Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.5%

Time: 12.2s

Alternatives: 14

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.5% 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 98.4%

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

   1. add-sqr-sqrt 97.6%

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

   2. pow1/2 97.6%

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

   3. pow1/2 97.6%

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

   4. pow-prod-down 98.4%

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

   5. swap-sqr 98.1%

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

   6. metadata-eval 98.6%

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

3. Applied egg-rr 98.6%

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

   1. unpow1/2 98.6%

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

5. Simplified 98.6%

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

6. Final simplification 98.6%

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

Alternative 2: 94.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.0035000001080334187:\\ \;\;\;\;\sqrt{39.47841760436263 \cdot \left(\frac{u1}{1 - u1} \cdot \left(u2 \cdot u2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 + u1 \cdot u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.0035000001080334187)
   (sqrt (* 39.47841760436263 (* (/ u1 (- 1.0 u1)) (* u2 u2))))
   (* (sin (* u2 6.28318530718)) (sqrt (+ u1 (* u1 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.0035000001080334187f) {
		tmp = sqrtf((39.47841760436263f * ((u1 / (1.0f - u1)) * (u2 * u2))));
	} else {
		tmp = sinf((u2 * 6.28318530718f)) * sqrtf((u1 + (u1 * 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) <= 0.0035000001080334187e0) then
        tmp = sqrt((39.47841760436263e0 * ((u1 / (1.0e0 - u1)) * (u2 * u2))))
    else
        tmp = sin((u2 * 6.28318530718e0)) * sqrt((u1 + (u1 * 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(0.0035000001080334187))
		tmp = sqrt(Float32(Float32(39.47841760436263) * Float32(Float32(u1 / Float32(Float32(1.0) - u1)) * Float32(u2 * u2))));
	else
		tmp = Float32(sin(Float32(u2 * Float32(6.28318530718))) * sqrt(Float32(u1 + Float32(u1 * u1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.0035000001080334187))
		tmp = sqrt((single(39.47841760436263) * ((u1 / (single(1.0) - u1)) * (u2 * u2))));
	else
		tmp = sin((u2 * single(6.28318530718))) * sqrt((u1 + (u1 * u1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.00350000011

   1. Initial program 98.3%

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

   2. Taylor expanded in u2 around 0 97.2%

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

      1. sqrt-div 97.1%

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

   4. Applied egg-rr 97.1%

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

      1. sqrt-undiv 97.2%

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

      2. add-cube-cbrt 97.0%

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

      3. unpow3 97.0%

         \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \sqrt{\color{blue}{{\left(\sqrt[3]{\frac{u1}{1 - u1}}\right)}^{3}}}\right) \]

      4. add-exp-log 92.1%

         \[\leadsto 6.28318530718 \cdot \color{blue}{e^{\log \left(u2 \cdot \sqrt{{\left(\sqrt[3]{\frac{u1}{1 - u1}}\right)}^{3}}\right)}} \]

      5. unpow3 92.1%

         \[\leadsto 6.28318530718 \cdot e^{\log \left(u2 \cdot \sqrt{\color{blue}{\left(\sqrt[3]{\frac{u1}{1 - u1}} \cdot \sqrt[3]{\frac{u1}{1 - u1}}\right) \cdot \sqrt[3]{\frac{u1}{1 - u1}}}}\right)} \]

      6. add-cube-cbrt 92.1%

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

      7. sqrt-undiv 92.2%

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

      8. clear-num 92.1%

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

      9. un-div-inv 92.1%

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

      10. sqrt-undiv 92.1%

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

   6. Applied egg-rr 92.1%

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

      1. add-sqr-sqrt 92.2%

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

      2. sqrt-unprod 92.1%

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

      3. *-commutative 92.1%

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

      4. *-commutative 92.1%

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

      5. swap-sqr 92.1%

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

   8. Applied egg-rr 97.9%

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

   if 0.00350000011 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 98.7%

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

      1. flip-- 98.6%

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

      2. associate-/r/ 98.6%

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

      3. metadata-eval 98.6%

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

      4. +-commutative 98.6%

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

   3. Applied egg-rr 98.6%

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

   4. Taylor expanded in u1 around 0 86.8%

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

      1. +-commutative 86.8%

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

      2. unpow2 86.8%

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

   6. Simplified 86.8%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.0035000001080334187:\\ \;\;\;\;\sqrt{39.47841760436263 \cdot \left(\frac{u1}{1 - u1} \cdot \left(u2 \cdot u2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 + u1 \cdot u1}\\ \end{array} \]

