Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.5%

Time: 13.2s

Alternatives: 11

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: 98.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(Float32(u1 / Float32(Float32(1.0) - u1)) * log(Float32(exp(1))))) * sin(Float32(Float32(6.28318530718) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(((u1 / (single(1.0) - u1)) * log(single(2.71828182845904523536)))) * sin((single(6.28318530718) * u2));
end

Derivation

1. Initial program 98.2%

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

   1. add-log-exp 50.8%

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

   2. *-un-lft-identity 50.8%

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

   3. exp-prod 50.8%

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

   4. log-pow 98.5%

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

3. Applied egg-rr 98.5%

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

4. Final simplification 98.5%

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

Alternative 2: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.006200000178068876:\\ \;\;\;\;\frac{6.28318530718}{\frac{\sqrt{\frac{1}{u1} + -1}}{u2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(6.28318530718 \cdot u2\right)}{\sqrt{\frac{1}{u1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.006200000178068876)
   (/ 6.28318530718 (/ (sqrt (+ (/ 1.0 u1) -1.0)) u2))
   (/ (sin (* 6.28318530718 u2)) (sqrt (/ 1.0 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.006200000178068876f) {
		tmp = 6.28318530718f / (sqrtf(((1.0f / u1) + -1.0f)) / u2);
	} else {
		tmp = sinf((6.28318530718f * u2)) / sqrtf((1.0f / u1));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((6.28318530718e0 * u2) <= 0.006200000178068876e0) then
        tmp = 6.28318530718e0 / (sqrt(((1.0e0 / u1) + (-1.0e0))) / u2)
    else
        tmp = sin((6.28318530718e0 * u2)) / sqrt((1.0e0 / u1))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(6.28318530718) * u2) <= Float32(0.006200000178068876))
		tmp = Float32(Float32(6.28318530718) / Float32(sqrt(Float32(Float32(Float32(1.0) / u1) + Float32(-1.0))) / u2));
	else
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) / sqrt(Float32(Float32(1.0) / u1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(6.28318530718) * u2) <= single(0.006200000178068876))
		tmp = single(6.28318530718) / (sqrt(((single(1.0) / u1) + single(-1.0))) / u2);
	else
		tmp = sin((single(6.28318530718) * u2)) / sqrt((single(1.0) / u1));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.00620000018

   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 97.3%

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

      1. *-commutative 97.3%

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

      2. associate-*r* 97.3%

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

   4. Simplified 97.3%

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

      1. clear-num 97.4%

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

      2. sqrt-div 97.4%

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

      3. metadata-eval 97.4%

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

      4. div-inv 97.5%

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

      5. associate-/l* 97.6%

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

      6. div-sub 97.5%

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

      7. *-inverses 97.5%

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

      8. sub-neg 97.5%

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

      9. metadata-eval 97.5%

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

   6. Applied egg-rr 97.5%

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

   if 0.00620000018 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.7%

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

      1. add-log-exp 51.4%

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

      2. *-un-lft-identity 51.4%

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

      3. exp-prod 51.4%

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

      4. log-pow 98.0%

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

   3. Applied egg-rr 98.0%

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

      1. *-commutative 98.0%

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

      2. rem-log-exp 97.7%

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

      3. associate-*l/ 97.7%

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

      4. sqrt-div 97.4%

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

      5. *-rgt-identity 97.4%

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

      6. clear-num 97.4%

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

      7. div-inv 97.6%

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

      8. sqrt-undiv 97.9%

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

   5. Applied egg-rr 97.9%

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

   6. Taylor expanded in u1 around 0 76.8%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.006200000178068876:\\ \;\;\;\;\frac{6.28318530718}{\frac{\sqrt{\frac{1}{u1} + -1}}{u2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(6.28318530718 \cdot u2\right)}{\sqrt{\frac{1}{u1}}}\\ \end{array} \]

Alternative 3: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.006200000178068876:\\ \;\;\;\;\frac{6.28318530718}{\frac{\sqrt{\frac{1}{u1} + -1}}{u2}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.006200000178068876)
   (/ 6.28318530718 (/ (sqrt (+ (/ 1.0 u1) -1.0)) u2))
   (* (sin (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.006200000178068876f) {
		tmp = 6.28318530718f / (sqrtf(((1.0f / u1) + -1.0f)) / 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) <= 0.006200000178068876e0) then
        tmp = 6.28318530718e0 / (sqrt(((1.0e0 / u1) + (-1.0e0))) / 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(0.006200000178068876))
		tmp = Float32(Float32(6.28318530718) / Float32(sqrt(Float32(Float32(Float32(1.0) / u1) + Float32(-1.0))) / 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(0.006200000178068876))
		tmp = single(6.28318530718) / (sqrt(((single(1.0) / u1) + single(-1.0))) / 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) < 0.00620000018

