Beckmann Sample, near normal, slope_y

Percentage Accurate: 58.3% → 98.4%

Time: 12.2s

Alternatives: 9

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: 58.3% 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, 0.6× 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 58.8%

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

   1. sub-neg 58.8%

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

   2. log1p-def 98.3%

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

   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: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log1p (- u1)))) (log1p (expm1 (sin (* 2.0 (* PI u2)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * log1pf(expm1f(sinf((2.0f * (((float) M_PI) * u2)))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * log1p(expm1(sin(Float32(Float32(2.0) * Float32(Float32(pi) * u2))))))
end

Derivation

1. Initial program 58.8%

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

   1. sub-neg 58.8%

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

   2. log1p-def 98.3%

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

   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. Step-by-step derivation

   1. cbrt-prod 98.3%

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

   2. unpow3 98.3%

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

   3. add-cbrt-cube 98.3%

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

   4. unpow3 98.3%

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

   5. add-cbrt-cube 98.3%

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

   6. pow1 98.3%

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

   7. metadata-eval 98.3%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left({\left(2 \cdot \left(\pi \cdot u2\right)\right)}^{\color{blue}{\left(3 \cdot 0.3333333333333333\right)}}\right) \]

   8. pow-pow 94.4%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left({\left({\left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}\right)}^{0.3333333333333333}\right)} \]

   9. log1p-expm1-u 94.3%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left({\left({\left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}\right)}^{0.3333333333333333}\right)\right)\right)} \]

   10. pow-pow 98.3%

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \color{blue}{\left({\left(2 \cdot \left(\pi \cdot u2\right)\right)}^{\left(3 \cdot 0.3333333333333333\right)}\right)}\right)\right) \]

   11. metadata-eval 98.3%

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left({\left(2 \cdot \left(\pi \cdot u2\right)\right)}^{\color{blue}{1}}\right)\right)\right) \]

   12. pow1 98.3%

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

7. Applied egg-rr 98.3%

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

8. Final simplification 98.3%

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

Alternative 3: 94.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(2 \cdot \pi\right)\\ \mathbf{if}\;t_0 \leq 0.0015999999595806003:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{u1 + 0.5 \cdot \left(u1 \cdot u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* 2.0 PI))))
   (if (<= t_0 0.0015999999595806003)
     (* (sqrt (- (log1p (- u1)))) t_0)
     (* (sin (* 2.0 (* PI u2))) (sqrt (+ u1 (* 0.5 (* u1 u1))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u2 * (2.0f * ((float) M_PI));
	float tmp;
	if (t_0 <= 0.0015999999595806003f) {
		tmp = sqrtf(-log1pf(-u1)) * t_0;
	} else {
		tmp = sinf((2.0f * (((float) M_PI) * u2))) * sqrtf((u1 + (0.5f * (u1 * u1))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u2 * Float32(Float32(2.0) * Float32(pi)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0015999999595806003))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * t_0);
	else
		tmp = Float32(sin(Float32(Float32(2.0) * Float32(Float32(pi) * u2))) * sqrt(Float32(u1 + Float32(Float32(0.5) * Float32(u1 * u1)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 2 (PI.f32)) u2) < 0.00159999996

   1. Initial program 61.8%

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

      1. sub-neg 61.8%

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

      2. log1p-def 98.3%

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

      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 98.4%

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

   5. Applied egg-rr 98.4%

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

   6. Taylor expanded in u2 around 0 98.0%

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

      1. associate-*r* 98.0%

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

      2. *-commutative 98.0%

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

      3. associate-*r* 98.0%

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

   8. Simplified 98.0%

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

   if 0.00159999996 < (*.f32 (*.f32 2 (PI.f32)) u2)

   1. Initial program 54.0%

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

      1. associate-*r* 54.0%

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

      2. add-cbrt-cube 54.0%

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

      3. pow1/3 53.6%

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \color{blue}{\left({\left(\left(\left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\right) \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)}^{0.3333333333333333}\right)} \]

      4. pow3 53.5%

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left({\color{blue}{\left({\left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}\right)}}^{0.3333333333333333}\right) \]

   3. Applied egg-rr 53.5%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \color{blue}{\left({\left({\left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}\right)}^{0.3333333333333333}\right)} \]

   4. Taylor expanded in u1 around 0 87.2%

      \[\leadsto \sqrt{-\color{blue}{\left(-1 \cdot u1 + -0.5 \cdot {u1}^{2}\right)}} \cdot \sin \left({\left({\left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}\right)}^{0.3333333333333333}\right) \]
   5. Step-by-step derivation