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. add-cbrt-cube 98.4%

      \[\leadsto \sqrt{\color{blue}{\sqrt[3]{\left(\frac{u1}{1 - u1} \cdot \frac{u1}{1 - u1}\right) \cdot \frac{u1}{1 - u1}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   2. pow3 98.4%

      \[\leadsto \sqrt{\sqrt[3]{\color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{3}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   3. clear-num 98.3%

      \[\leadsto \sqrt{\sqrt[3]{{\color{blue}{\left(\frac{1}{\frac{1 - u1}{u1}}\right)}}^{3}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   4. inv-pow 98.3%

      \[\leadsto \sqrt{\sqrt[3]{{\color{blue}{\left({\left(\frac{1 - u1}{u1}\right)}^{-1}\right)}}^{3}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   5. pow-pow 98.3%

      \[\leadsto \sqrt{\sqrt[3]{\color{blue}{{\left(\frac{1 - u1}{u1}\right)}^{\left(-1 \cdot 3\right)}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   6. div-sub 98.4%

      \[\leadsto \sqrt{\sqrt[3]{{\color{blue}{\left(\frac{1}{u1} - \frac{u1}{u1}\right)}}^{\left(-1 \cdot 3\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   7. pow1 98.4%

      \[\leadsto \sqrt{\sqrt[3]{{\left(\frac{1}{u1} - \frac{\color{blue}{{u1}^{1}}}{u1}\right)}^{\left(-1 \cdot 3\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   8. pow1 98.4%

      \[\leadsto \sqrt{\sqrt[3]{{\left(\frac{1}{u1} - \frac{{u1}^{1}}{\color{blue}{{u1}^{1}}}\right)}^{\left(-1 \cdot 3\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   9. pow-div 98.4%

      \[\leadsto \sqrt{\sqrt[3]{{\left(\frac{1}{u1} - \color{blue}{{u1}^{\left(1 - 1\right)}}\right)}^{\left(-1 \cdot 3\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   10. metadata-eval 98.4%

       \[\leadsto \sqrt{\sqrt[3]{{\left(\frac{1}{u1} - {u1}^{\color{blue}{0}}\right)}^{\left(-1 \cdot 3\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   11. metadata-eval 98.4%

       \[\leadsto \sqrt{\sqrt[3]{{\left(\frac{1}{u1} - \color{blue}{1}\right)}^{\left(-1 \cdot 3\right)}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   12. metadata-eval 98.4%

       \[\leadsto \sqrt{\sqrt[3]{{\left(\frac{1}{u1} - 1\right)}^{\color{blue}{-3}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

3. Applied egg-rr 98.4%

   \[\leadsto \sqrt{\color{blue}{\sqrt[3]{{\left(\frac{1}{u1} - 1\right)}^{-3}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

4. Taylor expanded in u2 around inf 98.4%

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

   1. *-commutative 98.4%

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

   2. sub-neg 98.4%

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

   3. metadata-eval 98.4%

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

   4. unpow-1 98.4%

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

   5. sqr-pow 98.5%

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

   6. rem-sqrt-square 98.5%

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

   7. metadata-eval 98.5%

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

   8. sqr-pow 97.8%

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

   9. fabs-sqr 97.8%

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

   10. sqr-pow 98.5%

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

   11. +-commutative 98.5%

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

6. Simplified 98.5%

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

7. Final simplification 98.5%

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

Alternative 4: 90.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.012000000104308128:\\ \;\;\;\;\sqrt{39.47841760436263 \cdot \left(\frac{u1}{1 - u1} \cdot \left(u2 \cdot u2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.012000000104308128)
   (sqrt (* 39.47841760436263 (* (/ u1 (- 1.0 u1)) (* u2 u2))))
   (* (sin (* u2 6.28318530718)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.012000000104308128f) {
		tmp = sqrtf((39.47841760436263f * ((u1 / (1.0f - u1)) * (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) <= 0.012000000104308128e0) then
        tmp = sqrt((39.47841760436263e0 * ((u1 / (1.0e0 - u1)) * (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(0.012000000104308128))
		tmp = sqrt(Float32(Float32(39.47841760436263) * Float32(Float32(u1 / Float32(Float32(1.0) - u1)) * 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(0.012000000104308128))
		tmp = sqrt((single(39.47841760436263) * ((u1 / (single(1.0) - u1)) * (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) < 0.0120000001

