Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.3%

Time: 11.6s

Alternatives: 9

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 93.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.0007200000109151006:\\ \;\;\;\;u2 \cdot \sqrt{\frac{39.47841760436263}{\frac{1}{u1} + -1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.0007200000109151006)
   (* u2 (sqrt (/ 39.47841760436263 (+ (/ 1.0 u1) -1.0))))
   (* (sin (* 6.28318530718 u2)) (sqrt (* u1 (+ u1 1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.0007200000109151006f) {
		tmp = u2 * sqrtf((39.47841760436263f / ((1.0f / u1) + -1.0f)));
	} else {
		tmp = sinf((6.28318530718f * u2)) * sqrtf((u1 * (u1 + 1.0f)));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((6.28318530718e0 * u2) <= 0.0007200000109151006e0) then
        tmp = u2 * sqrt((39.47841760436263e0 / ((1.0e0 / u1) + (-1.0e0))))
    else
        tmp = sin((6.28318530718e0 * u2)) * sqrt((u1 * (u1 + 1.0e0)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.0007200000109151006))
		tmp = Float32(u2 * sqrt(Float32(Float32(39.47841760436263) / Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(Float32(u1 * Float32(u1 + Float32(1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.0007200000109151006))
		tmp = u2 * sqrt((single(39.47841760436263) / ((single(1.0) / u1) + single(-1.0))));
	else
		tmp = sin((single(6.28318530718) * u2)) * sqrt((u1 * (u1 + single(1.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 7.20000011e-4

   1. Initial program 98.7%

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

      1. add-sqr-sqrt 98.0%

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

      2. sqrt-unprod 98.7%

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

      3. swap-sqr 98.5%

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

      4. add-sqr-sqrt 98.5%

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

      5. pow2 98.5%

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

   4. Applied egg-rr 98.5%

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

      1. *-lft-identity 98.5%

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

      2. associate-*l/ 98.4%

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

      3. associate-/r/ 98.5%

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

      4. associate-*l/ 98.5%

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

      5. *-lft-identity 98.5%

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

      6. div-sub 98.6%

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

      7. sub-neg 98.6%

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

      8. *-inverses 98.6%

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

      9. metadata-eval 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in u2 around 0 99.0%

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

      1. *-commutative 99.0%

         \[\leadsto \sqrt{\frac{\color{blue}{{u2}^{2} \cdot 39.47841760436263}}{\frac{1}{u1} + -1}} \]

   9. Simplified 99.0%

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

       1. pow1/2 99.0%

          \[\leadsto \color{blue}{{\left(\frac{{u2}^{2} \cdot 39.47841760436263}{\frac{1}{u1} + -1}\right)}^{0.5}} \]

       2. associate-/l* 98.8%

          \[\leadsto {\color{blue}{\left({u2}^{2} \cdot \frac{39.47841760436263}{\frac{1}{u1} + -1}\right)}}^{0.5} \]

       3. unpow-prod-down 98.9%

          \[\leadsto \color{blue}{{\left({u2}^{2}\right)}^{0.5} \cdot {\left(\frac{39.47841760436263}{\frac{1}{u1} + -1}\right)}^{0.5}} \]

       4. pow1/2 98.9%

          \[\leadsto \color{blue}{\sqrt{{u2}^{2}}} \cdot {\left(\frac{39.47841760436263}{\frac{1}{u1} + -1}\right)}^{0.5} \]

       5. sqrt-pow1 98.9%

          \[\leadsto \color{blue}{{u2}^{\left(\frac{2}{2}\right)}} \cdot {\left(\frac{39.47841760436263}{\frac{1}{u1} + -1}\right)}^{0.5} \]

