Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.4%

Time: 10.8s

Alternatives: 12

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 98.4%

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

   1. add-sqr-sqrt 97.8%

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

   2. sqrt-unprod 98.4%

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

   3. *-commutative 98.4%

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

   4. *-commutative 98.4%

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

   5. swap-sqr 98.1%

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

   6. pow2 98.1%

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

   7. metadata-eval 98.6%

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

4. Applied egg-rr 98.6%

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

Alternative 2: 93.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.0019450000254437327:\\ \;\;\;\;u2 \cdot \sqrt{\frac{u1}{1 - u1} \cdot 39.47841760436263}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.0019450000254437327)
   (* u2 (sqrt (* (/ u1 (- 1.0 u1)) 39.47841760436263)))
   (* (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) <= 0.0019450000254437327f) {
		tmp = u2 * sqrtf(((u1 / (1.0f - u1)) * 39.47841760436263f));
	} 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) <= 0.0019450000254437327e0) then
        tmp = u2 * sqrt(((u1 / (1.0e0 - u1)) * 39.47841760436263e0))
    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(0.0019450000254437327))
		tmp = Float32(u2 * sqrt(Float32(Float32(u1 / Float32(Float32(1.0) - u1)) * Float32(39.47841760436263))));
	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(0.0019450000254437327))
		tmp = u2 * sqrt(((u1 / (single(1.0) - u1)) * single(39.47841760436263)));
	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) < 0.00194500003

   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 97.9%

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

      1. add-sqr-sqrt 97.5%

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

      2. pow2 97.5%

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

      3. *-commutative 97.5%

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

      4. *-commutative 97.5%

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

      5. associate-*l* 97.6%

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

   5. Applied egg-rr 97.6%

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

      1. unpow2 97.6%

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

      2. add-sqr-sqrt 98.1%

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

      3. *-commutative 98.1%

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

      4. add-sqr-sqrt 97.5%

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

      5. sqrt-unprod 98.1%

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

      6. swap-sqr 97.8%

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

      7. add-sqr-sqrt 98.1%

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

      8. metadata-eval 98.5%

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

   7. Applied egg-rr 98.5%

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

   if 0.00194500003 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.1%

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

   3. Taylor expanded in u1 around 0 81.6%

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

      1. +-commutative 81.6%

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

   5. Simplified 81.6%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.0019450000254437327:\\ \;\;\;\;u2 \cdot \sqrt{\frac{u1}{1 - u1} \cdot 39.47841760436263}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.035999998450279236:\\ \;\;\;\;u2 \cdot \sqrt{\frac{u1}{1 - u1} \cdot 39.47841760436263}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.035999998450279236)
   (* u2 (sqrt (* (/ u1 (- 1.0 u1)) 39.47841760436263)))
   (* (sin (* u2 6.28318530718)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.035999998450279236f) {
		tmp = u2 * sqrtf(((u1 / (1.0f - u1)) * 39.47841760436263f));
	} 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) <= 0.035999998450279236e0) then
        tmp = u2 * sqrt(((u1 / (1.0e0 - u1)) * 39.47841760436263e0))
    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(0.035999998450279236))
		tmp = Float32(u2 * sqrt(Float32(Float32(u1 / Float32(Float32(1.0) - u1)) * Float32(39.47841760436263))));
	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(0.035999998450279236))
		tmp = u2 * sqrt(((u1 / (single(1.0) - u1)) * single(39.47841760436263)));
	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) < 0.0359999985

   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 93.8%

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

      1. add-sqr-sqrt 93.4%

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

      2. pow2 93.4%

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

      3. *-commutative 93.4%

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

      4. *-commutative 93.4%

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

      5. associate-*l* 93.6%

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

   5. Applied egg-rr 93.6%

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

      1. unpow2 93.6%

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

      2. add-sqr-sqrt 94.0%

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

      3. *-commutative 94.0%

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

      4. add-sqr-sqrt 93.5%

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

      5. sqrt-unprod 94.0%

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

      6. swap-sqr 93.7%

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

      7. add-sqr-sqrt 93.9%

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

      8. metadata-eval 94.3%

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

   7. Applied egg-rr 94.3%

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

   if 0.0359999985 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.6%

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

   3. Taylor expanded in u1 around 0 70.0%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.035999998450279236:\\ \;\;\;\;u2 \cdot \sqrt{\frac{u1}{1 - u1} \cdot 39.47841760436263}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

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. Add Preprocessing

3. Final simplification 98.4%

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

Alternative 5: 81.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u2 * sqrt(((u1 / (single(1.0) - 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. Add Preprocessing

