt{39.47841760436263}\right)\right)} - 1\right) \]

   7. metadata-eval 61.8%

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

5. Applied egg-rr 61.8%

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

   1. expm1-def 98.1%

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

   2. expm1-log1p-u 98.1%

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

   3. *-commutative 98.1%

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

   4. clear-num 98.1%

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

   5. sqrt-div 98.1%

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

   6. metadata-eval 98.1%

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

   7. sqrt-undiv 97.9%

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

   8. *-commutative 97.9%

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

   9. div-inv 97.9%

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

   10. *-commutative 97.9%

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

   11. sqrt-undiv 98.3%

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

7. Applied egg-rr 98.3%

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

8. Taylor expanded in u1 around 0 98.2%

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

9. Final simplification 98.2%

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

Alternative 6: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. add-sqr-sqrt 97.8%

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

   2. sqrt-unprod 98.1%

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

   3. swap-sqr 98.0%

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

   4. metadata-eval 98.4%

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

3. Applied egg-rr 98.4%

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

   1. expm1-log1p-u 98.4%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right)\right)\right)} \]

   2. expm1-udef 61.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right)\right)} - 1\right)} \]

   3. *-commutative 61.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(e^{\mathsf{log1p}\left(\sin \left(\sqrt{\color{blue}{\left(u2 \cdot u2\right) \cdot 39.47841760436263}}\right)\right)} - 1\right) \]

   4. sqrt-prod 61.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(e^{\mathsf{log1p}\left(\sin \color{blue}{\left(\sqrt{u2 \cdot u2} \cdot \sqrt{39.47841760436263}\right)}\right)} - 1\right) \]

   5. sqrt-prod 61.7%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(e^{\mathsf{log1p}\left(\sin \left(\color{blue}{\left(\sqrt{u2} \cdot \sqrt{u2}\right)} \cdot \sqrt{39.47841760436263}\right)\right)} - 1\right) \]

   6. add-sqr-sqrt 61.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(e^{\mathsf{log1p}\left(\sin \left(\color{blue}{u2} \cdot \sqrt{39.47841760436263}\right)\right)} - 1\right) \]

   7. metadata-eval 61.8%

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

5. Applied egg-rr 61.8%

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

   1. expm1-def 98.1%

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

   2. expm1-log1p-u 98.1%

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

   3. *-commutative 98.1%

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

   4. clear-num 98.1%

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

   5. sqrt-div 98.1%

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

   6. metadata-eval 98.1%

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

   7. sqrt-undiv 97.9%

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

   8. *-commutative 97.9%

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

   9. div-inv 97.9%

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

   10. *-commutative 97.9%

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

   11. sqrt-undiv 98.3%

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

7. Applied egg-rr 98.3%

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

8. Final simplification 98.3%

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

Alternative 7: 82.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((39.47841760436263e0 * (u2 * ((u1 / (1.0e0 - u1)) * u2))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

2. Taylor expanded in u2 around 0 78.7%

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

   1. add-sqr-sqrt 78.4%

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

   2. sqrt-unprod 78.7%

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

   3. swap-sqr 78.6%

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

   4. metadata-eval 78.8%

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

   5. swap-sqr 78.9%

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

   6. add-sqr-sqrt 79.0%

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

4. Applied egg-rr 79.0%

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

   1. associate-*l* 79.0%

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

6. Simplified 79.0%

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

7. Final simplification 79.0%

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

Alternative 8: 82.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt(((u1 / (1.0e0 - u1)) * (39.47841760436263e0 * (u2 * u2))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