   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 97.3%

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

      1. *-commutative 97.3%

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

      2. associate-*r* 97.3%

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

   4. Simplified 97.3%

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

      1. clear-num 97.4%

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

      2. sqrt-div 97.4%

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

      3. metadata-eval 97.4%

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

      4. div-inv 97.5%

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

      5. associate-/l* 97.6%

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

      6. div-sub 97.5%

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

      7. *-inverses 97.5%

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

      8. sub-neg 97.5%

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

      9. metadata-eval 97.5%

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

   6. Applied egg-rr 97.5%

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

   if 0.00620000018 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 97.7%

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

   2. Taylor expanded in u1 around 0 76.7%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.006200000178068876:\\ \;\;\;\;\frac{6.28318530718}{\frac{\sqrt{\frac{1}{u1} + -1}}{u2}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]

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 98.2%

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

2. Final simplification 98.2%

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

Alternative 5: 98.3% 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 98.2%

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

   1. add-log-exp 50.8%

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

   2. *-un-lft-identity 50.8%

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

   3. exp-prod 50.8%

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

   4. log-pow 98.5%

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

3. Applied egg-rr 98.5%

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

   1. rem-log-exp 98.2%

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

   2. associate-*l/ 98.2%

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

   3. sqrt-div 98.0%

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

   4. *-rgt-identity 98.0%

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

   5. clear-num 98.0%

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

   6. associate-/r/ 97.9%

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

   7. sqrt-undiv 98.3%

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

5. Applied egg-rr 98.3%

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

   1. associate-/l* 98.4%

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

   2. *-lft-identity 98.4%

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

   3. div-sub 98.3%

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

   4. sub-neg 98.3%

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

   5. *-inverses 98.3%

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

   6. metadata-eval 98.3%

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

7. Simplified 98.3%

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

8. Final simplification 98.3%

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

Alternative 6: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

   1. add-log-exp 50.8%

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

   2. *-un-lft-identity 50.8%

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

   3. exp-prod 50.8%

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

   4. log-pow 98.5%

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

3. Applied egg-rr 98.5%

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

   1. *-commutative 98.5%

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

   2. rem-log-exp 98.2%

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

   3. associate-*l/ 98.2%

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

   4. sqrt-div 98.0%

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

   5. *-rgt-identity 98.0%

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

   6. clear-num 98.0%

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

   7. div-inv 98.1%

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

   8. sqrt-undiv 98.4%

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

5. Applied egg-rr 98.4%

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

6. Final simplification 98.4%

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

Alternative 7: 81.4% 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 98.2%

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

2. Taylor expanded in u2 around 0 83.0%

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

3. Final simplification 83.0%

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

Alternative 8: 81.4% accurate, 1.9× speedup

Math

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

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(6.28318530718) * Float32(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 98.2%

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

2. Taylor expanded in u2 around 0 83.0%

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

   1. *-commutative 83.0%

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

   2. sqrt-div 82.9%

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

   3. associate-*r/ 82.9%

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

4. Applied egg-rr 82.9%

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

   1. associate-/l* 82.9%

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

6. Simplified 82.9%

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

   1. expm1-log1p-u 81.5%

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

   2. expm1-udef 81.6%

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

   3. sqrt-undiv 81.6%

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

8. Applied egg-rr 81.6%

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

   1. expm1-def 81.6%

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

   2. expm1-log1p 83.2%

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

10. Simplified 83.2%

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

11. Final simplification 83.2%

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

Alternative 9: 81.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

2. Taylor expanded in u2 around 0 83.0%

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

   1. *-commutative 83.0%

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

   2. associate-*r* 83.0%

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

4. Simplified 83.0%

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

   1. clear-num 83.1%

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

   2. sqrt-div 83.1%

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

   3. metadata-eval 83.1%

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

   4. div-inv 83.2%

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

   5. associate-/l* 83.3%

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

   6. div-sub 83.2%

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

   7. *-inverses 83.2%

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

   8. sub-neg 83.2%

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

   9. metadata-eval 83.2%

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

6. Applied egg-rr 83.2%

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

7. Final simplification 83.2%

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

Alternative 10: 65.0% 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 98.2%

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

2. Taylor expanded in u2 around 0 83.0%

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

3. Taylor expanded in u1 around 0 67.8%

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

4. Final simplification 67.8%

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

Alternative 11: 65.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.2%

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

2. Taylor expanded in u2 around 0 83.0%

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

   1. *-commutative 83.0%

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

   2. associate-*r* 83.0%

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

4. Simplified 83.0%

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

5. Taylor expanded in u1 around 0 67.8%

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

6. Final simplification 67.8%

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

Reproduce

herbie shell --seed 2023322 
(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))))