   1. Initial program 98.3%

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

   2. Taylor expanded in u2 around 0 96.0%

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

      1. sqrt-div 95.8%

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

   4. Applied egg-rr 95.8%

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

      1. sqrt-undiv 96.0%

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

      2. add-cube-cbrt 95.8%

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

      3. unpow3 95.8%

         \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \sqrt{\color{blue}{{\left(\sqrt[3]{\frac{u1}{1 - u1}}\right)}^{3}}}\right) \]

      4. add-exp-log 91.3%

         \[\leadsto 6.28318530718 \cdot \color{blue}{e^{\log \left(u2 \cdot \sqrt{{\left(\sqrt[3]{\frac{u1}{1 - u1}}\right)}^{3}}\right)}} \]

      5. unpow3 91.3%

         \[\leadsto 6.28318530718 \cdot e^{\log \left(u2 \cdot \sqrt{\color{blue}{\left(\sqrt[3]{\frac{u1}{1 - u1}} \cdot \sqrt[3]{\frac{u1}{1 - u1}}\right) \cdot \sqrt[3]{\frac{u1}{1 - u1}}}}\right)} \]

      6. add-cube-cbrt 91.3%

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

      7. sqrt-undiv 91.3%

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

      8. clear-num 91.3%

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

      9. un-div-inv 91.3%

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

      10. sqrt-undiv 91.3%

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

   6. Applied egg-rr 91.3%

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

      1. add-sqr-sqrt 91.3%

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

      2. sqrt-unprod 91.3%

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

      3. *-commutative 91.3%

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

      4. *-commutative 91.3%

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

      5. swap-sqr 91.2%

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

   8. Applied egg-rr 96.6%

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

   if 0.0120000001 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 98.7%

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

   2. Taylor expanded in u1 around 0 73.3%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.012000000104308128:\\ \;\;\;\;\sqrt{39.47841760436263 \cdot \left(\frac{u1}{1 - u1} \cdot \left(u2 \cdot u2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

2. Final simplification 98.4%

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

Alternative 6: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. clear-num 98.4%

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

   2. inv-pow 98.4%

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

   3. div-sub 98.4%

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

   4. pow1 98.4%

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

   5. pow1 98.4%

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

   6. pow-div 98.4%

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

   7. metadata-eval 98.4%

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

   8. metadata-eval 98.4%

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

3. Applied egg-rr 98.4%

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

   1. unpow-1 98.4%

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

   2. sub-neg 98.4%

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

   3. metadata-eval 98.4%

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

5. Simplified 98.4%

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

   1. *-commutative 98.4%

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

   2. *-commutative 98.4%

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

   3. sqrt-div 98.4%

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

   4. metadata-eval 98.4%

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

   5. un-div-inv 98.5%

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

   6. *-commutative 98.5%

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

7. Applied egg-rr 98.5%

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

8. Final simplification 98.5%

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

Alternative 7: 82.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

2. Taylor expanded in u2 around 0 82.6%

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

   1. sqrt-div 82.5%

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

4. Applied egg-rr 82.5%

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

   1. sqrt-undiv 82.6%

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

   2. add-cube-cbrt 82.4%

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

   3. unpow3 82.4%

      \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \sqrt{\color{blue}{{\left(\sqrt[3]{\frac{u1}{1 - u1}}\right)}^{3}}}\right) \]

   4. add-exp-log 79.1%

      \[\leadsto 6.28318530718 \cdot \color{blue}{e^{\log \left(u2 \cdot \sqrt{{\left(\sqrt[3]{\frac{u1}{1 - u1}}\right)}^{3}}\right)}} \]

   5. unpow3 79.1%

      \[\leadsto 6.28318530718 \cdot e^{\log \left(u2 \cdot \sqrt{\color{blue}{\left(\sqrt[3]{\frac{u1}{1 - u1}} \cdot \sqrt[3]{\frac{u1}{1 - u1}}\right) \cdot \sqrt[3]{\frac{u1}{1 - u1}}}}\right)} \]