      1. +-commutative 87.2%

         \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} + -1 \cdot u1\right)}} \cdot \sin \left({\left({\left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}\right)}^{0.3333333333333333}\right) \]

      2. mul-1-neg 87.2%

         \[\leadsto \sqrt{-\left(-0.5 \cdot {u1}^{2} + \color{blue}{\left(-u1\right)}\right)} \cdot \sin \left({\left({\left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}\right)}^{0.3333333333333333}\right) \]

      3. unsub-neg 87.2%

         \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} - u1\right)}} \cdot \sin \left({\left({\left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}\right)}^{0.3333333333333333}\right) \]

      4. unpow2 87.2%

         \[\leadsto \sqrt{-\left(-0.5 \cdot \color{blue}{\left(u1 \cdot u1\right)} - u1\right)} \cdot \sin \left({\left({\left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}\right)}^{0.3333333333333333}\right) \]

      5. associate-*r* 87.2%

         \[\leadsto \sqrt{-\left(\color{blue}{\left(-0.5 \cdot u1\right) \cdot u1} - u1\right)} \cdot \sin \left({\left({\left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}\right)}^{0.3333333333333333}\right) \]

   6. Simplified 87.2%

      \[\leadsto \sqrt{-\color{blue}{\left(\left(-0.5 \cdot u1\right) \cdot u1 - u1\right)}} \cdot \sin \left({\left({\left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}\right)}^{0.3333333333333333}\right) \]

   7. Taylor expanded in u2 around inf 87.2%

      \[\leadsto \color{blue}{\sqrt{u1 - -0.5 \cdot {u1}^{2}} \cdot \sin \left({\left({\left(2 \cdot \left(u2 \cdot \pi\right)\right)}^{3}\right)}^{0.3333333333333333}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 87.2%

         \[\leadsto \color{blue}{\sin \left({\left({\left(2 \cdot \left(u2 \cdot \pi\right)\right)}^{3}\right)}^{0.3333333333333333}\right) \cdot \sqrt{u1 - -0.5 \cdot {u1}^{2}}} \]

      2. unpow1/3 87.8%

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

      3. rem-cbrt-cube 87.8%

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

      4. cancel-sign-sub-inv 87.8%

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

      5. metadata-eval 87.8%

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

      6. unpow2 87.8%

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

   9. Simplified 87.8%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.0015999999595806003:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \left(2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{u1 + 0.5 \cdot \left(u1 \cdot u1\right)}\\ \end{array} \]

Alternative 4: 90.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(2 \cdot \pi\right)\\ \mathbf{if}\;t_0 \leq 0.014999999664723873:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* 2.0 PI))))
   (if (<= t_0 0.014999999664723873)
     (* (sqrt (- (log1p (- u1)))) t_0)
     (* (sqrt u1) (sin t_0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u2 * (2.0f * ((float) M_PI));
	float tmp;
	if (t_0 <= 0.014999999664723873f) {
		tmp = sqrtf(-log1pf(-u1)) * t_0;
	} else {
		tmp = sqrtf(u1) * sinf(t_0);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u2 * Float32(Float32(2.0) * Float32(pi)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.014999999664723873))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * t_0);
	else
		tmp = Float32(sqrt(u1) * sin(t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 2 (PI.f32)) u2) < 0.0149999997

   1. Initial program 61.0%

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

      1. sub-neg 61.0%

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

      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 98.4%

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

   5. Applied egg-rr 98.4%

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

   6. Taylor expanded in u2 around 0 96.2%

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

      1. associate-*r* 96.2%

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

      2. *-commutative 96.2%

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

      3. associate-*r* 96.2%

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

   8. Simplified 96.2%

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

   if 0.0149999997 < (*.f32 (*.f32 2 (PI.f32)) u2)

   1. Initial program 53.6%

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

   2. Taylor expanded in u1 around 0 79.0%

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

      1. mul-1-neg 79.0%

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

   4. Simplified 79.0%

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

   5. Taylor expanded in u2 around inf 79.0%

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

      1. *-commutative 79.0%

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

      2. *-commutative 79.0%

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

      3. associate-*r* 79.0%

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

      4. *-commutative 79.0%

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

      5. *-commutative 79.0%

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

      6. *-commutative 79.0%

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

   7. Simplified 79.0%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.014999999664723873:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \left(2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(u2 \cdot \left(2 \cdot \pi\right)\right)\\ \end{array} \]

