Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.4%

Time: 35.5s

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: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1.5} \cdot {\sin \left(6.28318530718 \cdot u2\right)}^{3}} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (cbrt (* (pow (/ u1 (- 1.0 u1)) 1.5) (pow (sin (* 6.28318530718 u2)) 3.0))))

C

float code(float cosTheta_i, float u1, float u2) {
	return cbrtf((powf((u1 / (1.0f - u1)), 1.5f) * powf(sinf((6.28318530718f * u2)), 3.0f)));
}

Julia

function code(cosTheta_i, u1, u2)
	return cbrt(Float32((Float32(u1 / Float32(Float32(1.0) - u1)) ^ Float32(1.5)) * (sin(Float32(Float32(6.28318530718) * u2)) ^ Float32(3.0))))
end

Derivation

1. Initial program 98.4%

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

   1. add-cbrt-cube 98.4%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \sqrt{\frac{u1}{1 - u1}}}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   2. add-cbrt-cube 98.4%

      \[\leadsto \sqrt[3]{\left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \sqrt{\frac{u1}{1 - u1}}} \cdot \color{blue}{\sqrt[3]{\left(\sin \left(6.28318530718 \cdot u2\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right) \cdot \sin \left(6.28318530718 \cdot u2\right)}} \]

   3. cbrt-unprod 98.2%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \left(\left(\sin \left(6.28318530718 \cdot u2\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right)}} \]

   4. add-sqr-sqrt 98.5%

      \[\leadsto \sqrt[3]{\left(\color{blue}{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \left(\left(\sin \left(6.28318530718 \cdot u2\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right)} \]

   5. pow1 98.5%

      \[\leadsto \sqrt[3]{\left(\color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \left(\left(\sin \left(6.28318530718 \cdot u2\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right)} \]

   6. pow1/2 98.5%

      \[\leadsto \sqrt[3]{\left({\left(\frac{u1}{1 - u1}\right)}^{1} \cdot \color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{0.5}}\right) \cdot \left(\left(\sin \left(6.28318530718 \cdot u2\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right)} \]

   7. pow-prod-up 98.4%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{u1}{1 - u1}\right)}^{\left(1 + 0.5\right)}} \cdot \left(\left(\sin \left(6.28318530718 \cdot u2\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right)} \]

   8. metadata-eval 98.4%

      \[\leadsto \sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{\color{blue}{1.5}} \cdot \left(\left(\sin \left(6.28318530718 \cdot u2\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right)} \]

   9. pow3 98.4%

      \[\leadsto \sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1.5} \cdot \color{blue}{{\sin \left(6.28318530718 \cdot u2\right)}^{3}}} \]

4. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{u1}{1 - u1}\right)}^{1.5} \cdot {\sin \left(6.28318530718 \cdot u2\right)}^{3}}} \]
5. Add Preprocessing

Alternative 2: 93.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{u1}{1 - u1} \leq 0.007000000216066837:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \frac{1}{\sqrt{\frac{1 - u1}{u1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (/ u1 (- 1.0 u1)) 0.007000000216066837)
   (* (sin (* 6.28318530718 u2)) (sqrt (* u1 (+ u1 1.0))))
   (* (* 6.28318530718 u2) (/ 1.0 (sqrt (/ (- 1.0 u1) u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u1 / (1.0f - u1)) <= 0.007000000216066837f) {
		tmp = sinf((6.28318530718f * u2)) * sqrtf((u1 * (u1 + 1.0f)));
	} else {
		tmp = (6.28318530718f * u2) * (1.0f / sqrtf(((1.0f - u1) / 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 ((u1 / (1.0e0 - u1)) <= 0.007000000216066837e0) then
        tmp = sin((6.28318530718e0 * u2)) * sqrt((u1 * (u1 + 1.0e0)))
    else
        tmp = (6.28318530718e0 * u2) * (1.0e0 / sqrt(((1.0e0 - u1) / u1)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u1 / Float32(Float32(1.0) - u1)) <= Float32(0.007000000216066837))
		tmp = Float32(sin(Float32(Float32(6.28318530718) * u2)) * sqrt(Float32(u1 * Float32(u1 + Float32(1.0)))));
	else
		tmp = Float32(Float32(Float32(6.28318530718) * u2) * Float32(Float32(1.0) / sqrt(Float32(Float32(Float32(1.0) - u1) / u1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u1 / (single(1.0) - u1)) <= single(0.007000000216066837))
		tmp = sin((single(6.28318530718) * u2)) * sqrt((u1 * (u1 + single(1.0))));
	else
		tmp = (single(6.28318530718) * u2) * (single(1.0) / sqrt(((single(1.0) - u1) / u1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 u1 (-.f32 #s(literal 1 binary32) u1)) < 0.00700000022

