Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.4%

Time: 58.8s

Alternatives: 15

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (sin (sqrt (* (pow u2 2.0) 39.47841760436263)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * sinf(sqrtf((powf(u2, 2.0f) * 39.47841760436263f)));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 / (1.0e0 - u1))) * sin(sqrt(((u2 ** 2.0e0) * 39.47841760436263e0)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * sin(sqrt(Float32((u2 ^ Float32(2.0)) * Float32(39.47841760436263)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * sin(sqrt(((u2 ^ single(2.0)) * single(39.47841760436263))));
end

Derivation

1. Initial program 98.5%

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

   1. add-sqr-sqrt 97.7%

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

   2. sqrt-unprod 98.5%

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

   3. *-commutative 98.5%

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

   4. *-commutative 98.5%

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

   5. swap-sqr 98.2%

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

   6. pow2 98.2%

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

   7. metadata-eval 98.7%

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

4. Applied egg-rr 98.7%

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

5. Final simplification 98.7%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right) \]
6. Add Preprocessing

Alternative 2: 93.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.001500000013038516:\\ \;\;\;\;\left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} + -1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.001500000013038516)
   (* (* u2 6.28318530718) (sqrt (/ 1.0 (+ (/ 1.0 u1) -1.0))))
   (* (sin (* u2 6.28318530718)) (sqrt (* u1 (+ u1 1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.001500000013038516f) {
		tmp = (u2 * 6.28318530718f) * sqrtf((1.0f / ((1.0f / u1) + -1.0f)));
	} else {
		tmp = sinf((u2 * 6.28318530718f)) * 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 ((u2 * 6.28318530718e0) <= 0.001500000013038516e0) then
        tmp = (u2 * 6.28318530718e0) * sqrt((1.0e0 / ((1.0e0 / u1) + (-1.0e0))))
    else
        tmp = sin((u2 * 6.28318530718e0)) * sqrt((u1 * (u1 + 1.0e0)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.001500000013038516))
		tmp = Float32(Float32(u2 * Float32(6.28318530718)) * sqrt(Float32(Float32(1.0) / Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)))));
	else
		tmp = Float32(sin(Float32(u2 * Float32(6.28318530718))) * 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 ((u2 * single(6.28318530718)) <= single(0.001500000013038516))
		tmp = (u2 * single(6.28318530718)) * sqrt((single(1.0) / ((single(1.0) / u1) + single(-1.0))));
	else
		tmp = sin((u2 * single(6.28318530718))) * 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) < 0.00150000001

   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. clear-num 98.7%

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

      2. inv-pow 98.7%

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

   4. Applied egg-rr 98.7%

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

      1. pow1 98.7%

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

      2. sqrt-pow1 98.6%

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

      3. div-sub 98.6%

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

      4. *-inverses 98.6%

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

      5. sub-neg 98.6%

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

      6. metadata-eval 98.6%

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

      7. metadata-eval 98.6%

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

   6. Applied egg-rr 98.6%

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

      1. unpow1 98.6%

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

      2. *-commutative 98.6%

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

   8. Simplified 98.6%

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

   9. Taylor expanded in u2 around 0 98.4%

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

       1. associate-*r* 98.5%

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

       2. *-commutative 98.5%

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

       3. sub-neg 98.5%

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

       4. metadata-eval 98.5%

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

   11. Simplified 98.5%

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

   if 0.00150000001 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.2%

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

   3. Taylor expanded in u1 around 0 87.5%

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

      1. +-commutative 52.4%

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

   5. Simplified 87.5%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.001500000013038516:\\ \;\;\;\;\left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} + -1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.003000000026077032:\\ \;\;\;\;\left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} + -1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.003000000026077032)
   (* (* u2 6.28318530718) (sqrt (/ 1.0 (+ (/ 1.0 u1) -1.0))))
   (* (sin (* u2 6.28318530718)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.003000000026077032f) {
		tmp = (u2 * 6.28318530718f) * sqrtf((1.0f / ((1.0f / u1) + -1.0f)));
	} 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) <= 0.003000000026077032e0) then
        tmp = (u2 * 6.28318530718e0) * sqrt((1.0e0 / ((1.0e0 / u1) + (-1.0e0))))
    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(0.003000000026077032))
		tmp = Float32(Float32(u2 * Float32(6.28318530718)) * sqrt(Float32(Float32(1.0) / Float32(Float32(Float32(1.0) / u1) + Float32(-1.0)))));
	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(0.003000000026077032))
		tmp = (u2 * single(6.28318530718)) * sqrt((single(1.0) / ((single(1.0) / u1) + single(-1.0))));
	else
		tmp = sin((u2 * single(6.28318530718))) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00300000003

   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. clear-num 98.8%

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

      2. inv-pow 98.8%

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

   4. Applied egg-rr 98.8%

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

      1. pow1 98.8%

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

      2. sqrt-pow1 98.7%

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

      3. div-sub 98.7%

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

      4. *-inverses 98.7%

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

      5. sub-neg 98.7%

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

      6. metadata-eval 98.7%

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

      7. metadata-eval 98.7%

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

   6. Applied egg-rr 98.7%

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

      1. unpow1 98.7%

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

      2. *-commutative 98.7%

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

   8. Simplified 98.7%

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

   9. Taylor expanded in u2 around 0 97.9%

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

       1. associate-*r* 98.0%

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

       2. *-commutative 98.0%

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

       3. sub-neg 98.0%

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

       4. metadata-eval 98.0%

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

   11. Simplified 98.0%

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

   if 0.00300000003 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.0%

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

   3. Taylor expanded in u1 around 0 75.1%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.003000000026077032:\\ \;\;\;\;\left(u2 \cdot 6.28318530718\right) \cdot \sqrt{\frac{1}{\frac{1}{u1} + -1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. clear-num 98.5%

