Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.2%

Time: 11.6s

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: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. *-commutative 98.3%

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

   2. sqrt-div 98.0%

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

   3. associate-*r/ 98.1%

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

3. Applied egg-rr 98.1%

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

   1. associate-/l* 98.4%

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

5. Simplified 98.4%

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

   1. frac-2neg 98.4%

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

   2. div-inv 98.2%

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

   3. sqrt-undiv 98.2%

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

7. Applied egg-rr 98.2%

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

   1. distribute-lft-neg-out 98.2%

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

   2. neg-mul-1 98.2%

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

   3. metadata-eval 98.2%

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

   4. associate-*r/ 98.5%

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

   5. *-rgt-identity 98.5%

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

   6. times-frac 98.5%

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

   7. neg-mul-1 98.5%

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

   8. remove-double-neg 98.5%

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

   9. *-lft-identity 98.5%

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

   10. div-sub 98.5%

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

   11. sub-neg 98.5%

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

   12. *-inverses 98.5%

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

   13. metadata-eval 98.5%

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

9. Simplified 98.5%

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

10. Final simplification 98.5%

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

Alternative 2: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.03799999877810478:\\ \;\;\;\;\sqrt{39.47841760436263 \cdot \frac{u2}{\frac{1 - u1}{u2 \cdot u1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(6.28318530718 \cdot u2\right)}{{u1}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.03799999877810478)
   (sqrt (* 39.47841760436263 (/ u2 (/ (- 1.0 u1) (* u2 u1)))))
   (/ (sin (* 6.28318530718 u2)) (pow u1 -0.5))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.03799999877810478f) {
		tmp = sqrtf((39.47841760436263f * (u2 / ((1.0f - u1) / (u2 * u1)))));
	} else {
		tmp = sinf((6.28318530718f * u2)) / powf(u1, -0.5f);
	}
	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) <= 0.03799999877810478e0) then
        tmp = sqrt((39.47841760436263e0 * (u2 / ((1.0e0 - u1) / (u2 * u1)))))
    else
        tmp = sin((6.28318530718e0 * u2)) / (u1 ** (-0.5e0))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.03799999877810478))
		tmp = sqrt(Float32(Float32(39.47841760436263) * Float32(u2 / Float32(Float32(Float32(1.0) - u1) / Float32(u2 * u1)))));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) / (u1 ^ Float32(-0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.03799999877810478))
		tmp = sqrt((single(39.47841760436263) * (u2 / ((single(1.0) - u1) / (u2 * u1)))));
	else
		tmp = sin((single(6.28318530718) * u2)) / (u1 ^ single(-0.5));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.0379999988

   1. Initial program 98.4%

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

   2. Taylor expanded in u2 around 0 95.2%

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

      1. add-sqr-sqrt 94.8%

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

      2. sqrt-unprod 95.2%

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

      3. swap-sqr 95.0%

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

      4. metadata-eval 95.3%

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

      5. *-commutative 95.3%

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

      6. *-commutative 95.3%

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

      7. swap-sqr 95.3%

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

      8. add-sqr-sqrt 95.6%

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

   4. Applied egg-rr 95.6%

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

      1. associate-*r* 95.7%

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

      2. associate-*l/ 95.6%

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

   6. Simplified 95.6%

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

      1. *-commutative 95.6%

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

      2. clear-num 95.6%

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

      3. un-div-inv 95.7%

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

      4. *-commutative 95.7%

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

   8. Applied egg-rr 95.7%

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

   if 0.0379999988 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.9%

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

   2. Taylor expanded in u1 around 0 76.1%

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

      1. *-commutative 76.1%

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

      2. pow1/2 76.1%

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

      3. metadata-eval 76.1%

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

      4. pow-flip 76.1%

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

      5. div-inv 76.2%

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

   4. Applied egg-rr 76.2%

      \[\leadsto \color{blue}{\frac{\sin \left(6.28318530718 \cdot u2\right)}{{u1}^{-0.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.03799999877810478:\\ \;\;\;\;\sqrt{39.47841760436263 \cdot \frac{u2}{\frac{1 - u1}{u2 \cdot u1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(6.28318530718 \cdot u2\right)}{{u1}^{-0.5}}\\ \end{array} \]