       6. metadata-eval 98.9%

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

       7. pow1 98.9%

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

   11. Applied egg-rr 98.9%

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

       1. unpow1/2 98.9%

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

   13. Simplified 98.9%

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

   if 7.20000011e-4 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.4%

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

   3. Taylor expanded in u1 around 0 87.7%

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

      1. +-commutative 87.7%

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

   5. Simplified 87.7%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.0007200000109151006:\\ \;\;\;\;u2 \cdot \sqrt{\frac{39.47841760436263}{\frac{1}{u1} + -1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.029999999329447746:\\ \;\;\;\;u2 \cdot \sqrt{\frac{39.47841760436263}{\frac{1}{u1} + -1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(6.28318530718 \cdot u2\right)}{\sqrt{\frac{1}{u1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.029999999329447746)
   (* u2 (sqrt (/ 39.47841760436263 (+ (/ 1.0 u1) -1.0))))
   (/ (sin (* 6.28318530718 u2)) (sqrt (/ 1.0 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.029999999329447746f) {
		tmp = u2 * sqrtf((39.47841760436263f / ((1.0f / u1) + -1.0f)));
	} else {
		tmp = sinf((6.28318530718f * u2)) / 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 ((6.28318530718e0 * u2) <= 0.029999999329447746e0) then
        tmp = u2 * sqrt((39.47841760436263e0 / ((1.0e0 / u1) + (-1.0e0))))
    else
        tmp = sin((6.28318530718e0 * u2)) / sqrt((1.0e0 / u1))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.029999999329447746))
		tmp = Float32(u2 * sqrt(Float32(Float32(39.47841760436263) / Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) / sqrt(Float32(Float32(1.0) / u1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.029999999329447746))
		tmp = u2 * sqrt((single(39.47841760436263) / ((single(1.0) / u1) + single(-1.0))));
	else
		tmp = sin((single(6.28318530718) * u2)) / sqrt((single(1.0) / u1));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.0299999993

   1. Initial program 98.6%

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

      1. add-sqr-sqrt 98.0%

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

      2. sqrt-unprod 98.6%

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

      3. swap-sqr 98.4%

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

      4. add-sqr-sqrt 98.7%

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

      5. pow2 98.7%

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

   4. Applied egg-rr 98.7%

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

      1. *-lft-identity 98.7%

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

      2. associate-*l/ 98.5%

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

      3. associate-/r/ 98.6%

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

      4. associate-*l/ 98.6%

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

      5. *-lft-identity 98.6%

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

      6. div-sub 98.7%

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

      7. sub-neg 98.7%

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

      8. *-inverses 98.7%

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

      9. metadata-eval 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in u2 around 0 95.2%

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

      1. *-commutative 95.2%

         \[\leadsto \sqrt{\frac{\color{blue}{{u2}^{2} \cdot 39.47841760436263}}{\frac{1}{u1} + -1}} \]

   9. Simplified 95.2%

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

       1. pow1/2 95.2%

          \[\leadsto \color{blue}{{\left(\frac{{u2}^{2} \cdot 39.47841760436263}{\frac{1}{u1} + -1}\right)}^{0.5}} \]

       2. associate-/l* 95.1%

          \[\leadsto {\color{blue}{\left({u2}^{2} \cdot \frac{39.47841760436263}{\frac{1}{u1} + -1}\right)}}^{0.5} \]

       3. unpow-prod-down 95.2%

          \[\leadsto \color{blue}{{\left({u2}^{2}\right)}^{0.5} \cdot {\left(\frac{39.47841760436263}{\frac{1}{u1} + -1}\right)}^{0.5}} \]

       4. pow1/2 95.2%

          \[\leadsto \color{blue}{\sqrt{{u2}^{2}}} \cdot {\left(\frac{39.47841760436263}{\frac{1}{u1} + -1}\right)}^{0.5} \]

       5. sqrt-pow1 95.2%

          \[\leadsto \color{blue}{{u2}^{\left(\frac{2}{2}\right)}} \cdot {\left(\frac{39.47841760436263}{\frac{1}{u1} + -1}\right)}^{0.5} \]

