Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.4% → 98.4%

Time: 18.3s

Alternatives: 13

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: 98.4% 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, 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 98.7%

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

2. Final simplification 98.7%

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

Alternative 2: 93.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{u1}{1 - u1}\\ \mathbf{if}\;t_0 \leq 0.004999999888241291:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{39.47841760436263 \cdot \left(t_0 \cdot \left(u2 \cdot u2\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (/ u1 (- 1.0 u1))))
   (if (<= t_0 0.004999999888241291)
     (* (sin (* 6.28318530718 u2)) (sqrt (* u1 (+ u1 1.0))))
     (sqrt (* 39.47841760436263 (* t_0 (* u2 u2)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u1 / (1.0f - u1);
	float tmp;
	if (t_0 <= 0.004999999888241291f) {
		tmp = sinf((6.28318530718f * u2)) * sqrtf((u1 * (u1 + 1.0f)));
	} else {
		tmp = sqrtf((39.47841760436263f * (t_0 * (u2 * u2))));
	}
	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) :: t_0
    real(4) :: tmp
    t_0 = u1 / (1.0e0 - u1)
    if (t_0 <= 0.004999999888241291e0) then
        tmp = sin((6.28318530718e0 * u2)) * sqrt((u1 * (u1 + 1.0e0)))
    else
        tmp = sqrt((39.47841760436263e0 * (t_0 * (u2 * u2))))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u1 / Float32(Float32(1.0) - u1))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.004999999888241291))
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(Float32(u1 * Float32(u1 + Float32(1.0)))));
	else
		tmp = sqrt(Float32(Float32(39.47841760436263) * Float32(t_0 * Float32(u2 * u2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = u1 / (single(1.0) - u1);
	tmp = single(0.0);
	if (t_0 <= single(0.004999999888241291))
		tmp = sin((single(6.28318530718) * u2)) * sqrt((u1 * (u1 + single(1.0))));
	else
		tmp = sqrt((single(39.47841760436263) * (t_0 * (u2 * u2))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 u1 (-.f32 1 u1)) < 0.00499999989

   1. Initial program 98.8%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. 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. associate-/r/ 98.5%

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

   3. Applied egg-rr 98.5%

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

   4. Taylor expanded in u1 around 0 97.3%

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

      1. +-commutative 97.3%

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

   6. Simplified 97.3%

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

   if 0.00499999989 < (/.f32 u1 (-.f32 1 u1))

   1. Initial program 98.3%

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

   2. Taylor expanded in u2 around 0 79.6%

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

      1. add-sqr-sqrt 79.1%

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

      2. sqrt-unprod 79.6%

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

      3. swap-sqr 79.6%

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

      4. metadata-eval 79.6%

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

      5. swap-sqr 80.0%

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

      6. add-sqr-sqrt 79.9%

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

   4. Applied egg-rr 79.9%

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

      1. *-commutative 79.9%

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

   6. Simplified 79.9%

      \[\leadsto \color{blue}{\sqrt{39.47841760436263 \cdot \left(\frac{u1}{1 - u1} \cdot \left(u2 \cdot u2\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{u1}{1 - u1} \leq 0.004999999888241291:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{39.47841760436263 \cdot \left(\frac{u1}{1 - u1} \cdot \left(u2 \cdot u2\right)\right)}\\ \end{array} \]

Alternative 3: 90.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.02199999988079071:\\ \;\;\;\;\sqrt{u2 \cdot \frac{u1 \cdot 39.47841760436263}{\frac{1 - u1}{u2}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.02199999988079071)
   (sqrt (* u2 (/ (* u1 39.47841760436263) (/ (- 1.0 u1) u2))))
   (* (sin (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.02199999988079071f) {
		tmp = sqrtf((u2 * ((u1 * 39.47841760436263f) / ((1.0f - u1) / u2))));
	} 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) <= 0.02199999988079071e0) then
        tmp = sqrt((u2 * ((u1 * 39.47841760436263e0) / ((1.0e0 - u1) / u2))))
    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(0.02199999988079071))
		tmp = sqrt(Float32(u2 * Float32(Float32(u1 * Float32(39.47841760436263)) / Float32(Float32(Float32(1.0) - u1) / u2))));
	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(0.02199999988079071))
		tmp = sqrt((u2 * ((u1 * single(39.47841760436263)) / ((single(1.0) - u1) / u2))));
	else
		tmp = sin((single(6.28318530718) * u2)) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.0219999999