2. Taylor expanded in u2 around 0 78.7%

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

   1. associate-*r* 78.7%

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

4. Simplified 78.7%

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

   1. add-sqr-sqrt 78.4%

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

   2. sqrt-unprod 78.7%

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

   3. *-commutative 78.7%

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

   4. *-commutative 78.7%

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

   5. swap-sqr 78.6%

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

   6. add-sqr-sqrt 78.6%

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

   7. pow2 78.6%

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

   8. *-commutative 78.6%

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

6. Applied egg-rr 78.6%

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

   1. unpow2 78.6%

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

   2. swap-sqr 78.7%

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

   3. metadata-eval 79.0%

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

8. Simplified 79.0%

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

9. Final simplification 79.0%

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

Alternative 9: 81.7% 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 98.1%

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

2. Taylor expanded in u2 around 0 78.7%

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

   1. clear-num 78.7%

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

   2. unpow-1 78.7%

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

   3. sqrt-pow1 78.7%

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

   4. div-sub 78.8%

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

   5. *-inverses 78.8%

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

   6. sub-neg 78.8%

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

   7. metadata-eval 78.8%

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

   8. metadata-eval 78.8%

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

4. Applied egg-rr 78.8%

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

5. Final simplification 78.8%

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

Alternative 10: 81.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return (u2 * 6.28318530718f) * 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 = (u2 * 6.28318530718e0) * (((1.0e0 / u1) + (-1.0e0)) ** (-0.5e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. add-sqr-sqrt 97.8%

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

   2. sqrt-unprod 98.1%

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

   3. swap-sqr 98.0%

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

   4. metadata-eval 98.4%

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

3. Applied egg-rr 98.4%

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

   1. clear-num 78.7%

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

   2. unpow-1 78.7%

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

   3. sqrt-pow1 78.7%

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

   4. div-sub 78.8%

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

   5. *-inverses 78.8%

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

   6. sub-neg 78.8%

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

   7. metadata-eval 78.8%

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

   8. metadata-eval 78.8%

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

5. Applied egg-rr 98.2%

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

6. Taylor expanded in u2 around 0 78.8%

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

   1. *-commutative 78.8%

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

8. Simplified 78.8%

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

9. Final simplification 78.8%

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

Alternative 11: 81.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

2. Taylor expanded in u2 around 0 78.7%

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

3. Final simplification 78.7%

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

Alternative 12: 81.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

2. Taylor expanded in u2 around 0 78.7%

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

   1. *-commutative 78.7%

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

   2. associate-*l* 78.8%

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

4. Simplified 78.8%

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

5. Final simplification 78.8%

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

Alternative 13: 81.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

   1. add-sqr-sqrt 97.8%

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

   2. sqrt-unprod 98.1%

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

   3. swap-sqr 98.0%

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

   4. metadata-eval 98.4%

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

3. Applied egg-rr 98.4%

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

   1. expm1-log1p-u 98.4%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right)\right)\right)} \]

   2. expm1-udef 61.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(\sqrt{39.47841760436263 \cdot \left(u2 \cdot u2\right)}\right)\right)} - 1\right)} \]

   3. *-commutative 61.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(e^{\mathsf{log1p}\left(\sin \left(\sqrt{\color{blue}{\left(u2 \cdot u2\right) \cdot 39.47841760436263}}\right)\right)} - 1\right) \]

   4. sqrt-prod 61.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(e^{\mathsf{log1p}\left(\sin \color{blue}{\left(\sqrt{u2 \cdot u2} \cdot \sqrt{39.47841760436263}\right)}\right)} - 1\right) \]

   5. sqrt-prod 61.7%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(e^{\mathsf{log1p}\left(\sin \left(\color{blue}{\left(\sqrt{u2} \cdot \sqrt{u2}\right)} \cdot \sqrt{39.47841760436263}\right)\right)} - 1\right) \]

   6. add-sqr-sqrt 61.8%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \left(e^{\mathsf{log1p}\left(\sin \left(\color{blue}{u2} \cdot \sqrt{39.47841760436263}\right)\right)} - 1\right) \]

