Beckmann Sample, near normal, slope_y

Percentage Accurate: 58.1% → 98.4%

Time: 12.2s

Alternatives: 8

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: 58.1% 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: 98.4% 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 61.1%

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

   1. sub-neg 61.1%

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

   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 2: 94.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.004000000189989805:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 - -0.5 \cdot \left(u1 \cdot u1\right)} \cdot \sin \left(\pi \cdot \left(u2 + u2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 (* 2.0 PI)) 0.004000000189989805)
   (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))
   (* (sqrt (- u1 (* -0.5 (* u1 u1)))) (sin (* PI (+ u2 u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * (2.0f * ((float) M_PI))) <= 0.004000000189989805f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	} else {
		tmp = sqrtf((u1 - (-0.5f * (u1 * u1)))) * sinf((((float) M_PI) * (u2 + u2)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(Float32(2.0) * Float32(pi))) <= Float32(0.004000000189989805))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(sqrt(Float32(u1 - Float32(Float32(-0.5) * Float32(u1 * u1)))) * sin(Float32(Float32(pi) * Float32(u2 + u2))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 2 (PI.f32)) u2) < 0.00400000019

   1. Initial program 61.1%

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

      1. sub-neg 61.1%

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

      2. log1p-def 98.6%

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

      3. associate-*l* 98.6%

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

   3. Simplified 98.6%

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

      1. add-sqr-sqrt 97.9%

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

      2. pow2 97.9%

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

   5. Applied egg-rr 97.9%

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

   6. Taylor expanded in u2 around 0 97.9%

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

   if 0.00400000019 < (*.f32 (*.f32 2 (PI.f32)) u2)

   1. Initial program 61.1%

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

      1. associate-*r* 61.1%

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

      2. add-cube-cbrt 60.9%

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

      3. pow3 61.0%

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

   3. Applied egg-rr 61.0%

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

   4. Taylor expanded in u1 around 0 85.8%

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

      1. +-commutative 85.8%

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

      2. mul-1-neg 85.8%

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

      3. unsub-neg 85.8%

         \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} - u1\right)}} \cdot {\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3} \]

      4. *-commutative 85.8%

         \[\leadsto \sqrt{-\left(\color{blue}{{u1}^{2} \cdot -0.5} - u1\right)} \cdot {\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3} \]

      5. unpow2 85.8%

         \[\leadsto \sqrt{-\left(\color{blue}{\left(u1 \cdot u1\right)} \cdot -0.5 - u1\right)} \cdot {\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3} \]

      6. associate-*l* 85.8%

         \[\leadsto \sqrt{-\left(\color{blue}{u1 \cdot \left(u1 \cdot -0.5\right)} - u1\right)} \cdot {\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3} \]

   6. Simplified 85.8%

      \[\leadsto \sqrt{-\color{blue}{\left(u1 \cdot \left(u1 \cdot -0.5\right) - u1\right)}} \cdot {\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3} \]

   7. Taylor expanded in u2 around inf 86.3%

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

      1. *-commutative 86.3%

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

      2. unpow2 86.3%

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

      3. count-2 86.3%

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

      4. distribute-rgt-out 86.3%

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

   9. Simplified 86.3%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.004000000189989805:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 - -0.5 \cdot \left(u1 \cdot u1\right)} \cdot \sin \left(\pi \cdot \left(u2 + u2\right)\right)\\ \end{array} \]

Alternative 3: 90.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(\pi \cdot u2\right)\\ \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.02500000037252903:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\sin t_0 \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* 2.0 (* PI u2))))
   (if (<= (* u2 (* 2.0 PI)) 0.02500000037252903)
     (* (sqrt (- (log1p (- 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 ((u2 * (2.0f * ((float) M_PI))) <= 0.02500000037252903f) {
		tmp = sqrtf(-log1pf(-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(u2 * Float32(Float32(2.0) * Float32(pi))) <= Float32(0.02500000037252903))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * t_0);
	else
		tmp = Float32(sin(t_0) * sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 2 (PI.f32)) u2) < 0.0250000004

   1. Initial program 61.2%

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

      1. sub-neg 61.2%

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

      2. log1p-def 98.5%

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

      3. associate-*l* 98.5%

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

   3. Simplified 98.5%

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

      1. add-sqr-sqrt 97.9%

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

      2. pow2 97.9%

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

   5. Applied egg-rr 97.9%

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

   6. Taylor expanded in u2 around 0 94.8%

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

   if 0.0250000004 < (*.f32 (*.f32 2 (PI.f32)) u2)

   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 97.9%

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

      3. associate-*l* 97.9%

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

   3. Simplified 97.9%

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

      1. neg-mul-1 97.9%

         \[\leadsto \sqrt{\color{blue}{-1 \cdot \mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(2 \cdot \left(\pi \cdot u2\right)\right) \]