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

   2. inv-pow 98.5%

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

4. Applied egg-rr 98.5%

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

   1. pow1 98.5%

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

   2. sqrt-pow1 98.5%

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

   3. div-sub 98.5%

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

   4. *-inverses 98.5%

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

   5. sub-neg 98.5%

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

   6. metadata-eval 98.5%

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

   7. metadata-eval 98.5%

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

6. Applied egg-rr 98.5%

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

   1. unpow1 98.5%

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

   2. *-commutative 98.5%

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

8. Simplified 98.5%

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

9. Final simplification 98.5%

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

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 98.5%

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

3. Final simplification 98.5%

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

Alternative 6: 81.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. clear-num 98.5%

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

   2. inv-pow 98.5%

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

4. Applied egg-rr 98.5%

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

   1. pow1 98.5%

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

   2. sqrt-pow1 98.5%

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

   3. div-sub 98.5%

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

   4. *-inverses 98.5%

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

   5. sub-neg 98.5%

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

   6. metadata-eval 98.5%

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

   7. metadata-eval 98.5%

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

6. Applied egg-rr 98.5%

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

   1. unpow1 98.5%

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

   2. *-commutative 98.5%

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

8. Simplified 98.5%

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

9. Taylor expanded in u2 around 0 82.3%

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

    1. associate-*r* 82.3%

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

    2. *-commutative 82.3%

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

    3. sub-neg 82.3%

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

    4. metadata-eval 82.3%

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

11. Simplified 82.3%

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

12. Final simplification 82.3%

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

Alternative 7: 73.0% 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 98.5%

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

3. Taylor expanded in u2 around 0 82.2%

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

4. Taylor expanded in u1 around 0 74.0%

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

   1. +-commutative 74.0%

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

6. Simplified 74.0%

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

7. Final simplification 74.0%

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

Alternative 8: 81.6% 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 98.5%

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

3. Taylor expanded in u2 around 0 82.2%

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

4. Final simplification 82.2%

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

Alternative 9: 81.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in u2 around 0 82.3%

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

4. Final simplification 82.3%

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

Alternative 10: 64.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in u2 around 0 82.2%

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

4. Taylor expanded in u1 around 0 65.9%

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

5. Final simplification 65.9%

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

Alternative 11: 64.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

   1. add-sqr-sqrt 97.7%

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

   2. sqrt-unprod 98.5%

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

   3. *-commutative 98.5%

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

   4. *-commutative 98.5%

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

   5. swap-sqr 98.2%

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

   6. pow2 98.2%

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

   7. metadata-eval 98.7%

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

4. Applied egg-rr 98.7%

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

5. Taylor expanded in u1 around 0 75.2%

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

6. Taylor expanded in u2 around 0 66.0%

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

   1. *-commutative 66.0%

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

8. Simplified 66.0%

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

9. Final simplification 66.0%

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

Alternative 12: 20.6% accurate, 19.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in u2 around 0 82.2%

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

4. Taylor expanded in u1 around 0 74.0%

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

   1. distribute-rgt-in 74.0%

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

   2. unpow2 74.0%

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

   3. *-lft-identity 74.0%

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

6. Simplified 74.0%

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

7. Taylor expanded in u1 around inf 20.2%

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

8. Final simplification 20.2%

   \[\leadsto 6.28318530718 \cdot \left(u1 \cdot \left(u2 + 0.5 \cdot \frac{u2}{u1}\right)\right) \]
9. Add Preprocessing

Alternative 13: 20.6% accurate, 19.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in u2 around 0 82.2%

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

4. Taylor expanded in u1 around 0 74.0%

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

   1. distribute-rgt-in 74.0%

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

   2. unpow2 74.0%

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

   3. *-lft-identity 74.0%

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

6. Simplified 74.0%

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

7. Taylor expanded in u1 around inf 20.2%

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

   1. associate-*r/ 20.2%

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

   2. metadata-eval 20.2%

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

9. Simplified 20.2%

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

10. Final simplification 20.2%

    \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \left(u1 \cdot \left(1 + \frac{0.5}{u1}\right)\right)\right) \]
11. Add Preprocessing

Alternative 14: 4.8% accurate, 41.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in u2 around 0 82.2%

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

4. Taylor expanded in u1 around 0 74.0%

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

   1. distribute-rgt-in 74.0%

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

   2. unpow2 74.0%

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

   3. *-lft-identity 74.0%

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

6. Simplified 74.0%

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

7. Taylor expanded in u1 around -inf 4.9%

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

8. Final simplification 4.9%

   \[\leadsto \left(u1 \cdot u2\right) \cdot -6.28318530718 \]
9. Add Preprocessing

Alternative 15: 19.5% accurate, 41.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in u2 around 0 82.2%

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

4. Taylor expanded in u1 around 0 74.0%

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

   1. distribute-rgt-in 74.0%

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

   2. unpow2 74.0%

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

   3. *-lft-identity 74.0%

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

6. Simplified 74.0%

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

7. Taylor expanded in u1 around inf 19.2%

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

8. Final simplification 19.2%

   \[\leadsto 6.28318530718 \cdot \left(u1 \cdot u2\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024115 
(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))))