Alternative 5: 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 58.8%

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

   1. sub-neg 58.8%

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

   2. log1p-def 98.3%

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

   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 6: 83.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.0002500000118743628:\\ \;\;\;\;2 \cdot \left(\pi \cdot \left(u2 \cdot \sqrt{u1 + 0.5 \cdot \left(u1 \cdot u1\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(u2 \cdot \left(2 \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.0002500000118743628)
   (* 2.0 (* PI (* u2 (sqrt (+ u1 (* 0.5 (* u1 u1)))))))
   (* (sqrt u1) (sin (* u2 (* 2.0 PI))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.0002500000118743628f) {
		tmp = 2.0f * (((float) M_PI) * (u2 * sqrtf((u1 + (0.5f * (u1 * u1))))));
	} else {
		tmp = sqrtf(u1) * sinf((u2 * (2.0f * ((float) M_PI))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.0002500000118743628))
		tmp = Float32(Float32(2.0) * Float32(Float32(pi) * Float32(u2 * sqrt(Float32(u1 + Float32(Float32(0.5) * Float32(u1 * u1)))))));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(u2 * Float32(Float32(2.0) * Float32(pi)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (u2 <= single(0.0002500000118743628))
		tmp = single(2.0) * (single(pi) * (u2 * sqrt((u1 + (single(0.5) * (u1 * u1))))));
	else
		tmp = sqrt(u1) * sin((u2 * (single(2.0) * single(pi))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u2 < 2.50000012e-4

   1. Initial program 61.8%

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

      1. associate-*r* 61.8%

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

      2. add-cbrt-cube 61.8%

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

      3. pow1/3 61.0%

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \color{blue}{\left({\left(\left(\left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\right) \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)}^{0.3333333333333333}\right)} \]

      4. pow3 61.0%

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left({\color{blue}{\left({\left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}\right)}}^{0.3333333333333333}\right) \]

   3. Applied egg-rr 61.0%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \color{blue}{\left({\left({\left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}\right)}^{0.3333333333333333}\right)} \]

   4. Taylor expanded in u1 around 0 81.9%

      \[\leadsto \sqrt{-\color{blue}{\left(-1 \cdot u1 + -0.5 \cdot {u1}^{2}\right)}} \cdot \sin \left({\left({\left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}\right)}^{0.3333333333333333}\right) \]
   5. Step-by-step derivation

      1. +-commutative 81.9%

         \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} + -1 \cdot u1\right)}} \cdot \sin \left({\left({\left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}\right)}^{0.3333333333333333}\right) \]

      2. mul-1-neg 81.9%

         \[\leadsto \sqrt{-\left(-0.5 \cdot {u1}^{2} + \color{blue}{\left(-u1\right)}\right)} \cdot \sin \left({\left({\left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}\right)}^{0.3333333333333333}\right) \]

      3. unsub-neg 81.9%

         \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} - u1\right)}} \cdot \sin \left({\left({\left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}\right)}^{0.3333333333333333}\right) \]

      4. unpow2 81.9%

         \[\leadsto \sqrt{-\left(-0.5 \cdot \color{blue}{\left(u1 \cdot u1\right)} - u1\right)} \cdot \sin \left({\left({\left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}\right)}^{0.3333333333333333}\right) \]

      5. associate-*r* 81.9%

         \[\leadsto \sqrt{-\left(\color{blue}{\left(-0.5 \cdot u1\right) \cdot u1} - u1\right)} \cdot \sin \left({\left({\left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}\right)}^{0.3333333333333333}\right) \]

   6. Simplified 81.9%

      \[\leadsto \sqrt{-\color{blue}{\left(\left(-0.5 \cdot u1\right) \cdot u1 - u1\right)}} \cdot \sin \left({\left({\left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}\right)}^{0.3333333333333333}\right) \]