Alternative 3: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.03799999877810478:\\ \;\;\;\;\sqrt{39.47841760436263 \cdot \frac{u2}{\frac{1 - u1}{u2 \cdot u1}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.03799999877810478)
   (sqrt (* 39.47841760436263 (/ u2 (/ (- 1.0 u1) (* u2 u1)))))
   (* (sin (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.03799999877810478f) {
		tmp = sqrtf((39.47841760436263f * (u2 / ((1.0f - u1) / (u2 * u1)))));
	} 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) <= 0.03799999877810478e0) then
        tmp = sqrt((39.47841760436263e0 * (u2 / ((1.0e0 - u1) / (u2 * u1)))))
    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(0.03799999877810478))
		tmp = sqrt(Float32(Float32(39.47841760436263) * Float32(u2 / Float32(Float32(Float32(1.0) - u1) / Float32(u2 * u1)))));
	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(0.03799999877810478))
		tmp = sqrt((single(39.47841760436263) * (u2 / ((single(1.0) - u1) / (u2 * u1)))));
	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) < 0.0379999988

   1. Initial program 98.4%

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

   2. Taylor expanded in u2 around 0 95.2%

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

      1. add-sqr-sqrt 94.8%

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

      2. sqrt-unprod 95.2%

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

      3. swap-sqr 95.0%

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

      4. metadata-eval 95.3%

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

      5. *-commutative 95.3%

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

      6. *-commutative 95.3%

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

      7. swap-sqr 95.3%

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

      8. add-sqr-sqrt 95.6%

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

   4. Applied egg-rr 95.6%

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

      1. associate-*r* 95.7%

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

      2. associate-*l/ 95.6%

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

   6. Simplified 95.6%

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

      1. *-commutative 95.6%

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

      2. clear-num 95.6%

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

      3. un-div-inv 95.7%

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

      4. *-commutative 95.7%

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

   8. Applied egg-rr 95.7%

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

   if 0.0379999988 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.9%

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

   2. Taylor expanded in u1 around 0 76.1%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.03799999877810478:\\ \;\;\;\;\sqrt{39.47841760436263 \cdot \frac{u2}{\frac{1 - u1}{u2 \cdot u1}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

2. Final simplification 98.3%

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

Alternative 5: 81.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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 80.4%

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

   1. add-sqr-sqrt 80.1%

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

   2. sqrt-unprod 80.4%

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

   3. swap-sqr 80.3%

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

   4. metadata-eval 80.5%

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

   5. *-commutative 80.5%

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

   6. *-commutative 80.5%

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

   7. swap-sqr 80.5%

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

   8. add-sqr-sqrt 80.7%

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

4. Applied egg-rr 80.7%

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

   1. associate-*r* 80.8%

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

   2. associate-*l/ 80.7%

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

6. Simplified 80.7%

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

7. Taylor expanded in u2 around 0 80.7%

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

   1. associate-*r/ 80.8%

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

9. Simplified 80.8%

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

10. Final simplification 80.8%

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

Alternative 6: 81.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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 80.4%

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

   1. add-sqr-sqrt 80.1%

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

   2. sqrt-unprod 80.4%

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

   3. swap-sqr 80.3%

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

   4. metadata-eval 80.5%

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

   5. *-commutative 80.5%

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

   6. *-commutative 80.5%

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

   7. swap-sqr 80.5%

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

   8. add-sqr-sqrt 80.7%

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

4. Applied egg-rr 80.7%

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

   1. associate-*r* 80.8%

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

   2. associate-*l/ 80.7%

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

6. Simplified 80.7%

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

   1. *-commutative 80.7%

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

   2. clear-num 80.7%

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

   3. un-div-inv 80.8%

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

   4. *-commutative 80.8%

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

8. Applied egg-rr 80.8%

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

9. Final simplification 80.8%

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

Alternative 7: 72.6% 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 98.3%

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

2. Taylor expanded in u2 around 0 80.4%

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

3. Taylor expanded in u1 around 0 73.9%

   \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \sqrt{\color{blue}{{u1}^{2} + \left({u1}^{3} + u1\right)}}\right) \]
4. Step-by-step derivation

   1. associate-+r+ 73.9%

      \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \sqrt{\color{blue}{\left({u1}^{2} + {u1}^{3}\right) + u1}}\right) \]

