Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.9% → 98.1%

Time: 14.0s

Alternatives: 12

Speedup: 2.9×

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: 57.9% 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: 98.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\sqrt[3]{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(2 \cdot \pi\right)}^{3}\right)\right) \cdot {u2}^{3}}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (- (log1p (- u1))))
  (sin (cbrt (* (expm1 (log1p (pow (* 2.0 PI) 3.0))) (pow u2 3.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * sinf(cbrtf((expm1f(log1pf(powf((2.0f * ((float) M_PI)), 3.0f))) * powf(u2, 3.0f))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * sin(cbrt(Float32(expm1(log1p((Float32(Float32(2.0) * Float32(pi)) ^ Float32(3.0)))) * (u2 ^ Float32(3.0))))))
end

Derivation

1. Initial program 57.2%

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

   1. sub-neg 57.2%

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

   2. log1p-define 98.2%

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

3. Simplified 98.2%

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

   1. add-cbrt-cube 98.2%

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

   2. add-cbrt-cube 98.2%

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

   3. cbrt-unprod 98.3%

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

   4. pow3 98.3%

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

   5. pow3 98.3%

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

6. Applied egg-rr 98.3%

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

   1. expm1-log1p-u 98.3%

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

   2. expm1-undefine 98.3%

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

   3. unpow-prod-down 98.3%

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

   4. metadata-eval 98.3%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\sqrt[3]{\left(e^{\mathsf{log1p}\left(\color{blue}{8} \cdot {\pi}^{3}\right)} - 1\right) \cdot {u2}^{3}}\right) \]

8. Applied egg-rr 98.3%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\sqrt[3]{\color{blue}{\left(e^{\mathsf{log1p}\left(8 \cdot {\pi}^{3}\right)} - 1\right)} \cdot {u2}^{3}}\right) \]
9. Step-by-step derivation

   1. expm1-define 98.3%

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

   2. metadata-eval 98.3%

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

   3. cube-prod 98.3%

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

10. Simplified 98.3%

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

Alternative 2: 98.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.2%

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

   1. sub-neg 57.2%

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

   2. log1p-define 98.2%

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

3. Simplified 98.2%

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

   1. add-cbrt-cube 98.2%

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

   2. add-cbrt-cube 98.2%

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

   3. cbrt-unprod 98.3%

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

   4. pow3 98.3%

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

   5. pow3 98.3%

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

6. Applied egg-rr 98.3%

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

Alternative 3: 98.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.2%

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

   1. sub-neg 57.2%

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

   2. log1p-define 98.2%

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

3. Simplified 98.2%

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

   1. add-cbrt-cube 98.2%

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

   2. pow3 98.2%

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

   3. associate-*l* 98.2%

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

6. Applied egg-rr 98.2%

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

Alternative 4: 98.4% accurate, 1.0× speedup

Math

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

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(Float32(2.0) * Float32(pi)) * u2)))
end

Derivation

1. Initial program 57.2%

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

   1. sub-neg 57.2%

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

   2. log1p-define 98.2%

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

3. Simplified 98.2%

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

Alternative 5: 95.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.0006500000017695129:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.0006500000017695129)
     (* (sqrt (- (log1p (- u1)))) t_0)
     (*
      (sin t_0)
      (sqrt (* u1 (+ 1.0 (* u1 (+ 0.5 (* u1 0.3333333333333333))))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.0006500000017695129f) {
		tmp = sqrtf(-log1pf(-u1)) * t_0;
	} else {
		tmp = sinf(t_0) * sqrtf((u1 * (1.0f + (u1 * (0.5f + (u1 * 0.3333333333333333f))))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(2.0) * Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0006500000017695129))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * t_0);
	else
		tmp = Float32(sin(t_0) * sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(u1 * Float32(Float32(0.5) + Float32(u1 * Float32(0.3333333333333333))))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 6.50000002e-4

   1. Initial program 59.6%

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

      1. sub-neg 59.6%

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

      2. log1p-define 98.4%

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

   3. Simplified 98.4%

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

      1. add-cbrt-cube 98.4%

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

      2. pow1/3 92.5%

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

      3. pow3 92.5%

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

      4. associate-*l* 92.5%

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

   6. Applied egg-rr 92.5%

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

   7. Taylor expanded in u2 around 0 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-*r* 98.3%

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

      3. *-commutative 98.3%

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

   9. Simplified 98.3%

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

   if 6.50000002e-4 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 53.8%

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

   3. Taylor expanded in u1 around 0 93.5%

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

      1. *-commutative 93.5%

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

   5. Simplified 93.5%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.0006500000017695129:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.00419999985024333:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.00419999985024333)
     (* (sqrt (- (log1p (- u1)))) t_0)
     (* (sin t_0) (sqrt (* u1 (+ 1.0 (* u1 0.5))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.00419999985024333f) {
		tmp = sqrtf(-log1pf(-u1)) * t_0;
	} else {
		tmp = sinf(t_0) * sqrtf((u1 * (1.0f + (u1 * 0.5f))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(2.0) * Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.00419999985024333))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * t_0);
	else
		tmp = Float32(sin(t_0) * sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(u1 * Float32(0.5))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00419999985