   6. add-cube-cbrt 79.1%

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

   7. sqrt-undiv 79.1%

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

   8. clear-num 79.1%

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

   9. un-div-inv 79.1%

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

   10. sqrt-undiv 79.1%

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

6. Applied egg-rr 79.1%

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

   1. add-sqr-sqrt 79.2%

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

   2. sqrt-unprod 79.1%

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

   3. *-commutative 79.1%

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

   4. *-commutative 79.1%

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

   5. swap-sqr 79.1%

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

8. Applied egg-rr 83.1%

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

9. Final simplification 83.1%

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

Alternative 8: 82.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

2. Taylor expanded in u2 around 0 82.6%

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

3. Final simplification 82.6%

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

Alternative 9: 82.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

2. Taylor expanded in u2 around 0 82.6%

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

   1. sqrt-div 82.5%

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

4. Applied egg-rr 82.5%

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

   1. expm1-log1p-u 82.5%

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

   2. expm1-udef 26.9%

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

   3. sqrt-undiv 26.9%

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

   4. add-cube-cbrt 26.9%

      \[\leadsto 6.28318530718 \cdot \left(e^{\mathsf{log1p}\left(u2 \cdot \sqrt{\color{blue}{\left(\sqrt[3]{\frac{u1}{1 - u1}} \cdot \sqrt[3]{\frac{u1}{1 - u1}}\right) \cdot \sqrt[3]{\frac{u1}{1 - u1}}}}\right)} - 1\right) \]

   5. unpow3 26.9%

      \[\leadsto 6.28318530718 \cdot \left(e^{\mathsf{log1p}\left(u2 \cdot \sqrt{\color{blue}{{\left(\sqrt[3]{\frac{u1}{1 - u1}}\right)}^{3}}}\right)} - 1\right) \]

   6. unpow3 26.9%

      \[\leadsto 6.28318530718 \cdot \left(e^{\mathsf{log1p}\left(u2 \cdot \sqrt{\color{blue}{\left(\sqrt[3]{\frac{u1}{1 - u1}} \cdot \sqrt[3]{\frac{u1}{1 - u1}}\right) \cdot \sqrt[3]{\frac{u1}{1 - u1}}}}\right)} - 1\right) \]

   7. add-cube-cbrt 26.9%

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

   8. sqrt-undiv 26.9%

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

   9. clear-num 26.9%

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

   10. un-div-inv 26.9%

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

   11. sqrt-undiv 26.9%

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

6. Applied egg-rr 26.9%

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

   1. expm1-def 82.7%

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

   2. expm1-log1p 82.7%

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

   3. div-sub 82.7%

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

   4. sub-neg 82.7%

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

   5. *-inverses 82.7%

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

   6. metadata-eval 82.7%

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

8. Simplified 82.7%

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

9. Final simplification 82.7%

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

Alternative 10: 82.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

2. Taylor expanded in u2 around 0 82.6%

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

   1. sqrt-div 82.5%

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

4. Applied egg-rr 82.5%

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

   1. sqrt-undiv 82.6%

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

   2. add-cube-cbrt 82.4%

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

   3. unpow3 82.4%

      \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \sqrt{\color{blue}{{\left(\sqrt[3]{\frac{u1}{1 - u1}}\right)}^{3}}}\right) \]

   4. add-exp-log 79.1%

      \[\leadsto 6.28318530718 \cdot \color{blue}{e^{\log \left(u2 \cdot \sqrt{{\left(\sqrt[3]{\frac{u1}{1 - u1}}\right)}^{3}}\right)}} \]

   5. unpow3 79.1%

      \[\leadsto 6.28318530718 \cdot e^{\log \left(u2 \cdot \sqrt{\color{blue}{\left(\sqrt[3]{\frac{u1}{1 - u1}} \cdot \sqrt[3]{\frac{u1}{1 - u1}}\right) \cdot \sqrt[3]{\frac{u1}{1 - u1}}}}\right)} \]

   6. add-cube-cbrt 79.1%

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

   7. sqrt-undiv 79.1%

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

   8. clear-num 79.1%

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

   9. un-div-inv 79.1%

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

   10. sqrt-undiv 79.1%

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

6. Applied egg-rr 79.1%

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

   1. add-exp-log 82.7%

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

   2. clear-num 82.5%

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

   3. un-div-inv 82.6%

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

   4. div-sub 82.7%

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

   5. *-inverses 82.7%

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

   6. sub-neg 82.7%

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

   7. metadata-eval 82.7%

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

8. Applied egg-rr 82.7%

   \[\leadsto \color{blue}{\frac{6.28318530718}{\frac{\sqrt{\frac{1}{u1} + -1}}{u2}}} \]
9. Step-by-step derivation