   1. Initial program 98.6%

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

   2. Taylor expanded in u2 around 0 94.4%

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

      1. add-sqr-sqrt 93.9%

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

      2. sqrt-unprod 94.4%

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

      3. swap-sqr 94.4%

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

      4. add-sqr-sqrt 94.5%

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

   4. Applied egg-rr 94.5%

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

      1. associate-*l* 94.4%

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

      2. associate-*r/ 94.5%

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

   6. Simplified 94.5%

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

      1. add-sqr-sqrt 93.9%

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

      2. sqrt-unprod 94.5%

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

      3. *-commutative 94.5%

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

      4. *-commutative 94.5%

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

      5. swap-sqr 94.3%

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

      6. add-sqr-sqrt 94.3%

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

      7. *-commutative 94.3%

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

      8. metadata-eval 94.8%

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

   8. Applied egg-rr 94.8%

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

      1. associate-*l* 94.9%

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

      2. associate-/l* 94.8%

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

      3. associate-*l/ 94.9%

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

   10. Simplified 94.9%

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

   if 0.0219999999 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 98.8%

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

   2. Taylor expanded in u1 around 0 72.1%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.02199999988079071:\\ \;\;\;\;\sqrt{u2 \cdot \frac{u1 \cdot 39.47841760436263}{\frac{1 - u1}{u2}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 4: 81.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

