Trowbridge-Reitz Sample, near normal, slope_x

Percentage Accurate: 99.0% → 99.0%

Time: 12.7s

Alternatives: 8

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{u1}{1 - u1}\right)}^{0.5} \cdot \cos \left(6.28318530718 \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (pow (/ u1 (- 1.0 u1)) 0.5) (cos (* 6.28318530718 u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return powf((u1 / (1.0f - u1)), 0.5f) * cosf((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 = ((u1 / (1.0e0 - u1)) ** 0.5e0) * cos((6.28318530718e0 * u2))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32((Float32(u1 / Float32(Float32(1.0) - u1)) ^ Float32(0.5)) * cos(Float32(Float32(6.28318530718) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = ((u1 / (single(1.0) - u1)) ^ single(0.5)) * cos((single(6.28318530718) * u2));
end

Derivation

1. Initial program 98.9%

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

   1. pow1/2 98.9%

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

4. Applied egg-rr 98.9%

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

5. Final simplification 98.9%

   \[\leadsto {\left(\frac{u1}{1 - u1}\right)}^{0.5} \cdot \cos \left(6.28318530718 \cdot u2\right) \]
6. Add Preprocessing

Alternative 2: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.005100000184029341:\\ \;\;\;\;{\left(\frac{u1}{1 - u1}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(6.28318530718 \cdot u2\right)}{{u1}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.005100000184029341)
   (pow (/ u1 (- 1.0 u1)) 0.5)
   (/ (cos (* 6.28318530718 u2)) (pow u1 -0.5))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.005100000184029341f) {
		tmp = powf((u1 / (1.0f - u1)), 0.5f);
	} else {
		tmp = cosf((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) <= 0.005100000184029341e0) then
        tmp = (u1 / (1.0e0 - u1)) ** 0.5e0
    else
        tmp = cos((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(0.005100000184029341))
		tmp = Float32(u1 / Float32(Float32(1.0) - u1)) ^ Float32(0.5);
	else
		tmp = Float32(cos(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(0.005100000184029341))
		tmp = (u1 / (single(1.0) - u1)) ^ single(0.5);
	else
		tmp = cos((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) < 0.00510000018

   1. Initial program 99.3%

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

   3. Taylor expanded in u2 around 0 97.7%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
   4. Step-by-step derivation

      1. pow1/2 99.4%

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

   5. Applied egg-rr 97.7%

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

   if 0.00510000018 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 98.1%

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

      1. *-commutative 98.1%

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

      2. sqrt-div 97.9%

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

      3. associate-*r/ 97.7%

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

   4. Applied egg-rr 97.7%

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

      1. associate-/l* 97.8%

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

   6. Simplified 97.8%

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

      1. clear-num 97.8%

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

      2. sqrt-div 98.0%

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

      3. pow1/2 98.0%

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

      4. pow-flip 98.3%

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

      5. metadata-eval 98.3%

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

   8. Applied egg-rr 98.3%

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

   9. Taylor expanded in u1 around 0 74.4%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.005100000184029341:\\ \;\;\;\;{\left(\frac{u1}{1 - u1}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(6.28318530718 \cdot u2\right)}{{u1}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.005100000184029341:\\ \;\;\;\;{\left(\frac{u1}{1 - u1}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* 6.28318530718 u2) 0.005100000184029341)
   (pow (/ u1 (- 1.0 u1)) 0.5)
   (* (cos (* 6.28318530718 u2)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((6.28318530718f * u2) <= 0.005100000184029341f) {
		tmp = powf((u1 / (1.0f - u1)), 0.5f);
	} else {
		tmp = cosf((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) <= 0.005100000184029341e0) then
        tmp = (u1 / (1.0e0 - u1)) ** 0.5e0
    else
        tmp = cos((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(0.005100000184029341))
		tmp = Float32(u1 / Float32(Float32(1.0) - u1)) ^ Float32(0.5);
	else
		tmp = Float32(cos(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(0.005100000184029341))
		tmp = (u1 / (single(1.0) - u1)) ^ single(0.5);
	else
		tmp = cos((single(6.28318530718) * u2)) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 314159265359/50000000000 u2) < 0.00510000018

   1. Initial program 99.3%

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

   3. Taylor expanded in u2 around 0 97.7%

      \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
   4. Step-by-step derivation

      1. pow1/2 99.4%

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

   5. Applied egg-rr 97.7%

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

   if 0.00510000018 < (*.f32 314159265359/50000000000 u2)

   1. Initial program 98.1%

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

   3. Taylor expanded in u1 around 0 74.3%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;6.28318530718 \cdot u2 \leq 0.005100000184029341:\\ \;\;\;\;{\left(\frac{u1}{1 - u1}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(6.28318530718 \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Final simplification 98.9%

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

Alternative 5: 80.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{u1}{1 - u1}\right)}^{0.5} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (pow (/ u1 (- 1.0 u1)) 0.5))

C

float code(float cosTheta_i, float u1, float u2) {
	return powf((u1 / (1.0f - u1)), 0.5f);
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = (u1 / (1.0e0 - u1)) ** 0.5e0
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u1 / Float32(Float32(1.0) - u1)) ^ Float32(0.5)
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (u1 / (single(1.0) - u1)) ^ single(0.5);
end

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u2 around 0 75.9%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
4. Step-by-step derivation

   1. pow1/2 98.9%

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

5. Applied egg-rr 76.0%

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

6. Final simplification 76.0%

   \[\leadsto {\left(\frac{u1}{1 - u1}\right)}^{0.5} \]
7. Add Preprocessing

Alternative 6: 80.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u2 around 0 75.9%

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

4. Final simplification 75.9%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \]
5. Add Preprocessing

Alternative 7: 63.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {u1}^{0.5} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (pow u1 0.5))

C

float code(float cosTheta_i, float u1, float u2) {
	return powf(u1, 0.5f);
}

Fortran

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

Julia

function code(cosTheta_i, u1, u2)
	return u1 ^ Float32(0.5)
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u1 ^ single(0.5);
end

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u2 around 0 75.9%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1}}} \]
4. Step-by-step derivation

   1. pow1/2 98.9%

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

5. Applied egg-rr 76.0%

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

6. Taylor expanded in u1 around 0 61.9%

   \[\leadsto {\color{blue}{u1}}^{0.5} \]

7. Final simplification 61.9%

   \[\leadsto {u1}^{0.5} \]
8. Add Preprocessing

Alternative 8: 63.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return 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 = sqrt(u1)
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(u1)
end

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in u2 around 0 75.9%

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

4. Taylor expanded in u1 around 0 61.9%

   \[\leadsto \sqrt{\color{blue}{u1}} \]

5. Final simplification 61.9%

   \[\leadsto \sqrt{u1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024027 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_x"
  :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))) (cos (* 6.28318530718 u2))))