3. Taylor expanded in u2 around 0 80.6%

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

   1. add-sqr-sqrt 80.4%

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

   2. pow2 80.4%

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

   3. *-commutative 80.4%

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

   4. *-commutative 80.4%

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

   5. associate-*l* 80.5%

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

5. Applied egg-rr 80.5%

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

   1. unpow2 80.5%

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

   2. add-sqr-sqrt 80.8%

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

   3. *-commutative 80.8%

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

   4. add-sqr-sqrt 80.4%

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

   5. sqrt-unprod 80.8%

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

   6. swap-sqr 80.6%

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

   7. add-sqr-sqrt 80.8%

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

   8. metadata-eval 81.0%

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

7. Applied egg-rr 81.0%

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

8. Final simplification 81.0%

   \[\leadsto u2 \cdot \sqrt{\frac{u1}{1 - u1} \cdot 39.47841760436263} \]
9. 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 98.4%

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

3. Taylor expanded in u2 around 0 80.6%

   \[\leadsto \color{blue}{6.28318530718 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)} \]
4. 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 98.4%

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

3. Taylor expanded in u2 around 0 80.6%

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

4. Taylor expanded in u1 around 0 72.0%

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

   1. +-commutative 84.4%

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

6. Simplified 72.0%

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

7. Final simplification 72.0%

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

Alternative 8: 64.6% 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 98.4%

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

3. Taylor expanded in u2 around 0 80.6%

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

4. Taylor expanded in u1 around 0 64.2%

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

5. Taylor expanded in u1 around -inf -0.0%

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

   1. *-commutative -0.0%

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

   2. associate-*r* -0.0%

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

   3. *-commutative -0.0%

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

   4. unpow2 -0.0%

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

   5. rem-square-sqrt 64.3%

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

   6. neg-mul-1 64.3%

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

   7. distribute-rgt-neg-in 64.3%

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

   8. distribute-lft-neg-in 64.3%

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

   9. metadata-eval 64.3%

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

   10. associate-*r* 64.2%

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

   11. *-commutative 64.2%

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

   12. associate-*r* 64.3%

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

   13. *-commutative 64.3%

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

   14. associate-*r* 64.3%

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

7. Simplified 64.3%

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

8. Final simplification 64.3%

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

Alternative 9: 64.6% 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. Add Preprocessing

3. Taylor expanded in u2 around 0 80.6%

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

4. Taylor expanded in u1 around 0 64.2%

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

5. Final simplification 64.2%

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

Alternative 10: 20.5% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ u1 \cdot \left(u2 \cdot 6.28318530718 + 3.14159265359 \cdot \frac{u2}{u1}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* u1 (+ (* u2 6.28318530718) (* 3.14159265359 (/ u2 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return u1 * ((u2 * 6.28318530718f) + (3.14159265359f * (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 = u1 * ((u2 * 6.28318530718e0) + (3.14159265359e0 * (u2 / u1)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u1 * Float32(Float32(u2 * Float32(6.28318530718)) + Float32(Float32(3.14159265359) * Float32(u2 / u1))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u1 * ((u2 * single(6.28318530718)) + (single(3.14159265359) * (u2 / u1)));
end

Derivation

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 84.4%

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

   1. distribute-lft-in 84.3%

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

   2. *-rgt-identity 84.3%

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

   3. unpow2 84.3%

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

5. Simplified 84.3%

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

6. Taylor expanded in u2 around 0 71.9%

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

   1. +-commutative 71.9%

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

   2. unpow2 71.9%

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

   3. rem-square-sqrt 71.9%

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

   4. hypot-undefine 72.0%

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

8. Simplified 72.0%

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

9. Taylor expanded in u1 around inf 20.6%

   \[\leadsto \color{blue}{u1 \cdot \left(3.14159265359 \cdot \frac{u2}{u1} + 6.28318530718 \cdot u2\right)} \]

10. Final simplification 20.6%

    \[\leadsto u1 \cdot \left(u2 \cdot 6.28318530718 + 3.14159265359 \cdot \frac{u2}{u1}\right) \]
11. Add Preprocessing

Alternative 11: 20.5% accurate, 19.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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 84.4%

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

   1. distribute-lft-in 84.3%

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

   2. *-rgt-identity 84.3%

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

   3. unpow2 84.3%

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

5. Simplified 84.3%

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

6. Taylor expanded in u2 around 0 71.9%

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

   1. +-commutative 71.9%

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

   2. unpow2 71.9%

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

   3. rem-square-sqrt 71.9%

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

   4. hypot-undefine 72.0%

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

8. Simplified 72.0%

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

9. Taylor expanded in u1 around inf 20.6%

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

10. Final simplification 20.6%

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

Alternative 12: 19.4% accurate, 41.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u1 around 0 84.4%

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

   1. distribute-lft-in 84.3%

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

   2. *-rgt-identity 84.3%

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

   3. unpow2 84.3%

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

5. Simplified 84.3%

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

6. Taylor expanded in u2 around 0 71.9%

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

   1. +-commutative 71.9%

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

   2. unpow2 71.9%

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

   3. rem-square-sqrt 71.9%

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

   4. hypot-undefine 72.0%

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

8. Simplified 72.0%

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

9. Taylor expanded in u1 around inf 19.3%

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

10. Final simplification 19.3%

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

Reproduce

herbie shell --seed 2024103 
(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))))