2. Taylor expanded in u2 around 0 79.8%

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

   1. add-sqr-sqrt 79.3%

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

   2. sqrt-unprod 79.8%

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

   3. swap-sqr 79.7%

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

   4. metadata-eval 79.9%

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

   5. swap-sqr 79.8%

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

   6. add-sqr-sqrt 79.9%

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

4. Applied egg-rr 79.9%

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

   1. *-commutative 79.9%

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

6. Simplified 79.9%

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

7. Final simplification 79.9%

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

Alternative 5: 81.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

2. Taylor expanded in u2 around 0 79.8%

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

   1. add-sqr-sqrt 79.4%

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

   2. sqrt-unprod 79.8%

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

   3. swap-sqr 79.8%

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

   4. add-sqr-sqrt 79.9%

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

4. Applied egg-rr 79.9%

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

   1. associate-*l* 79.8%

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

   2. associate-*r/ 79.8%

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

6. Simplified 79.8%

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

   1. add-sqr-sqrt 79.4%

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

   2. sqrt-unprod 79.8%

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

   3. *-commutative 79.8%

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

   4. *-commutative 79.8%

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

   5. swap-sqr 79.7%

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

   6. add-sqr-sqrt 79.7%

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

   7. *-commutative 79.7%

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

   8. metadata-eval 80.1%

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

8. Applied egg-rr 80.1%

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

   1. associate-*l* 80.1%

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

   2. *-commutative 80.1%

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

   3. associate-*r/ 80.0%

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

10. Simplified 80.0%

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

11. Final simplification 80.0%

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

Alternative 6: 81.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

2. Taylor expanded in u2 around 0 79.8%

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

   1. add-sqr-sqrt 79.3%

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

   2. sqrt-unprod 79.8%

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

   3. swap-sqr 79.7%

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

   4. metadata-eval 79.9%

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

   5. swap-sqr 79.8%

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

   6. add-sqr-sqrt 79.9%

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

4. Applied egg-rr 79.9%

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

   1. *-commutative 79.9%

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

   2. associate-*l* 80.0%

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

   3. associate-*l* 80.0%

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

   4. associate-*r/ 80.1%

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

6. Simplified 80.1%

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

7. Final simplification 80.1%

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

Alternative 7: 81.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

2. Taylor expanded in u2 around 0 79.8%

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

   1. add-sqr-sqrt 79.4%

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

   2. sqrt-unprod 79.8%

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

   3. swap-sqr 79.8%

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

   4. add-sqr-sqrt 79.9%

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

4. Applied egg-rr 79.9%

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

   1. associate-*l* 79.8%

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

   2. associate-*r/ 79.8%

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

6. Simplified 79.8%

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

   1. add-sqr-sqrt 79.4%

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

   2. sqrt-unprod 79.8%

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

   3. *-commutative 79.8%

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

   4. *-commutative 79.8%

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

   5. swap-sqr 79.7%

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

   6. add-sqr-sqrt 79.7%

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

   7. *-commutative 79.7%

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

   8. metadata-eval 80.1%

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

8. Applied egg-rr 80.1%

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

   1. associate-*l* 80.1%

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

   2. associate-/l* 80.1%

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

   3. associate-*l/ 80.1%

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

10. Simplified 80.1%

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

11. Final simplification 80.1%

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

Alternative 8: 73.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

2. Taylor expanded in u2 around 0 79.8%

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

   1. flip-- 79.7%

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

   2. associate-/r/ 79.7%

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

   3. metadata-eval 79.7%

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

   4. +-commutative 79.7%

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

4. Applied egg-rr 79.7%

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

5. Taylor expanded in u1 around 0 72.0%

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

   1. +-commutative 72.0%

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

   2. unpow2 72.0%

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

7. Simplified 72.0%

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

8. Final simplification 72.0%

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

Alternative 9: 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 98.7%

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

2. Taylor expanded in u2 around 0 79.8%

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

3. Final simplification 79.8%

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

Alternative 10: 81.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

2. Taylor expanded in u2 around 0 79.8%

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

   1. associate-*r* 79.8%

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

4. Simplified 79.8%

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

5. Final simplification 79.8%

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

Alternative 11: 64.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

2. Taylor expanded in u2 around 0 79.8%

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

3. Taylor expanded in u1 around 0 63.5%

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

   1. expm1-log1p-u 63.5%

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

   2. expm1-udef 28.9%

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

5. Applied egg-rr 28.9%

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

   1. expm1-def 63.5%

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

   2. expm1-log1p 63.5%

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

   3. *-commutative 63.5%

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

   4. associate-*l* 63.4%

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

7. Simplified 63.4%

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

   1. add-sqr-sqrt 63.3%

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

   2. sqrt-unprod 63.4%

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

   3. swap-sqr 63.4%

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

   4. swap-sqr 63.4%

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

   5. add-sqr-sqrt 63.5%

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

   6. metadata-eval 63.5%

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

9. Applied egg-rr 63.5%

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

    1. associate-*l* 63.5%

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

11. Simplified 63.5%

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

12. Final simplification 63.5%

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

Alternative 12: 64.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

2. Taylor expanded in u2 around 0 79.8%

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

   1. add-sqr-sqrt 79.4%

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

   2. sqrt-unprod 79.8%

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

   3. swap-sqr 79.8%

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

   4. add-sqr-sqrt 79.9%

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

4. Applied egg-rr 79.9%

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

   1. associate-*l* 79.8%

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

   2. associate-*r/ 79.8%

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

6. Simplified 79.8%

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

7. Taylor expanded in u1 around 0 63.5%

   \[\leadsto 6.28318530718 \cdot \sqrt{\color{blue}{{u2}^{2} \cdot u1}} \]
8. Step-by-step derivation

   1. *-commutative 63.5%

      \[\leadsto 6.28318530718 \cdot \sqrt{\color{blue}{u1 \cdot {u2}^{2}}} \]

   2. unpow2 63.5%

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

9. Simplified 63.5%

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

10. Final simplification 63.5%

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

Alternative 13: 64.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

2. Taylor expanded in u2 around 0 79.8%

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

3. Taylor expanded in u1 around 0 63.5%

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

4. Final simplification 63.5%

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

Reproduce

herbie shell --seed 2023274 
(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))))