Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.8% → 98.4%

Time: 21.0s

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: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3} \cdot {\left(-\mathsf{log1p}\left(-u1\right)\right)}^{1.5}} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (cbrt (* (pow (sin (* 2.0 (* PI u2))) 3.0) (pow (- (log1p (- u1))) 1.5))))

C

float code(float cosTheta_i, float u1, float u2) {
	return cbrtf((powf(sinf((2.0f * (((float) M_PI) * u2))), 3.0f) * powf(-log1pf(-u1), 1.5f)));
}

Julia

function code(cosTheta_i, u1, u2)
	return cbrt(Float32((sin(Float32(Float32(2.0) * Float32(Float32(pi) * u2))) ^ Float32(3.0)) * (Float32(-log1p(Float32(-u1))) ^ Float32(1.5))))
end

Derivation

1. Initial program 58.6%

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

   1. sub-neg 58.6%

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

   2. log1p-def 98.4%

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

   3. associate-*l* 98.4%

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

3. Simplified 98.4%

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

   1. expm1-log1p-u 98.4%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot u2\right)\right)}\right) \]

5. Applied egg-rr 98.4%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot u2\right)\right)}\right) \]
6. Step-by-step derivation

   1. add-cube-cbrt 98.1%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)} \cdot u2\right)\right)\right) \]

   2. pow3 98.1%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{3}} \cdot u2\right)\right)\right) \]

7. Applied egg-rr 98.1%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{3}} \cdot u2\right)\right)\right) \]
8. Step-by-step derivation

   1. *-commutative 98.1%

      \[\leadsto \color{blue}{\sin \left(2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt[3]{\pi}\right)}^{3} \cdot u2\right)\right)\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}} \]

   2. sin-2 98.0%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt[3]{\pi}\right)}^{3} \cdot u2\right)\right)\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt[3]{\pi}\right)}^{3} \cdot u2\right)\right)\right)\right)\right)} \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)} \]

   3. expm1-log1p-u 98.0%

      \[\leadsto \left(2 \cdot \left(\sin \color{blue}{\left({\left(\sqrt[3]{\pi}\right)}^{3} \cdot u2\right)} \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt[3]{\pi}\right)}^{3} \cdot u2\right)\right)\right)\right)\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)} \]

   4. rem-cube-cbrt 98.3%

      \[\leadsto \left(2 \cdot \left(\sin \left(\color{blue}{\pi} \cdot u2\right) \cdot \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt[3]{\pi}\right)}^{3} \cdot u2\right)\right)\right)\right)\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)} \]

   5. expm1-log1p-u 98.3%

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

   6. rem-cube-cbrt 98.3%

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

   7. sin-2 98.4%

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

   8. add-cbrt-cube 98.4%

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

   9. add-cbrt-cube 98.4%

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

9. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\sqrt[3]{{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3} \cdot {\left(-\mathsf{log1p}\left(-u1\right)\right)}^{1.5}}} \]

10. Final simplification 98.5%

    \[\leadsto \sqrt[3]{{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3} \cdot {\left(-\mathsf{log1p}\left(-u1\right)\right)}^{1.5}} \]

Alternative 2: 83.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.0019000000320374966:\\ \;\;\;\;\left(\pi \cdot \left(2 \cdot u2\right)\right) \cdot \sqrt{u1 + 0.5 \cdot \left(u1 \cdot u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 (* 2.0 PI)) 0.0019000000320374966)
   (* (* PI (* 2.0 u2)) (sqrt (+ u1 (* 0.5 (* u1 u1)))))
   (* (sin (* 2.0 (* PI u2))) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * (2.0f * ((float) M_PI))) <= 0.0019000000320374966f) {
		tmp = (((float) M_PI) * (2.0f * u2)) * sqrtf((u1 + (0.5f * (u1 * u1))));
	} 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 (Float32(u2 * Float32(Float32(2.0) * Float32(pi))) <= Float32(0.0019000000320374966))
		tmp = Float32(Float32(Float32(pi) * Float32(Float32(2.0) * u2)) * sqrt(Float32(u1 + Float32(Float32(0.5) * Float32(u1 * u1)))));
	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(2.0) * single(pi))) <= single(0.0019000000320374966))
		tmp = (single(pi) * (single(2.0) * u2)) * sqrt((u1 + (single(0.5) * (u1 * u1))));
	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 (*.f32 (*.f32 2 (PI.f32)) u2) < 0.00190000003

