Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.8% → 98.3%

Time: 12.5s

Alternatives: 6

Speedup: 2.0×

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(2 \cdot \sqrt[3]{{\pi}^{3} \cdot {u2}^{3}}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (- (log1p (- u1))))
  (sin (* 2.0 (cbrt (* (pow PI 3.0) (pow u2 3.0)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * sinf((2.0f * cbrtf((powf(((float) M_PI), 3.0f) * powf(u2, 3.0f)))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * sin(Float32(Float32(2.0) * cbrt(Float32((Float32(pi) ^ Float32(3.0)) * (u2 ^ Float32(3.0)))))))
end

Derivation

1. Initial program 60.9%

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

   1. sub-neg 60.9%

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

   2. log1p-def 98.3%

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

   3. associate-*l* 98.3%

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

3. Simplified 98.3%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(2 \cdot \left(\pi \cdot u2\right)\right)} \]
4. Step-by-step derivation

   1. add-cbrt-cube 98.3%

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

   2. add-cbrt-cube 98.3%

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

   3. cbrt-unprod 98.3%

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

   4. pow3 98.3%

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

   5. pow3 98.3%

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

5. Applied egg-rr 98.3%

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

6. Final simplification 98.3%

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

Alternative 2: 93.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\log \left(1 - u1\right)\\ \mathbf{if}\;t_0 \leq 0.004000000189989805:\\ \;\;\;\;\sqrt{u1 - u1 \cdot \left(u1 \cdot -0.5\right)} \cdot \sin \left(u2 \cdot \left(2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_0} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (- (log (- 1.0 u1)))))
   (if (<= t_0 0.004000000189989805)
     (* (sqrt (- u1 (* u1 (* u1 -0.5)))) (sin (* u2 (* 2.0 PI))))
     (* (sqrt t_0) (* 2.0 (* PI u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = -logf((1.0f - u1));
	float tmp;
	if (t_0 <= 0.004000000189989805f) {
		tmp = sqrtf((u1 - (u1 * (u1 * -0.5f)))) * sinf((u2 * (2.0f * ((float) M_PI))));
	} else {
		tmp = sqrtf(t_0) * (2.0f * (((float) M_PI) * u2));
	}
	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(0.004000000189989805))
		tmp = Float32(sqrt(Float32(u1 - Float32(u1 * Float32(u1 * Float32(-0.5))))) * sin(Float32(u2 * Float32(Float32(2.0) * Float32(pi)))));
	else
		tmp = Float32(sqrt(t_0) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	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(0.004000000189989805))
		tmp = sqrt((u1 - (u1 * (u1 * single(-0.5))))) * sin((u2 * (single(2.0) * single(pi))));
	else
		tmp = sqrt(t_0) * (single(2.0) * (single(pi) * u2));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 1 u1))) < 0.00400000019

   1. Initial program 46.3%

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

   2. Taylor expanded in u1 around 0 97.4%

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

      1. +-commutative 97.4%

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

      2. mul-1-neg 97.4%

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

      3. unsub-neg 97.4%

         \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} - u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. unpow2 97.4%

         \[\leadsto \sqrt{-\left(-0.5 \cdot \color{blue}{\left(u1 \cdot u1\right)} - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. associate-*r* 97.4%

         \[\leadsto \sqrt{-\left(\color{blue}{\left(-0.5 \cdot u1\right) \cdot u1} - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. Simplified 97.4%

      \[\leadsto \sqrt{-\color{blue}{\left(\left(-0.5 \cdot u1\right) \cdot u1 - u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   if 0.00400000019 < (neg.f32 (log.f32 (-.f32 1 u1)))

   1. Initial program 94.8%

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

      1. associate-*r* 94.8%

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

      2. add-cube-cbrt 94.3%

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

      3. pow3 94.5%

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

   3. Applied egg-rr 94.5%

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

   4. Taylor expanded in u2 around 0 80.1%

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

      1. associate-*r* 80.1%

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

      2. *-commutative 80.1%

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

      3. associate-*r* 80.1%

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

   6. Simplified 80.1%

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

      1. rem-cube-cbrt 80.3%

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

      2. associate-*r* 80.3%

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

      3. *-commutative 80.3%

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

      4. associate-*r* 80.3%

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

   8. Applied egg-rr 80.3%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.004000000189989805:\\ \;\;\;\;\sqrt{u1 - u1 \cdot \left(u1 \cdot -0.5\right)} \cdot \sin \left(u2 \cdot \left(2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \end{array} \]