       6. metadata-eval 95.2%

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

       7. pow1 95.2%

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

   11. Applied egg-rr 95.2%

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

       1. unpow1/2 95.2%

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

   13. Simplified 95.2%

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

   if 0.0299999993 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.4%

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

      1. clear-num 98.6%

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

      2. inv-pow 98.6%

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

   4. Applied egg-rr 98.6%

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

      1. *-commutative 98.6%

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

      2. pow1 98.6%

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

      3. metadata-eval 98.6%

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

      4. sqrt-pow1 92.6%

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

      5. div-sub 92.2%

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

      6. *-inverses 92.2%

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

      7. sub-neg 92.2%

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

      8. metadata-eval 92.2%

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

      9. inv-pow 92.2%

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

      10. sqrt-prod 92.0%

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

      11. div-inv 91.9%

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

      12. sqrt-div 91.9%

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

      13. sqrt-pow1 97.8%

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

      14. metadata-eval 97.8%

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

      15. pow1 97.8%

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

      16. +-commutative 97.8%

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

   6. Applied egg-rr 97.8%

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

      1. +-commutative 97.8%

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

   8. Simplified 97.8%

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

   9. Taylor expanded in u1 around 0 74.2%

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

Alternative 4: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.029999999329447746:\\ \;\;\;\;u2 \cdot \sqrt{\frac{39.47841760436263}{\frac{1}{u1} + -1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.029999999329447746)
   (* u2 (sqrt (/ 39.47841760436263 (+ (/ 1.0 u1) -1.0))))
   (* (sin (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.029999999329447746f) {
		tmp = u2 * sqrtf((39.47841760436263f / ((1.0f / u1) + -1.0f)));
	} else {
		tmp = sinf((6.28318530718f * u2)) * sqrtf(u1);
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((6.28318530718e0 * u2) <= 0.029999999329447746e0) then
        tmp = u2 * sqrt((39.47841760436263e0 / ((1.0e0 / u1) + (-1.0e0))))
    else
        tmp = sin((6.28318530718e0 * u2)) * sqrt(u1)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.029999999329447746))
		tmp = Float32(u2 * sqrt(Float32(Float32(39.47841760436263) / Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.029999999329447746))
		tmp = u2 * sqrt((single(39.47841760436263) / ((single(1.0) / u1) + single(-1.0))));
	else
		tmp = sin((single(6.28318530718) * u2)) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.0299999993

   1. Initial program 98.6%

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

      1. add-sqr-sqrt 98.0%

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

      2. sqrt-unprod 98.6%

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

      3. swap-sqr 98.4%

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

      4. add-sqr-sqrt 98.7%

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

      5. pow2 98.7%

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

   4. Applied egg-rr 98.7%

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

      1. *-lft-identity 98.7%

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

      2. associate-*l/ 98.5%

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

      3. associate-/r/ 98.6%

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

      4. associate-*l/ 98.6%

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

      5. *-lft-identity 98.6%

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

      6. div-sub 98.7%

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

      7. sub-neg 98.7%

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

      8. *-inverses 98.7%

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

      9. metadata-eval 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in u2 around 0 95.2%

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

      1. *-commutative 95.2%

         \[\leadsto \sqrt{\frac{\color{blue}{{u2}^{2} \cdot 39.47841760436263}}{\frac{1}{u1} + -1}} \]

   9. Simplified 95.2%

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

       1. pow1/2 95.2%

          \[\leadsto \color{blue}{{\left(\frac{{u2}^{2} \cdot 39.47841760436263}{\frac{1}{u1} + -1}\right)}^{0.5}} \]

       2. associate-/l* 95.1%

          \[\leadsto {\color{blue}{\left({u2}^{2} \cdot \frac{39.47841760436263}{\frac{1}{u1} + -1}\right)}}^{0.5} \]

       3. unpow-prod-down 95.2%

          \[\leadsto \color{blue}{{\left({u2}^{2}\right)}^{0.5} \cdot {\left(\frac{39.47841760436263}{\frac{1}{u1} + -1}\right)}^{0.5}} \]

       4. pow1/2 95.2%

          \[\leadsto \color{blue}{\sqrt{{u2}^{2}}} \cdot {\left(\frac{39.47841760436263}{\frac{1}{u1} + -1}\right)}^{0.5} \]

       5. sqrt-pow1 95.2%

          \[\leadsto \color{blue}{{u2}^{\left(\frac{2}{2}\right)}} \cdot {\left(\frac{39.47841760436263}{\frac{1}{u1} + -1}\right)}^{0.5} \]

       6. metadata-eval 95.2%

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

       7. pow1 95.2%

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

   11. Applied egg-rr 95.2%

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

       1. unpow1/2 95.2%

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

   13. Simplified 95.2%

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

   if 0.0299999993 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.4%

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

   3. Taylor expanded in u1 around 0 74.1%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.029999999329447746:\\ \;\;\;\;u2 \cdot \sqrt{\frac{39.47841760436263}{\frac{1}{u1} + -1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