   7. Taylor expanded in u2 around 0 85.5%

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

      1. associate-*r* 85.6%

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

      2. *-commutative 85.6%

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

      3. *-commutative 85.6%

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

      4. cancel-sign-sub-inv 85.6%

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

      5. metadata-eval 85.6%

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

      6. unpow2 85.6%

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

   9. Simplified 85.6%

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

   if 2.50000012e-4 < u2

   1. Initial program 54.0%

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

   2. Taylor expanded in u1 around 0 78.4%

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

      1. mul-1-neg 78.4%

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

   4. Simplified 78.4%

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

   5. Taylor expanded in u2 around inf 78.4%

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

      1. *-commutative 78.4%

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

      2. *-commutative 78.4%

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

      3. associate-*r* 78.4%

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

      4. *-commutative 78.4%

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

      5. *-commutative 78.4%

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

      6. *-commutative 78.4%

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

   7. Simplified 78.4%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \leq 0.0002500000118743628:\\ \;\;\;\;2 \cdot \left(\pi \cdot \left(u2 \cdot \sqrt{u1 + 0.5 \cdot \left(u1 \cdot u1\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(u2 \cdot \left(2 \cdot \pi\right)\right)\\ \end{array} \]

Alternative 7: 74.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.8%

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

   1. associate-*r* 58.8%

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

   2. add-cbrt-cube 58.8%

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

   3. pow1/3 58.2%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \color{blue}{\left({\left(\left(\left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\right) \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)}^{0.3333333333333333}\right)} \]

   4. pow3 58.2%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left({\color{blue}{\left({\left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}\right)}}^{0.3333333333333333}\right) \]

3. Applied egg-rr 58.2%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \color{blue}{\left({\left({\left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}\right)}^{0.3333333333333333}\right)} \]

4. Taylor expanded in u1 around 0 83.9%

   \[\leadsto \sqrt{-\color{blue}{\left(-1 \cdot u1 + -0.5 \cdot {u1}^{2}\right)}} \cdot \sin \left({\left({\left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}\right)}^{0.3333333333333333}\right) \]
5. Step-by-step derivation

   1. +-commutative 83.9%

      \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} + -1 \cdot u1\right)}} \cdot \sin \left({\left({\left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}\right)}^{0.3333333333333333}\right) \]

   2. mul-1-neg 83.9%

      \[\leadsto \sqrt{-\left(-0.5 \cdot {u1}^{2} + \color{blue}{\left(-u1\right)}\right)} \cdot \sin \left({\left({\left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}\right)}^{0.3333333333333333}\right) \]

   3. unsub-neg 83.9%

      \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} - u1\right)}} \cdot \sin \left({\left({\left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}\right)}^{0.3333333333333333}\right) \]

   4. unpow2 83.9%

      \[\leadsto \sqrt{-\left(-0.5 \cdot \color{blue}{\left(u1 \cdot u1\right)} - u1\right)} \cdot \sin \left({\left({\left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}\right)}^{0.3333333333333333}\right) \]

   5. associate-*r* 83.9%

      \[\leadsto \sqrt{-\left(\color{blue}{\left(-0.5 \cdot u1\right) \cdot u1} - u1\right)} \cdot \sin \left({\left({\left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}\right)}^{0.3333333333333333}\right) \]

6. Simplified 83.9%

   \[\leadsto \sqrt{-\color{blue}{\left(\left(-0.5 \cdot u1\right) \cdot u1 - u1\right)}} \cdot \sin \left({\left({\left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}\right)}^{0.3333333333333333}\right) \]

7. Taylor expanded in u2 around 0 72.2%

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

   1. associate-*r* 72.2%

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

   2. *-commutative 72.2%

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

   3. *-commutative 72.2%

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

   4. cancel-sign-sub-inv 72.2%

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

   5. metadata-eval 72.2%

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

   6. unpow2 72.2%

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

9. Simplified 72.2%

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

10. Final simplification 72.2%

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

Alternative 8: 66.2% accurate, 2.0× speedup

Math

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

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(Float32(2.0) * 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 58.8%

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

2. Taylor expanded in u1 around 0 75.5%

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

   1. mul-1-neg 75.5%

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

4. Simplified 75.5%

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

5. Taylor expanded in u2 around 0 64.2%

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

   1. associate-*l* 64.2%

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

   2. associate-*r* 64.2%

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

7. Simplified 64.2%

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

8. Final simplification 64.2%

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

Alternative 9: 66.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.8%

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

2. Taylor expanded in u1 around 0 75.5%

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

   1. mul-1-neg 75.5%

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

4. Simplified 75.5%

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

5. Taylor expanded in u2 around 0 64.2%

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

   1. associate-*r* 64.2%

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

   2. associate-*r* 64.2%

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

   3. *-commutative 64.2%

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

   4. associate-*r* 64.2%

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

7. Simplified 64.2%

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

8. Final simplification 64.2%

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

Reproduce

herbie shell --seed 2023185 
(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))))