   1. Initial program 98.3%

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

   3. Taylor expanded in u1 around 0 96.2%

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

      1. +-commutative 96.2%

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

   5. Simplified 96.2%

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

   if 0.00700000022 < (/.f32 u1 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 98.7%

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

      1. clear-num 98.4%

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

      2. sqrt-div 98.5%

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

      3. metadata-eval 98.5%

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in u2 around 0 82.9%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{u1}{1 - u1} \leq 0.007000000216066837:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \frac{1}{\sqrt{\frac{1 - u1}{u1}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.012000000104308128:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(6.28318530718 \cdot u2\right)}{\sqrt{\frac{1}{u1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.012000000104308128)
   (* (* 6.28318530718 u2) (sqrt (/ u1 (- 1.0 u1))))
   (/ (sin (* 6.28318530718 u2)) (sqrt (/ 1.0 u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.012000000104308128f) {
		tmp = (6.28318530718f * u2) * sqrtf((u1 / (1.0f - u1)));
	} 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) <= 0.012000000104308128e0) then
        tmp = (6.28318530718e0 * u2) * sqrt((u1 / (1.0e0 - u1)))
    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(0.012000000104308128))
		tmp = Float32(Float32(Float32(6.28318530718) * u2) * sqrt(Float32(u1 / Float32(Float32(1.0) - u1))));
	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(0.012000000104308128))
		tmp = (single(6.28318530718) * u2) * sqrt((u1 / (single(1.0) - u1)));
	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 #s(literal 314159265359/50000000000 binary32) u2) < 0.0120000001

   1. Initial program 98.6%

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

   3. Taylor expanded in u2 around 0 94.8%

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

   if 0.0120000001 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.1%

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

      1. clear-num 98.1%

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

      2. sqrt-div 97.8%

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

      3. metadata-eval 97.8%

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

   4. Applied egg-rr 97.8%

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

      1. div-sub 97.8%

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

      2. sub-neg 97.8%

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

      3. *-inverses 97.8%

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

      4. metadata-eval 97.8%

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

   6. Simplified 97.8%

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

      1. associate-*l/ 98.1%

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

      2. *-un-lft-identity 98.1%

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

   8. Applied egg-rr 98.1%

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

      1. *-commutative 98.1%

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

   10. Simplified 98.1%

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

   11. Taylor expanded in u1 around 0 73.8%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.012000000104308128:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(6.28318530718 \cdot u2\right)}{\sqrt{\frac{1}{u1}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.012000000104308128:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.012000000104308128)
   (* (* 6.28318530718 u2) (sqrt (/ u1 (- 1.0 u1))))
   (* (sin (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.012000000104308128f) {
		tmp = (6.28318530718f * u2) * sqrtf((u1 / (1.0f - 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) <= 0.012000000104308128e0) then
        tmp = (6.28318530718e0 * u2) * sqrt((u1 / (1.0e0 - 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(0.012000000104308128))
		tmp = Float32(Float32(Float32(6.28318530718) * u2) * sqrt(Float32(u1 / Float32(Float32(1.0) - 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(0.012000000104308128))
		tmp = (single(6.28318530718) * u2) * sqrt((u1 / (single(1.0) - 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 #s(literal 314159265359/50000000000 binary32) u2) < 0.0120000001