   2. unpow2 73.9%

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

   3. cube-mult 73.9%

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

   4. unpow2 73.9%

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

   5. distribute-lft-out 73.9%

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

   6. +-commutative 73.9%

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

   7. unpow2 73.9%

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

   8. fma-udef 73.9%

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

   9. +-commutative 73.9%

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

5. Simplified 73.9%

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

6. Taylor expanded in u1 around 0 71.0%

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

   1. unpow2 71.0%

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

8. Simplified 71.0%

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

9. Final simplification 71.0%

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

Alternative 8: 80.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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 80.4%

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

3. Final simplification 80.4%

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

Alternative 9: 80.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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 80.4%

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

   1. *-commutative 80.4%

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

   2. associate-*l* 80.4%

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

4. Simplified 80.4%

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

5. Final simplification 80.4%

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

Alternative 10: 80.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. *-commutative 98.3%

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

   2. sqrt-div 98.0%

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

   3. associate-*r/ 98.1%

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

3. Applied egg-rr 98.1%

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

   1. associate-/l* 98.4%

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

5. Simplified 98.4%

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

   1. frac-2neg 98.4%

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

   2. div-inv 98.2%

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

   3. sqrt-undiv 98.2%

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

7. Applied egg-rr 98.2%

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

   1. distribute-lft-neg-out 98.2%

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

   2. neg-mul-1 98.2%

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

   3. metadata-eval 98.2%

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

   4. associate-*r/ 98.5%

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

   5. *-rgt-identity 98.5%

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

   6. times-frac 98.5%

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

   7. neg-mul-1 98.5%

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

   8. remove-double-neg 98.5%

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

   9. *-lft-identity 98.5%

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

   10. div-sub 98.5%

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

   11. sub-neg 98.5%

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

   12. *-inverses 98.5%

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

   13. metadata-eval 98.5%

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

9. Simplified 98.5%

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

10. Taylor expanded in u2 around 0 80.5%

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

11. Final simplification 80.5%

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

Alternative 11: 80.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

   1. *-commutative 98.3%

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

   2. sqrt-div 98.0%

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

   3. associate-*r/ 98.1%

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

3. Applied egg-rr 98.1%

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

   1. associate-/l* 98.4%

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

5. Simplified 98.4%

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

   1. expm1-log1p-u 96.2%

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

   2. expm1-udef 96.1%

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

   3. sqrt-undiv 96.2%

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

7. Applied egg-rr 96.2%

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

   1. expm1-def 96.2%

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

   2. expm1-log1p 98.5%

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

9. Simplified 98.5%

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

10. Taylor expanded in u2 around 0 80.5%

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

11. Final simplification 80.5%

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

Alternative 12: 64.3% 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 98.3%

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

2. Taylor expanded in u2 around 0 80.4%

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

3. Taylor expanded in u1 around 0 63.0%

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

   1. add-sqr-sqrt 62.8%

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

   2. sqrt-unprod 63.0%

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

   3. *-commutative 63.0%

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

   4. *-commutative 63.0%

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

   5. swap-sqr 63.1%

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

   6. add-sqr-sqrt 63.1%

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

5. Applied egg-rr 63.1%

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

6. Final simplification 63.1%

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

Alternative 13: 64.3% 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 98.3%

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

2. Taylor expanded in u2 around 0 80.4%

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

3. Taylor expanded in u1 around 0 63.0%

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

4. Final simplification 63.0%

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

Alternative 14: 64.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.3%

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

2. Taylor expanded in u1 around 0 74.2%

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

3. Taylor expanded in u2 around 0 63.0%

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

   1. *-commutative 63.0%

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

   2. associate-*l* 63.1%

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

5. Simplified 63.1%

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

6. Final simplification 63.1%

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

Alternative 15: -0.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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 80.4%

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

   1. add-sqr-sqrt 80.1%

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

   2. sqrt-unprod 80.4%

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

   3. swap-sqr 80.3%

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

   4. metadata-eval 80.5%

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

   5. *-commutative 80.5%

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

   6. *-commutative 80.5%

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

   7. swap-sqr 80.5%

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

   8. add-sqr-sqrt 80.7%

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

4. Applied egg-rr 80.7%

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

   1. associate-*r* 80.8%

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

   2. associate-*l/ 80.7%

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

6. Simplified 80.7%

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

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

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

8. Final simplification -0.0%

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

Reproduce

herbie shell --seed 2023238 
(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))))