   7. metadata-eval 61.8%

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

5. Applied egg-rr 61.8%

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

   1. expm1-def 98.1%

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

   2. expm1-log1p-u 98.1%

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

   3. *-commutative 98.1%

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

   4. clear-num 98.1%

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

   5. sqrt-div 98.1%

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

   6. metadata-eval 98.1%

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

   7. sqrt-undiv 97.9%

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

   8. *-commutative 97.9%

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

   9. div-inv 97.9%

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

   10. *-commutative 97.9%

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

   11. sqrt-undiv 98.3%

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

7. Applied egg-rr 98.3%

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

8. Taylor expanded in u2 around 0 78.8%

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

9. Final simplification 78.8%

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

Alternative 14: 64.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

2. Taylor expanded in u2 around 0 78.7%

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

3. Taylor expanded in u1 around 0 63.3%

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

4. Final simplification 63.3%

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

Alternative 15: 64.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

2. Taylor expanded in u2 around 0 78.7%

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

3. Taylor expanded in u1 around 0 63.3%

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

   1. expm1-log1p-u 63.3%

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

   2. expm1-udef 27.7%

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

   3. associate-*r* 27.7%

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

   4. *-commutative 27.7%

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

5. Applied egg-rr 27.7%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\right)} - 1} \]
6. Step-by-step derivation

   1. expm1-def 63.4%

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

   2. expm1-log1p 63.4%

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

   3. associate-*l* 63.4%

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

7. Simplified 63.4%

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

8. Final simplification 63.4%

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

Alternative 16: 64.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

2. Taylor expanded in u2 around 0 78.7%

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

   1. associate-*r* 78.7%

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

4. Simplified 78.7%

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

5. Taylor expanded in u1 around 0 63.4%

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

6. Final simplification 63.4%

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

Alternative 17: 20.5% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

2. Taylor expanded in u2 around 0 78.7%

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

3. Taylor expanded in u1 around 0 71.1%

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

   1. unpow2 71.1%

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

   2. fma-udef 71.1%

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

5. Simplified 71.1%

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

6. Taylor expanded in u1 around inf 19.9%

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

7. Final simplification 19.9%

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

Alternative 18: 20.5% accurate, 29.9× speedup

Math

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

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(6.28318530718) * Float32(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 98.1%

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

2. Taylor expanded in u2 around 0 78.7%

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

3. Taylor expanded in u1 around 0 71.1%

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

   1. unpow2 71.1%

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

   2. fma-udef 71.1%

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

5. Simplified 71.1%

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

6. Taylor expanded in u1 around inf 19.9%

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

   1. +-commutative 19.9%

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

   2. *-commutative 19.9%

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

   3. distribute-lft-out 19.9%

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

8. Simplified 19.9%

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

9. Final simplification 19.9%

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

Alternative 19: 4.8% accurate, 41.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

2. Taylor expanded in u2 around 0 78.7%

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

3. Taylor expanded in u1 around 0 71.1%

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

   1. unpow2 71.1%

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

   2. fma-udef 71.1%

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

5. Simplified 71.1%

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

6. Taylor expanded in u1 around -inf 5.2%

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

7. Final simplification 5.2%

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

Alternative 20: 19.4% accurate, 41.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

2. Taylor expanded in u2 around 0 78.7%

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

3. Taylor expanded in u1 around 0 71.1%

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

   1. unpow2 71.1%

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

   2. fma-udef 71.1%

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

5. Simplified 71.1%

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

6. Taylor expanded in u1 around inf 18.8%

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

7. Final simplification 18.8%

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

Alternative 21: 19.4% accurate, 41.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.1%

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

2. Taylor expanded in u2 around 0 78.7%

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

3. Taylor expanded in u1 around 0 71.1%

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

   1. unpow2 71.1%

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

   2. fma-udef 71.1%

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

5. Simplified 71.1%

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

6. Taylor expanded in u1 around inf 18.8%

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

   1. *-commutative 18.8%

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

   2. *-commutative 18.8%

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

   3. associate-*l* 18.8%

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

   4. *-commutative 18.8%

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

8. Simplified 18.8%

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

9. Final simplification 18.8%

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

Reproduce

herbie shell --seed 2023188 
(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))))