   1. Initial program 98.6%

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

   3. Taylor expanded in u2 around 0 94.8%

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

   if 0.0120000001 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 98.1%

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

   3. Taylor expanded in u1 around 0 73.6%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.012000000104308128:\\ \;\;\;\;\left(6.28318530718 \cdot u2\right) \cdot \sqrt{\frac{u1}{1 - u1}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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 98.4%

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

3. Final simplification 98.4%

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

Alternative 6: 81.4% accurate, 1.9× speedup

Math

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

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(Float32(6.28318530718) * 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 98.4%

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

3. Taylor expanded in u2 around 0 79.3%

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

4. Final simplification 79.3%

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

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 98.4%

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

3. Taylor expanded in u2 around 0 79.3%

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

4. Final simplification 79.3%

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

Alternative 8: 72.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0 79.3%

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

4. Taylor expanded in u1 around 0 70.5%

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

   1. +-commutative 85.2%

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

6. Simplified 70.5%

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

7. Final simplification 70.5%

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

Alternative 9: 64.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0 79.3%

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

4. Taylor expanded in u1 around 0 62.9%

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

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

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

   1. associate-*r* -0.0%

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

   2. unpow2 -0.0%

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

   3. rem-square-sqrt 62.9%

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

7. Simplified 62.9%

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

8. Final simplification 62.9%

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

Alternative 10: 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 98.4%

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

3. Taylor expanded in u2 around 0 79.3%

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

4. Taylor expanded in u1 around 0 62.9%

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

5. Final simplification 62.9%

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

Alternative 11: 14.8% accurate, 29.9× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return 6.28318530718f * (u2 * (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 * (u1 * u1))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0 79.3%

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

4. Taylor expanded in u1 around 0 75.0%

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

5. Taylor expanded in u1 around inf 15.0%

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

   1. unpow2 15.0%

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

7. Applied egg-rr 15.0%

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

8. Final simplification 15.0%

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

Alternative 12: 4.8% accurate, 41.8× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (* u2 (* u1 -3.14159265359)))

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 * (u1 * -3.14159265359f);
}

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 * (u1 * (-3.14159265359e0))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u2 * Float32(u1 * Float32(-3.14159265359)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u2 * (u1 * single(-3.14159265359));
end

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0 79.3%

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

4. Taylor expanded in u1 around 0 75.0%

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

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

   \[\leadsto 6.28318530718 \cdot \left(\color{blue}{\left({u1}^{2} \cdot \left({\left(\sqrt{-1}\right)}^{2} - 0.5 \cdot \frac{1}{u1}\right)\right)} \cdot u2\right) \]
6. Step-by-step derivation

   1. sub-neg -0.0%

      \[\leadsto 6.28318530718 \cdot \left(\left({u1}^{2} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} + \left(-0.5 \cdot \frac{1}{u1}\right)\right)}\right) \cdot u2\right) \]

   2. unpow2 -0.0%

      \[\leadsto 6.28318530718 \cdot \left(\left({u1}^{2} \cdot \left(\color{blue}{\sqrt{-1} \cdot \sqrt{-1}} + \left(-0.5 \cdot \frac{1}{u1}\right)\right)\right) \cdot u2\right) \]

   3. rem-square-sqrt 4.6%

      \[\leadsto 6.28318530718 \cdot \left(\left({u1}^{2} \cdot \left(\color{blue}{-1} + \left(-0.5 \cdot \frac{1}{u1}\right)\right)\right) \cdot u2\right) \]

   4. associate-*r/ 4.6%

      \[\leadsto 6.28318530718 \cdot \left(\left({u1}^{2} \cdot \left(-1 + \left(-\color{blue}{\frac{0.5 \cdot 1}{u1}}\right)\right)\right) \cdot u2\right) \]

   5. metadata-eval 4.6%

      \[\leadsto 6.28318530718 \cdot \left(\left({u1}^{2} \cdot \left(-1 + \left(-\frac{\color{blue}{0.5}}{u1}\right)\right)\right) \cdot u2\right) \]