   1. Initial program 57.6%

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

      1. associate-*r* 57.6%

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

      2. add-sqr-sqrt 57.6%

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

      3. pow2 57.6%

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

   3. Applied egg-rr 57.6%

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

   4. Taylor expanded in u1 around 0 88.1%

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

      1. +-commutative 88.1%

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

      2. mul-1-neg 88.1%

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

      3. unsub-neg 88.1%

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

      4. *-commutative 88.1%

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

      5. unpow2 88.1%

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

      6. associate-*l* 88.1%

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

   6. Simplified 88.1%

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

   7. Taylor expanded in u2 around 0 88.3%

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

      1. associate-*r* 88.3%

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

      2. associate-*r* 88.3%

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

      3. *-commutative 88.3%

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

      4. cancel-sign-sub-inv 88.3%

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

      5. metadata-eval 88.3%

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

      6. unpow2 88.3%

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

   9. Simplified 88.3%

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

   if 0.00190000003 < (*.f32 (*.f32 2 (PI.f32)) u2)

   1. Initial program 60.3%

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

      1. sub-neg 60.3%

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

      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 60.3%

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

      2. sub-neg 60.3%

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

      3. add-sqr-sqrt 60.2%

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

      4. pow2 60.2%

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

   5. Applied egg-rr 75.1%

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

   6. Taylor expanded in u1 around 0 77.3%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.0019000000320374966:\\ \;\;\;\;\left(\pi \cdot \left(2 \cdot u2\right)\right) \cdot \sqrt{u1 + 0.5 \cdot \left(u1 \cdot u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 3: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.6%

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

   1. sub-neg 58.6%

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

   2. log1p-def 98.4%

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

   3. associate-*l* 98.4%

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

3. Simplified 98.4%

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

4. Final simplification 98.4%

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

Alternative 4: 88.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \sin \left(\pi \cdot \left(2 \cdot u2\right)\right) \cdot \sqrt{u1 + 0.5 \cdot \left(u1 \cdot u1\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sin (* PI (* 2.0 u2))) (sqrt (+ u1 (* 0.5 (* u1 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf((((float) M_PI) * (2.0f * u2))) * sqrtf((u1 + (0.5f * (u1 * u1))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sin(Float32(Float32(pi) * Float32(Float32(2.0) * u2))) * sqrt(Float32(u1 + Float32(Float32(0.5) * Float32(u1 * u1)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sin((single(pi) * (single(2.0) * u2))) * sqrt((u1 + (single(0.5) * (u1 * u1))));
end

Derivation

1. Initial program 58.6%

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

   1. associate-*r* 58.6%

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

   2. add-sqr-sqrt 57.6%

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

   3. pow2 57.6%

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

3. Applied egg-rr 57.6%

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

4. Taylor expanded in u1 around 0 86.6%

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

   1. +-commutative 86.6%

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

   2. mul-1-neg 86.6%

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

   3. unsub-neg 86.6%

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

   4. *-commutative 86.6%

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

   5. unpow2 86.6%

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

   6. associate-*l* 86.6%

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

6. Simplified 86.6%

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

7. Taylor expanded in u2 around inf 88.6%

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

   1. associate-*r* 88.6%

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

   2. *-commutative 88.6%

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

   3. cancel-sign-sub-inv 88.6%

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

   4. metadata-eval 88.6%

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

   5. unpow2 88.6%

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

9. Simplified 88.6%

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

10. Final simplification 88.6%

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

Alternative 5: 74.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(\pi \cdot \left(2 \cdot u2\right)\right) \cdot \sqrt{u1 + 0.5 \cdot \left(u1 \cdot u1\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (* PI (* 2.0 u2)) (sqrt (+ u1 (* 0.5 (* u1 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return (((float) M_PI) * (2.0f * u2)) * sqrtf((u1 + (0.5f * (u1 * u1))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(Float32(pi) * Float32(Float32(2.0) * u2)) * sqrt(Float32(u1 + Float32(Float32(0.5) * Float32(u1 * u1)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (single(pi) * (single(2.0) * u2)) * sqrt((u1 + (single(0.5) * (u1 * u1))));
end