      2. log1p-udef 60.9%

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

      3. sub-neg 60.9%

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

      4. neg-mul-1 60.9%

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

      5. add-sqr-sqrt 61.0%

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

      6. pow2 61.0%

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

   5. Applied egg-rr 73.3%

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

   6. Taylor expanded in u1 around 0 75.5%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.02500000037252903:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 4: 83.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.000699999975040555:\\ \;\;\;\;\pi \cdot \left(\sqrt{u1 - -0.5 \cdot \left(u1 \cdot u1\right)} \cdot \left(u2 + u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.000699999975040555)
   (* PI (* (sqrt (- u1 (* -0.5 (* u1 u1)))) (+ u2 u2)))
   (* (sin (* 2.0 (* PI u2))) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.000699999975040555f) {
		tmp = ((float) M_PI) * (sqrtf((u1 - (-0.5f * (u1 * u1)))) * (u2 + u2));
	} else {
		tmp = sinf((2.0f * (((float) M_PI) * u2))) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.000699999975040555))
		tmp = Float32(Float32(pi) * Float32(sqrt(Float32(u1 - Float32(Float32(-0.5) * Float32(u1 * u1)))) * Float32(u2 + u2)));
	else
		tmp = Float32(sin(Float32(Float32(2.0) * Float32(Float32(pi) * u2))) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (u2 <= single(0.000699999975040555))
		tmp = single(pi) * (sqrt((u1 - (single(-0.5) * (u1 * u1)))) * (u2 + u2));
	else
		tmp = sin((single(2.0) * (single(pi) * u2))) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u2 < 6.99999975e-4

   1. Initial program 61.3%

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

      1. associate-*r* 61.3%

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

      2. add-cube-cbrt 61.2%

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

      3. pow3 61.2%

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

   3. Applied egg-rr 61.2%

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

   4. Taylor expanded in u1 around 0 88.0%

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

      1. +-commutative 88.0%

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

      2. mul-1-neg 88.0%

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

      3. unsub-neg 88.0%

         \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} - u1\right)}} \cdot {\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3} \]

      4. *-commutative 88.0%

         \[\leadsto \sqrt{-\left(\color{blue}{{u1}^{2} \cdot -0.5} - u1\right)} \cdot {\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3} \]

      5. unpow2 88.0%

         \[\leadsto \sqrt{-\left(\color{blue}{\left(u1 \cdot u1\right)} \cdot -0.5 - u1\right)} \cdot {\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3} \]

      6. associate-*l* 88.0%

         \[\leadsto \sqrt{-\left(\color{blue}{u1 \cdot \left(u1 \cdot -0.5\right)} - u1\right)} \cdot {\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3} \]

   6. Simplified 88.0%

      \[\leadsto \sqrt{-\color{blue}{\left(u1 \cdot \left(u1 \cdot -0.5\right) - u1\right)}} \cdot {\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3} \]

   7. Taylor expanded in u2 around 0 88.1%

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

      1. associate-*r* 88.1%

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

      2. associate-*r* 88.1%

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

      3. *-commutative 88.1%

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

      4. *-commutative 88.1%

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

      5. associate-*l* 88.2%

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

      6. *-commutative 88.2%

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

      7. rem-log-exp 34.7%

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

      8. log-pow 34.8%

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

      9. unpow2 34.8%

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

      10. log-prod 34.7%

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

      11. rem-log-exp 44.0%

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

      12. rem-log-exp 88.2%

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

      13. unpow2 88.2%

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

   9. Simplified 88.2%

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

   if 6.99999975e-4 < u2

   1. Initial program 60.8%

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

      1. sub-neg 60.8%

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

      2. log1p-def 98.0%

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

      3. associate-*l* 98.0%

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

   3. Simplified 98.0%

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

      1. neg-mul-1 98.0%

         \[\leadsto \sqrt{\color{blue}{-1 \cdot \mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(2 \cdot \left(\pi \cdot u2\right)\right) \]

      2. log1p-udef 60.8%

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

      3. sub-neg 60.8%

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

      4. neg-mul-1 60.8%

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

      5. add-sqr-sqrt 61.0%

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

      6. pow2 61.0%

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

   5. Applied egg-rr 72.4%

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

   6. Taylor expanded in u1 around 0 74.6%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \leq 0.000699999975040555:\\ \;\;\;\;\pi \cdot \left(\sqrt{u1 - -0.5 \cdot \left(u1 \cdot u1\right)} \cdot \left(u2 + u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 5: 74.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \pi \cdot \left(\sqrt{u1 - -0.5 \cdot \left(u1 \cdot u1\right)} \cdot \left(u2 + u2\right)\right) \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return ((float) M_PI) * (sqrtf((u1 - (-0.5f * (u1 * u1)))) * (u2 + u2));
}