   1. associate-/r/ 82.9%

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

   2. +-commutative 82.9%

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

10. Simplified 82.9%

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

11. Final simplification 82.9%

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

Alternative 11: 64.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. flip-- 98.4%

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

   2. associate-/r/ 98.4%

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

   3. metadata-eval 98.4%

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

   4. +-commutative 98.4%

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

3. Applied egg-rr 98.4%

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

4. Taylor expanded in u2 around 0 82.6%

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

   1. associate-*r* 82.5%

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

   2. *-commutative 82.5%

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

   3. associate-*l* 82.5%

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

   4. +-commutative 82.5%

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

   5. distribute-lft1-in 82.7%

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

   6. fma-def 82.7%

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

   7. unpow2 82.7%

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

6. Simplified 82.7%

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

7. Taylor expanded in u1 around 0 64.6%

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

   1. add-sqr-sqrt 64.6%

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

   2. sqrt-unprod 64.6%

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

   3. *-commutative 64.6%

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

   4. *-commutative 64.6%

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

   5. swap-sqr 64.6%

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

   6. *-commutative 64.6%

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

   7. *-commutative 64.6%

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

   8. swap-sqr 64.6%

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

   9. add-sqr-sqrt 64.6%

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

   10. metadata-eval 64.6%

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

9. Applied egg-rr 64.6%

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

    1. *-commutative 64.6%

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

    2. *-commutative 64.6%

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

    3. metadata-eval 64.6%

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

    4. rem-square-sqrt 64.6%

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

    5. swap-sqr 64.6%

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

    6. associate-*l* 64.6%

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

    7. swap-sqr 64.6%

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

    8. metadata-eval 64.6%

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

    9. rem-square-sqrt 64.7%

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

    10. *-commutative 64.7%

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

11. Simplified 64.7%

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

12. Final simplification 64.7%

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

Alternative 12: 64.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

2. Taylor expanded in u2 around 0 82.6%

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

3. Taylor expanded in u1 around 0 64.6%

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

4. Final simplification 64.6%

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

Alternative 13: 64.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

   1. flip-- 98.4%

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

   2. associate-/r/ 98.4%

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

   3. metadata-eval 98.4%

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

   4. +-commutative 98.4%

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

3. Applied egg-rr 98.4%

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

4. Taylor expanded in u2 around 0 82.6%

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

   1. associate-*r* 82.5%

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

   2. *-commutative 82.5%

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

   3. associate-*l* 82.5%

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

   4. +-commutative 82.5%

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

   5. distribute-lft1-in 82.7%

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

   6. fma-def 82.7%

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

   7. unpow2 82.7%

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

6. Simplified 82.7%

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

7. Taylor expanded in u1 around 0 64.6%

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

   1. add-sqr-sqrt 64.5%

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

   2. sqrt-unprod 64.6%

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

   3. *-commutative 64.6%

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

   4. *-commutative 64.6%

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

   5. swap-sqr 64.6%

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

   6. add-sqr-sqrt 64.7%

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

   7. metadata-eval 64.6%

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

9. Applied egg-rr 64.6%

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

10. Final simplification 64.6%

    \[\leadsto u2 \cdot \sqrt{u1 \cdot 39.47841760436263} \]

Alternative 14: 64.8% 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 98.4%

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

   1. flip-- 98.4%

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

   2. associate-/r/ 98.4%

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

   3. metadata-eval 98.4%

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

   4. +-commutative 98.4%

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

3. Applied egg-rr 98.4%

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

4. Taylor expanded in u2 around 0 82.6%

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

   1. associate-*r* 82.5%

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

   2. *-commutative 82.5%

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

   3. associate-*l* 82.5%

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

   4. +-commutative 82.5%

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

   5. distribute-lft1-in 82.7%

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

   6. fma-def 82.7%

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

   7. unpow2 82.7%

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

6. Simplified 82.7%

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

7. Taylor expanded in u1 around 0 64.6%

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

8. Final simplification 64.6%

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

Reproduce

herbie shell --seed 2023268 
(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))))