   1. add-sqr-sqrt 96.4%

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

   2. sqrt-unprod 97.2%

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

   3. swap-sqr 96.9%

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

   4. add-sqr-sqrt 97.1%

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

   5. pow2 97.1%

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

4. Applied egg-rr 97.1%

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

   1. *-lft-identity 97.1%

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

   2. associate-*l/ 96.9%

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

   3. associate-/r/ 97.0%

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

   4. associate-*l/ 97.1%

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

   5. *-lft-identity 97.1%

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

   6. div-sub 97.0%

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

   7. sub-neg 97.0%

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

   8. *-inverses 97.0%

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

   9. metadata-eval 97.0%

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

6. Simplified 97.0%

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

7. Taylor expanded in u2 around 0 82.2%

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

   1. *-commutative 82.2%

      \[\leadsto \sqrt{\frac{\color{blue}{{u2}^{2} \cdot 39.47841760436263}}{\frac{1}{u1} + -1}} \]

9. Simplified 82.2%

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

    1. pow1/2 82.2%

       \[\leadsto \color{blue}{{\left(\frac{{u2}^{2} \cdot 39.47841760436263}{\frac{1}{u1} + -1}\right)}^{0.5}} \]

    2. associate-/l* 82.1%

       \[\leadsto {\color{blue}{\left({u2}^{2} \cdot \frac{39.47841760436263}{\frac{1}{u1} + -1}\right)}}^{0.5} \]

    3. unpow-prod-down 82.1%

       \[\leadsto \color{blue}{{\left({u2}^{2}\right)}^{0.5} \cdot {\left(\frac{39.47841760436263}{\frac{1}{u1} + -1}\right)}^{0.5}} \]

    4. pow1/2 82.1%

       \[\leadsto \color{blue}{\sqrt{{u2}^{2}}} \cdot {\left(\frac{39.47841760436263}{\frac{1}{u1} + -1}\right)}^{0.5} \]

    5. sqrt-pow1 82.1%

       \[\leadsto \color{blue}{{u2}^{\left(\frac{2}{2}\right)}} \cdot {\left(\frac{39.47841760436263}{\frac{1}{u1} + -1}\right)}^{0.5} \]

    6. metadata-eval 82.1%

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

    7. pow1 82.1%

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

11. Applied egg-rr 82.1%

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

    1. unpow1/2 82.1%

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

13. Simplified 82.1%

    \[\leadsto \color{blue}{u2 \cdot \sqrt{\frac{39.47841760436263}{\frac{1}{u1} + -1}}} \]
14. Add Preprocessing

Alternative 6: 81.4% 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 98.6%

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

3. Taylor expanded in u2 around 0 81.9%

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

Alternative 7: 73.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in u2 around 0 81.9%

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

4. Taylor expanded in u1 around 0 75.0%

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

   1. +-commutative 87.9%

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

6. Simplified 75.0%

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

7. Final simplification 75.0%

   \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\right) \]
8. Add Preprocessing

Alternative 8: 64.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in u2 around 0 81.9%

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

   1. *-commutative 81.9%

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

   2. sqrt-div 81.8%

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

   3. associate-*r/ 81.9%

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

5. Applied egg-rr 81.9%

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

   1. *-commutative 81.9%

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

   2. associate-/l* 81.8%

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

7. Simplified 81.8%

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

8. Taylor expanded in u1 around 0 66.9%

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

   1. associate-*r* 66.9%

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

   2. *-commutative 66.9%

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

   3. *-commutative 66.9%

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

10. Simplified 66.9%

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

11. Taylor expanded in u2 around 0 66.9%

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

    1. *-commutative 66.9%

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

    2. associate-*l* 67.0%

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

    3. *-commutative 67.0%

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

13. Simplified 67.0%

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

14. Final simplification 67.0%

    \[\leadsto \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1} \]
15. Add Preprocessing

Alternative 9: 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 98.6%

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

3. Taylor expanded in u2 around 0 81.9%

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

4. Taylor expanded in u1 around 0 66.9%

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

5. Final simplification 66.9%

   \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \sqrt{u1}\right) \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024090 
(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))))