   6. distribute-neg-frac 4.6%

      \[\leadsto 6.28318530718 \cdot \left(\left({u1}^{2} \cdot \left(-1 + \color{blue}{\frac{-0.5}{u1}}\right)\right) \cdot u2\right) \]

   7. metadata-eval 4.6%

      \[\leadsto 6.28318530718 \cdot \left(\left({u1}^{2} \cdot \left(-1 + \frac{\color{blue}{-0.5}}{u1}\right)\right) \cdot u2\right) \]

7. Simplified 4.6%

   \[\leadsto 6.28318530718 \cdot \left(\color{blue}{\left({u1}^{2} \cdot \left(-1 + \frac{-0.5}{u1}\right)\right)} \cdot u2\right) \]

8. Taylor expanded in u1 around 0 4.6%

   \[\leadsto \color{blue}{-3.14159265359 \cdot \left(u1 \cdot u2\right)} \]
9. Step-by-step derivation

   1. associate-*r* 4.6%

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

10. Simplified 4.6%

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

11. Final simplification 4.6%

    \[\leadsto u2 \cdot \left(u1 \cdot -3.14159265359\right) \]
12. Add Preprocessing

Alternative 13: 4.8% accurate, 41.8× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (* -3.14159265359 (* u1 u2)))

C

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

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(-3.14159265359) * Float32(u1 * u2))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(-3.14159265359) * (u1 * u2);
end

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in u2 around 0 79.3%

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

4. Taylor expanded in u1 around 0 75.0%

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

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

   \[\leadsto 6.28318530718 \cdot \left(\color{blue}{\left({u1}^{2} \cdot \left({\left(\sqrt{-1}\right)}^{2} - 0.5 \cdot \frac{1}{u1}\right)\right)} \cdot u2\right) \]
6. Step-by-step derivation

   1. sub-neg -0.0%

      \[\leadsto 6.28318530718 \cdot \left(\left({u1}^{2} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} + \left(-0.5 \cdot \frac{1}{u1}\right)\right)}\right) \cdot u2\right) \]

   2. unpow2 -0.0%

      \[\leadsto 6.28318530718 \cdot \left(\left({u1}^{2} \cdot \left(\color{blue}{\sqrt{-1} \cdot \sqrt{-1}} + \left(-0.5 \cdot \frac{1}{u1}\right)\right)\right) \cdot u2\right) \]

   3. rem-square-sqrt 4.6%

      \[\leadsto 6.28318530718 \cdot \left(\left({u1}^{2} \cdot \left(\color{blue}{-1} + \left(-0.5 \cdot \frac{1}{u1}\right)\right)\right) \cdot u2\right) \]

   4. associate-*r/ 4.6%

      \[\leadsto 6.28318530718 \cdot \left(\left({u1}^{2} \cdot \left(-1 + \left(-\color{blue}{\frac{0.5 \cdot 1}{u1}}\right)\right)\right) \cdot u2\right) \]

   5. metadata-eval 4.6%

      \[\leadsto 6.28318530718 \cdot \left(\left({u1}^{2} \cdot \left(-1 + \left(-\frac{\color{blue}{0.5}}{u1}\right)\right)\right) \cdot u2\right) \]

   6. distribute-neg-frac 4.6%

      \[\leadsto 6.28318530718 \cdot \left(\left({u1}^{2} \cdot \left(-1 + \color{blue}{\frac{-0.5}{u1}}\right)\right) \cdot u2\right) \]

   7. metadata-eval 4.6%

      \[\leadsto 6.28318530718 \cdot \left(\left({u1}^{2} \cdot \left(-1 + \frac{\color{blue}{-0.5}}{u1}\right)\right) \cdot u2\right) \]

7. Simplified 4.6%

   \[\leadsto 6.28318530718 \cdot \left(\color{blue}{\left({u1}^{2} \cdot \left(-1 + \frac{-0.5}{u1}\right)\right)} \cdot u2\right) \]

8. Taylor expanded in u1 around 0 4.6%

   \[\leadsto \color{blue}{-3.14159265359 \cdot \left(u1 \cdot u2\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024116 
(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))))