Derivation

1. Initial program 58.6%

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

   1. associate-*r* 58.6%

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

   2. add-sqr-sqrt 57.6%

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

   3. pow2 57.6%

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

3. Applied egg-rr 57.6%

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

4. Taylor expanded in u1 around 0 86.6%

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

   1. +-commutative 86.6%

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

   2. mul-1-neg 86.6%

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

   3. unsub-neg 86.6%

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

   4. *-commutative 86.6%

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

   5. unpow2 86.6%

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

   6. associate-*l* 86.6%

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

6. Simplified 86.6%

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

7. Taylor expanded in u2 around 0 76.7%

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

   1. associate-*r* 76.7%

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

   2. associate-*r* 76.7%

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

   3. *-commutative 76.7%

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

   4. cancel-sign-sub-inv 76.7%

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

   5. metadata-eval 76.7%

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

   6. unpow2 76.7%

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

9. Simplified 76.7%

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

10. Final simplification 76.7%

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

Alternative 6: 66.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ u2 \cdot \left(\pi \cdot \sqrt{u1 \cdot 4}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (* u2 (* PI (sqrt (* u1 4.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 * (((float) M_PI) * sqrtf((u1 * 4.0f)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u2 * Float32(Float32(pi) * sqrt(Float32(u1 * Float32(4.0)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = u2 * (single(pi) * sqrt((u1 * single(4.0))));
end

Derivation

1. Initial program 58.6%

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

2. Taylor expanded in u1 around 0 76.9%

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

   1. mul-1-neg 76.9%

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

4. Simplified 76.9%

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

5. Taylor expanded in u2 around 0 68.5%

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

   1. associate-*r* 68.5%

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

7. Simplified 68.5%

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

8. Taylor expanded in u2 around 0 68.5%

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

   1. *-commutative 68.5%

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

   2. associate-*r* 68.5%

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

   3. associate-*l* 68.5%

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

   4. *-commutative 68.5%

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

   5. *-commutative 68.5%

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

10. Simplified 68.5%

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

    1. associate-*r* 68.5%

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

    2. associate-*l* 68.5%

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

    3. *-commutative 68.5%

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

    4. *-commutative 68.5%

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

    5. associate-*r* 68.5%

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

    6. rem-cube-cbrt 68.3%

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

    7. expm1-log1p-u 68.3%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(2 \cdot \sqrt{u1}\right) \cdot \left(u2 \cdot {\left(\sqrt[3]{\pi}\right)}^{3}\right)\right)\right)} \]

    8. expm1-udef 29.7%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(2 \cdot \sqrt{u1}\right) \cdot \left(u2 \cdot {\left(\sqrt[3]{\pi}\right)}^{3}\right)\right)} - 1} \]

12. Applied egg-rr 29.7%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(u2 \cdot \left(\sqrt{u1 \cdot 4} \cdot \pi\right)\right)} - 1} \]
13. Step-by-step derivation

    1. expm1-def 68.5%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(u2 \cdot \left(\sqrt{u1 \cdot 4} \cdot \pi\right)\right)\right)} \]

    2. expm1-log1p 68.5%

       \[\leadsto \color{blue}{u2 \cdot \left(\sqrt{u1 \cdot 4} \cdot \pi\right)} \]

    3. *-commutative 68.5%

       \[\leadsto u2 \cdot \color{blue}{\left(\pi \cdot \sqrt{u1 \cdot 4}\right)} \]

14. Simplified 68.5%

    \[\leadsto \color{blue}{u2 \cdot \left(\pi \cdot \sqrt{u1 \cdot 4}\right)} \]

15. Final simplification 68.5%

    \[\leadsto u2 \cdot \left(\pi \cdot \sqrt{u1 \cdot 4}\right) \]

Reproduce

herbie shell --seed 2023275 
(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))))