Beckmann Sample, near normal, slope_x

Percentage Accurate: 58.0% → 99.0%

Time: 19.0s

Alternatives: 6

Speedup: 4.0×

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

\[\begin{array}{l} \\ \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf((1.0f - u1))) * cosf(((2.0f * ((float) M_PI)) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(-log((single(1.0) - u1))) * cos(((single(2.0) * single(pi)) * u2));
end

Initial Program: 58.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf((1.0f - u1))) * cosf(((2.0f * ((float) M_PI)) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(-log((single(1.0) - u1))) * cos(((single(2.0) * single(pi)) * u2));
end

Alternative 1: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log1p (- u1)))) (cos (* 2.0 (* PI u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * cosf((2.0f * (((float) M_PI) * u2)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * cos(Float32(Float32(2.0) * Float32(Float32(pi) * u2))))
end

Derivation

1. Initial program 57.9%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Step-by-step derivation

   1. sub-neg 57.9%

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. log1p-def 99.1%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. associate-*l* 99.1%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \color{blue}{\left(2 \cdot \left(\pi \cdot u2\right)\right)} \]

3. Simplified 99.1%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)} \]

4. Final simplification 99.1%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \]

Alternative 2: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\log \left(1 - u1\right)\\ \mathbf{if}\;t_0 \leq 0.0007999999797903001:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (- (log (- 1.0 u1)))))
   (if (<= t_0 0.0007999999797903001)
     (* (cos (* PI (* 2.0 u2))) (sqrt u1))
     (sqrt t_0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = -logf((1.0f - u1));
	float tmp;
	if (t_0 <= 0.0007999999797903001f) {
		tmp = cosf((((float) M_PI) * (2.0f * u2))) * sqrtf(u1);
	} else {
		tmp = sqrtf(t_0);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(-log(Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0007999999797903001))
		tmp = Float32(cos(Float32(Float32(pi) * Float32(Float32(2.0) * u2))) * sqrt(u1));
	else
		tmp = sqrt(t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = -log((single(1.0) - u1));
	tmp = single(0.0);
	if (t_0 <= single(0.0007999999797903001))
		tmp = cos((single(pi) * (single(2.0) * u2))) * sqrt(u1);
	else
		tmp = sqrt(t_0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 1 u1))) < 7.9999998e-4

   1. Initial program 42.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Step-by-step derivation

      1. add-cube-cbrt 42.8%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)} \cdot \sqrt[3]{\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)}\right) \cdot \sqrt[3]{\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)}} \]

      2. pow3 42.8%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)}\right)}^{3}} \]

   3. Applied egg-rr 87.5%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{\mathsf{log1p}\left(u1\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3}} \]

   4. Taylor expanded in u1 around 0 89.7%

      \[\leadsto \color{blue}{{1}^{0.16666666666666666} \cdot \left(\sqrt{u1} \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)} \]
   5. Step-by-step derivation

      1. pow-base-1 89.7%

         \[\leadsto \color{blue}{1} \cdot \left(\sqrt{u1} \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right) \]

      2. *-lft-identity 89.7%

         \[\leadsto \color{blue}{\sqrt{u1} \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)} \]

      3. *-commutative 89.7%

         \[\leadsto \color{blue}{\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \sqrt{u1}} \]

      4. associate-*r* 89.7%

         \[\leadsto \cos \color{blue}{\left(\left(2 \cdot u2\right) \cdot \pi\right)} \cdot \sqrt{u1} \]

      5. *-commutative 89.7%

         \[\leadsto \cos \left(\color{blue}{\left(u2 \cdot 2\right)} \cdot \pi\right) \cdot \sqrt{u1} \]

      6. *-commutative 89.7%

         \[\leadsto \cos \color{blue}{\left(\pi \cdot \left(u2 \cdot 2\right)\right)} \cdot \sqrt{u1} \]

   6. Simplified 89.7%

      \[\leadsto \color{blue}{\cos \left(\pi \cdot \left(u2 \cdot 2\right)\right) \cdot \sqrt{u1}} \]

   if 7.9999998e-4 < (neg.f32 (log.f32 (-.f32 1 u1)))

   1. Initial program 92.3%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u2 around 0 75.0%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.0007999999797903001:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)}\\ \end{array} \]

Alternative 3: 93.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.984000027179718:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 - u1 \cdot \left(u1 \cdot -0.5\right)} \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.984000027179718)
   (sqrt (- (log (- 1.0 u1))))
   (* (sqrt (- u1 (* u1 (* u1 -0.5)))) (cos (* u2 (* 2.0 PI))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((1.0f - u1) <= 0.984000027179718f) {
		tmp = sqrtf(-logf((1.0f - u1)));
	} else {
		tmp = sqrtf((u1 - (u1 * (u1 * -0.5f)))) * cosf((u2 * (2.0f * ((float) M_PI))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.984000027179718))
		tmp = sqrt(Float32(-log(Float32(Float32(1.0) - u1))));
	else
		tmp = Float32(sqrt(Float32(u1 - Float32(u1 * Float32(u1 * Float32(-0.5))))) * cos(Float32(u2 * Float32(Float32(2.0) * Float32(pi)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(1.0) - u1) <= single(0.984000027179718))
		tmp = sqrt(-log((single(1.0) - u1)));
	else
		tmp = sqrt((u1 - (u1 * (u1 * single(-0.5))))) * cos((u2 * (single(2.0) * single(pi))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 1 u1) < 0.984000027