Alternative 3: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.9%

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

   1. sub-neg 60.9%

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

   2. log1p-def 98.3%

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

   3. associate-*l* 98.3%

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

3. Simplified 98.3%

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

4. Final simplification 98.3%

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

Alternative 4: 86.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(\pi \cdot u2\right)\\ \mathbf{if}\;1 - u1 \leq 0.9995399713516235:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\sin t_0 \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* 2.0 (* PI u2))))
   (if (<= (- 1.0 u1) 0.9995399713516235)
     (* (sqrt (- (log (- 1.0 u1)))) t_0)
     (* (sin t_0) (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = 2.0f * (((float) M_PI) * u2);
	float tmp;
	if ((1.0f - u1) <= 0.9995399713516235f) {
		tmp = sqrtf(-logf((1.0f - u1))) * t_0;
	} else {
		tmp = sinf(t_0) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(2.0) * Float32(Float32(pi) * u2))
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.9995399713516235))
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * t_0);
	else
		tmp = Float32(sin(t_0) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = single(2.0) * (single(pi) * u2);
	tmp = single(0.0);
	if ((single(1.0) - u1) <= single(0.9995399713516235))
		tmp = sqrt(-log((single(1.0) - u1))) * t_0;
	else
		tmp = sin(t_0) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 1 u1) < 0.999539971

   1. Initial program 91.9%

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

      1. associate-*r* 91.9%

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

      2. add-cube-cbrt 91.5%

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

      3. pow3 91.6%

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

   3. Applied egg-rr 91.6%

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

   4. Taylor expanded in u2 around 0 79.1%

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

      1. associate-*r* 79.1%

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

      2. *-commutative 79.1%

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

      3. associate-*r* 79.1%

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

   6. Simplified 79.1%

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

      1. rem-cube-cbrt 79.3%

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

      2. associate-*r* 79.3%

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

      3. *-commutative 79.3%

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

      4. associate-*r* 79.3%

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

   8. Applied egg-rr 79.3%

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

   if 0.999539971 < (-.f32 1 u1)

   1. Initial program 40.4%

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

      1. sub-neg 40.4%

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

      2. log1p-def 98.2%

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

      3. associate-*l* 98.2%

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

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(2 \cdot \left(\pi \cdot u2\right)\right)} \]
   4. Step-by-step derivation

      1. log1p-udef 40.4%

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

      2. sub-neg 40.4%

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

      3. add-sqr-sqrt 40.4%

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

      4. pow2 40.4%

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

   5. Applied egg-rr 89.3%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{0.25}\right)}^{2}} \cdot \sin \left(2 \cdot \left(\pi \cdot u2\right)\right) \]

   6. Taylor expanded in u1 around 0 91.3%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9995399713516235:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 5: 76.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \sin \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{u1} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sin (* 2.0 (* PI u2))) (sqrt u1)))

C

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

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sin(Float32(Float32(2.0) * Float32(Float32(pi) * u2))) * sqrt(u1))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sin((single(2.0) * (single(pi) * u2))) * sqrt(u1);
end

Derivation

1. Initial program 60.9%

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

   1. sub-neg 60.9%

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

   2. log1p-def 98.3%

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

   3. associate-*l* 98.3%

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

3. Simplified 98.3%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(2 \cdot \left(\pi \cdot u2\right)\right)} \]
4. Step-by-step derivation

   1. log1p-udef 60.9%

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

   2. sub-neg 60.9%

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

   3. add-sqr-sqrt 60.9%

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

   4. pow2 60.9%

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

5. Applied egg-rr 72.2%

   \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{0.25}\right)}^{2}} \cdot \sin \left(2 \cdot \left(\pi \cdot u2\right)\right) \]

6. Taylor expanded in u1 around 0 74.6%

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

7. Final simplification 74.6%

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

Alternative 6: 66.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.9%

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

2. Taylor expanded in u1 around 0 74.6%

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

   1. mul-1-neg 74.6%

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

4. Simplified 74.6%

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

5. Taylor expanded in u2 around 0 65.1%

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

   1. associate-*l* 65.1%

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

7. Simplified 65.1%

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

8. Final simplification 65.1%

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

Reproduce

herbie shell --seed 2023228 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann 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 (- (log (- 1.0 u1)))) (sin (* (* 2.0 PI) u2))))