Julia

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

MATLAB

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

Derivation

1. Initial program 61.1%

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

   1. associate-*r* 61.1%

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

   2. add-cube-cbrt 61.0%

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

   3. pow3 61.0%

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

3. Applied egg-rr 61.0%

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

4. Taylor expanded in u1 around 0 87.1%

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

   1. +-commutative 87.1%

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

   2. mul-1-neg 87.1%

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

   3. unsub-neg 87.1%

      \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} - u1\right)}} \cdot {\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3} \]

   4. *-commutative 87.1%

      \[\leadsto \sqrt{-\left(\color{blue}{{u1}^{2} \cdot -0.5} - u1\right)} \cdot {\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3} \]

   5. unpow2 87.1%

      \[\leadsto \sqrt{-\left(\color{blue}{\left(u1 \cdot u1\right)} \cdot -0.5 - u1\right)} \cdot {\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3} \]

   6. associate-*l* 87.1%

      \[\leadsto \sqrt{-\left(\color{blue}{u1 \cdot \left(u1 \cdot -0.5\right)} - u1\right)} \cdot {\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3} \]

6. Simplified 87.1%

   \[\leadsto \sqrt{-\color{blue}{\left(u1 \cdot \left(u1 \cdot -0.5\right) - u1\right)}} \cdot {\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3} \]

7. Taylor expanded in u2 around 0 72.6%

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

   1. associate-*r* 72.6%

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

   2. associate-*r* 72.6%

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

   3. *-commutative 72.6%

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

   4. *-commutative 72.6%

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

   5. associate-*l* 72.7%

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

   6. *-commutative 72.7%

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

   7. rem-log-exp 39.0%

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

   8. log-pow 39.0%

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

   9. unpow2 39.0%

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

   10. log-prod 39.0%

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

   11. rem-log-exp 44.9%

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

   12. rem-log-exp 72.7%

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

   13. unpow2 72.7%

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

9. Simplified 72.7%

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

10. Final simplification 72.7%

    \[\leadsto \pi \cdot \left(\sqrt{u1 - -0.5 \cdot \left(u1 \cdot u1\right)} \cdot \left(u2 + u2\right)\right) \]

Alternative 6: 65.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.1%

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

2. Taylor expanded in u1 around 0 75.7%

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

   1. mul-1-neg 75.7%

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

4. Simplified 75.7%

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

5. Taylor expanded in u2 around 0 64.6%

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

   1. *-commutative 64.6%

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

   2. *-commutative 64.6%

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

   3. associate-*r* 64.6%

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

   4. *-commutative 64.6%

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

7. Simplified 64.6%

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

8. Taylor expanded in u2 around 0 64.6%

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

   1. *-commutative 64.6%

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

   2. *-commutative 64.6%

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

   3. *-commutative 64.6%

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

   4. associate-*l* 64.5%

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

   5. associate-*l* 64.5%

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

10. Simplified 64.5%

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

11. Taylor expanded in u2 around 0 64.6%

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

    1. associate-*r* 64.6%

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

    2. *-commutative 64.6%

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

    3. associate-*r* 64.5%

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

    4. *-commutative 64.5%

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

    5. associate-*l* 64.5%

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

    6. *-commutative 64.5%

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

13. Simplified 64.5%

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

14. Final simplification 64.5%

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

Alternative 7: 65.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.1%

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

2. Taylor expanded in u1 around 0 75.7%

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

   1. mul-1-neg 75.7%

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

4. Simplified 75.7%

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

5. Taylor expanded in u2 around 0 64.6%

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

   1. *-commutative 64.6%

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

   2. *-commutative 64.6%

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

   3. associate-*r* 64.6%

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

   4. *-commutative 64.6%

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

7. Simplified 64.6%

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

8. Taylor expanded in u2 around 0 64.6%

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

   1. *-commutative 64.6%

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

   2. *-commutative 64.6%

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

   3. *-commutative 64.6%

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

   4. associate-*l* 64.5%

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

   5. associate-*l* 64.5%

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

10. Simplified 64.5%

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

11. Final simplification 64.5%

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

Alternative 8: 65.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 61.1%

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

2. Taylor expanded in u1 around 0 75.7%

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

   1. mul-1-neg 75.7%

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

4. Simplified 75.7%

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

5. Taylor expanded in u2 around 0 64.6%

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

   1. *-commutative 64.6%

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

   2. *-commutative 64.6%

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

   3. associate-*r* 64.6%

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

   4. *-commutative 64.6%

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

7. Simplified 64.6%

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

8. Final simplification 64.6%

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

Reproduce

herbie shell --seed 2023271 
(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))))