   1. Initial program 97.7%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u2 around 0 79.4%

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

   if 0.984000027 < (-.f32 1 u1)

   1. Initial program 50.1%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u1 around 0 96.5%

      \[\leadsto \sqrt{-\color{blue}{\left(-1 \cdot u1 + -0.5 \cdot {u1}^{2}\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   3. Step-by-step derivation

      1. +-commutative 73.6%

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

      2. mul-1-neg 73.6%

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

      3. unsub-neg 73.6%

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

      4. *-commutative 73.6%

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

      5. unpow2 73.6%

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

      6. associate-*l* 73.6%

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

   4. Simplified 96.5%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.984000027179718:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 - u1 \cdot \left(u1 \cdot -0.5\right)} \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\\ \end{array} \]

Alternative 4: 78.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\log \left(1 - u1\right)\\ \mathbf{if}\;t_0 \leq 0.0034000000450760126:\\ \;\;\;\;\sqrt{u1 - u1 \cdot \left(u1 \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (- (log (- 1.0 u1)))))
   (if (<= t_0 0.0034000000450760126)
     (sqrt (- u1 (* u1 (* u1 -0.5))))
     (sqrt t_0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = -logf((1.0f - u1));
	float tmp;
	if (t_0 <= 0.0034000000450760126f) {
		tmp = sqrtf((u1 - (u1 * (u1 * -0.5f))));
	} else {
		tmp = sqrtf(t_0);
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -log((1.0e0 - u1))
    if (t_0 <= 0.0034000000450760126e0) then
        tmp = sqrt((u1 - (u1 * (u1 * (-0.5e0)))))
    else
        tmp = sqrt(t_0)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(-log(Float32(Float32(1.0) - u1)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0034000000450760126))
		tmp = sqrt(Float32(u1 - Float32(u1 * Float32(u1 * Float32(-0.5)))));
	else
		tmp = sqrt(t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = -log((single(1.0) - u1));
	tmp = single(0.0);
	if (t_0 <= single(0.0034000000450760126))
		tmp = sqrt((u1 - (u1 * (u1 * single(-0.5)))));
	else
		tmp = sqrt(t_0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 1 u1))) < 0.00340000005

   1. Initial program 46.5%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u2 around 0 40.0%

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

   3. Taylor expanded in u1 around 0 74.5%

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

      1. +-commutative 74.5%

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

      2. mul-1-neg 74.5%

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

      3. unsub-neg 74.5%

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

      4. *-commutative 74.5%

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

      5. unpow2 74.5%

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

      6. associate-*l* 74.5%

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

   5. Simplified 74.5%

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

   if 0.00340000005 < (neg.f32 (log.f32 (-.f32 1 u1)))

   1. Initial program 95.1%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u2 around 0 76.4%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.0034000000450760126:\\ \;\;\;\;\sqrt{u1 - u1 \cdot \left(u1 \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)}\\ \end{array} \]

Alternative 5: 72.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1 - u1 \cdot \left(u1 \cdot -0.5\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (sqrt (- u1 (* u1 (* u1 -0.5)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 - (u1 * (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 = sqrt((u1 - (u1 * (u1 * (-0.5e0)))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(u1 - Float32(u1 * Float32(u1 * Float32(-0.5)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 - (u1 * (u1 * single(-0.5)))));
end

Derivation

1. Initial program 57.9%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

2. Taylor expanded in u2 around 0 48.5%

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

3. Taylor expanded in u1 around 0 69.2%

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

   1. +-commutative 69.2%

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

   2. mul-1-neg 69.2%

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

   3. unsub-neg 69.2%

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

   4. *-commutative 69.2%

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

   5. unpow2 69.2%

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

   6. associate-*l* 69.2%

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

5. Simplified 69.2%

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

6. Final simplification 69.2%

   \[\leadsto \sqrt{u1 - u1 \cdot \left(u1 \cdot -0.5\right)} \]

Alternative 6: 64.5% accurate, 4.0× 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 57.9%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

2. Taylor expanded in u2 around 0 48.5%

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

   1. sub-neg 48.5%

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

   2. log1p-udef 75.9%

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

   3. add-cbrt-cube 75.9%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}}} \cdot 1 \]

   4. pow1/3 74.3%

      \[\leadsto \color{blue}{{\left(\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right)}^{0.3333333333333333}} \cdot 1 \]

4. Applied egg-rr 59.4%

   \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot 1 \]

5. Taylor expanded in u1 around 0 61.6%

   \[\leadsto \color{blue}{\sqrt{u1}} \cdot 1 \]

6. Final simplification 61.6%

   \[\leadsto \sqrt{u1} \]

Reproduce

herbie shell --seed 2023278 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann 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 (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))