   1. Initial program 60.0%

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

      1. sub-neg 60.0%

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

      2. log1p-define 98.3%

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

   3. Simplified 98.3%

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

      1. add-cbrt-cube 98.3%

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

      2. pow1/3 92.8%

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

      3. pow3 92.8%

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

      4. associate-*l* 92.8%

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

   6. Applied egg-rr 92.8%

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

   7. Taylor expanded in u2 around 0 97.4%

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

      1. *-commutative 97.4%

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

      2. associate-*r* 97.4%

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

      3. *-commutative 97.4%

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

   9. Simplified 97.4%

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

   if 0.00419999985 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 51.7%

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

   3. Taylor expanded in u1 around 0 90.9%

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

      1. *-commutative 90.9%

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

   5. Simplified 90.9%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot 0.5\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.00419999985024333:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 + u1 \cdot 0.25\right)\right)\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sin (* (* 2.0 PI) u2))
  (sqrt
   (* u1 (+ 1.0 (* u1 (+ 0.5 (* u1 (+ 0.3333333333333333 (* u1 0.25))))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sinf(((2.0f * ((float) M_PI)) * u2)) * sqrtf((u1 * (1.0f + (u1 * (0.5f + (u1 * (0.3333333333333333f + (u1 * 0.25f))))))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)) * sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(u1 * Float32(Float32(0.5) + Float32(u1 * Float32(Float32(0.3333333333333333) + Float32(u1 * Float32(0.25))))))))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sin(((single(2.0) * single(pi)) * u2)) * sqrt((u1 * (single(1.0) + (u1 * (single(0.5) + (u1 * (single(0.3333333333333333) + (u1 * single(0.25)))))))));
end

Derivation

1. Initial program 57.2%

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

3. Taylor expanded in u1 around 0 94.2%

   \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 + 0.25 \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
4. Step-by-step derivation

   1. *-commutative 94.2%

      \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 + \color{blue}{u1 \cdot 0.25}\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

5. Simplified 94.2%

   \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 + u1 \cdot 0.25\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

6. Final simplification 94.2%

   \[\leadsto \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 + u1 \cdot 0.25\right)\right)\right)} \]
7. Add Preprocessing

Alternative 8: 90.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.005200000014156103:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.005200000014156103)
     (* (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 (t_0 <= 0.005200000014156103f) {
		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(Float32(2.0) * Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.005200000014156103))
		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 #s(literal 2 binary32) (PI.f32)) u2) < 0.00520000001

   1. Initial program 60.2%

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

      1. sub-neg 60.2%

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

      2. log1p-define 98.3%

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

   3. Simplified 98.3%

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

      1. add-cbrt-cube 98.3%

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

      2. pow1/3 92.9%

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

      3. pow3 92.9%

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

      4. associate-*l* 92.9%

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

   6. Applied egg-rr 92.9%

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

   7. Taylor expanded in u2 around 0 97.2%

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

      1. *-commutative 97.2%

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

      2. associate-*r* 97.2%

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

      3. *-commutative 97.2%

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

   9. Simplified 97.2%

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

   if 0.00520000001 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 51.0%

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

   3. Taylor expanded in u1 around 0 79.8%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.005200000014156103:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.2%

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

3. Taylor expanded in u1 around 0 77.7%

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

4. Final simplification 77.7%

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

Alternative 10: 66.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.2%

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

3. Taylor expanded in u1 around 0 77.7%

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

4. Taylor expanded in u2 around 0 66.4%

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

   1. associate-*r* 66.4%

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

6. Simplified 66.4%

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

7. Final simplification 66.4%

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

Alternative 11: 66.4% accurate, 2.9× speedup

Math

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

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(u2 * Float32(Float32(2.0) * Float32(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 57.2%

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

3. Taylor expanded in u1 around 0 77.7%

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

4. Taylor expanded in u2 around 0 71.3%

   \[\leadsto \color{blue}{u2 \cdot \left(-1.3333333333333333 \cdot \left(\sqrt{u1} \cdot \left({u2}^{2} \cdot {\pi}^{3}\right)\right) + 2 \cdot \left(\sqrt{u1} \cdot \pi\right)\right)} \]

5. Taylor expanded in u2 around 0 66.4%

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

6. Final simplification 66.4%

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

Alternative 12: 66.4% accurate, 2.9× speedup

Math

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

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(2.0) * Float32(Float32(pi) * Float32(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 57.2%

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

3. Taylor expanded in u1 around 0 77.7%

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

4. Taylor expanded in u2 around 0 71.3%

   \[\leadsto \color{blue}{u2 \cdot \left(-1.3333333333333333 \cdot \left(\sqrt{u1} \cdot \left({u2}^{2} \cdot {\pi}^{3}\right)\right) + 2 \cdot \left(\sqrt{u1} \cdot \pi\right)\right)} \]

5. Taylor expanded in u2 around 0 66.4%

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

6. Taylor expanded in u2 around 0 66.4%

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

   1. associate-*r* 66.4%

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

8. Simplified 66.4%

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

9. Final simplification 66.4%

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

Reproduce

herbie shell --seed 2024123 
(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))))