Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.3% → 98.4%

Time: 11.2s

Alternatives: 10

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: 57.3% 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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

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

   1. sub-neg 57.7%

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

   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 90.3%

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

   3. pow3 90.3%

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

   4. *-commutative 90.3%

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

   5. associate-*r* 90.3%

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

6. Applied egg-rr 90.3%

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

   1. unpow1/3 98.3%

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

   2. rem-cbrt-cube 98.3%

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

   3. *-commutative 98.3%

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

   4. rem-cbrt-cube 98.3%

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

   5. add-cbrt-cube 98.3%

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

   6. cbrt-unprod 98.2%

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

   7. *-commutative 98.2%

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

   8. *-commutative 98.2%

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

8. Applied egg-rr 98.4%

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

Alternative 2: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

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

   1. sub-neg 57.7%

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

   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. Final simplification 98.3%

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

Alternative 3: 96.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(\pi \cdot 2\right)\\ \mathbf{if}\;t\_0 \leq 0.0006500000017695129:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 + u1 \cdot 0.25\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* PI 2.0))))
   (if (<= t_0 0.0006500000017695129)
     (* (sqrt (- (log1p (- u1)))) t_0)
     (*
      (sin t_0)
      (sqrt
       (*
        u1
        (+ 1.0 (* u1 (+ 0.5 (* u1 (+ 0.3333333333333333 (* u1 0.25))))))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u2 * (((float) M_PI) * 2.0f);
	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 + (u1 * 0.25f))))))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u2 * Float32(Float32(pi) * Float32(2.0)))
	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(Float32(0.3333333333333333) + Float32(u1 * Float32(0.25))))))))));
	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 59.2%

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

      1. sub-neg 59.2%

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

      2. log1p-define 98.4%

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

   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. *-commutative 92.5%

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

      5. associate-*r* 92.5%

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

   6. Applied egg-rr 92.5%

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

   7. Taylor expanded in u2 around 0 98.4%

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

      1. *-commutative 98.4%

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

      2. associate-*r* 98.4%

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

   9. Simplified 98.4%

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

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

   1. Initial program 55.5%

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

   3. Taylor expanded in u1 around 0 96.8%

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

      1. *-commutative 96.8%

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

   5. Simplified 96.8%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 + u1 \cdot 0.25\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.0006500000017695129:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \left(\pi \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 + u1 \cdot 0.25\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.5% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* PI 2.0))))
   (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 = u2 * (((float) M_PI) * 2.0f);
	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(u2 * Float32(Float32(pi) * Float32(2.0)))
	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 59.2%

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

      1. sub-neg 59.2%

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

      2. log1p-define 98.4%

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

   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. *-commutative 92.5%

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

      5. associate-*r* 92.5%

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

   6. Applied egg-rr 92.5%

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

   7. Taylor expanded in u2 around 0 98.4%

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

      1. *-commutative 98.4%

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

      2. associate-*r* 98.4%

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

   9. Simplified 98.4%

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

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

   1. Initial program 55.5%

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

   3. Taylor expanded in u1 around 0 95.1%

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

      1. *-commutative 95.1%

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

   5. Simplified 95.1%

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

4. Final simplification 97.1%

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

Alternative 5: 94.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(\pi \cdot 2\right)\\ \mathbf{if}\;t\_0 \leq 0.00279999990016222:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* PI 2.0))))
   (if (<= t_0 0.00279999990016222)
     (* (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 = u2 * (((float) M_PI) * 2.0f);
	float tmp;
	if (t_0 <= 0.00279999990016222f) {
		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(u2 * Float32(Float32(pi) * Float32(2.0)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.00279999990016222))
		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) < 0.0027999999

   1. Initial program 58.9%

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

      1. sub-neg 58.9%

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

      2. log1p-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 92.8%

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

      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. *-commutative 92.8%

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

      5. associate-*r* 92.8%

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

   6. Applied egg-rr 92.8%

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

   7. Taylor expanded in u2 around 0 97.7%

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

      1. *-commutative 97.7%

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

      2. associate-*r* 97.7%

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

   9. Simplified 97.7%

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

   if 0.0027999999 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 55.2%

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

   3. Taylor expanded in u1 around 0 91.8%

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

      1. *-commutative 91.8%

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

   5. Simplified 91.8%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.00279999990016222:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \left(\pi \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(\pi \cdot 2\right)\\ \mathbf{if}\;t\_0 \leq 0.0031999999191612005:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* PI 2.0))))
   (if (<= t_0 0.0031999999191612005)
     (* (sqrt (- (log1p (- u1)))) t_0)
     (* (sin t_0) (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u2 * (((float) M_PI) * 2.0f);
	float tmp;
	if (t_0 <= 0.0031999999191612005f) {
		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(u2 * Float32(Float32(pi) * Float32(2.0)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0031999999191612005))
		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) < 0.00319999992

   1. Initial program 58.7%

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

      1. sub-neg 58.7%

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

      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 92.8%

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

      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. *-commutative 92.8%

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

      5. associate-*r* 92.8%

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

   6. Applied egg-rr 92.8%

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

   7. Taylor expanded in u2 around 0 97.6%

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

      1. *-commutative 97.6%

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

      2. associate-*r* 97.6%

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

   9. Simplified 97.6%

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

   if 0.00319999992 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 55.6%

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

   3. Taylor expanded in u1 around 0 78.8%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.0031999999191612005:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \left(\pi \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(\pi \cdot 2\right)\\ \mathbf{if}\;t\_0 \leq 0.0031999999191612005:\\ \;\;\;\;2 \cdot \left(\sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right)\right)} \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* PI 2.0))))
   (if (<= t_0 0.0031999999191612005)
     (*
      2.0
      (*
       (sqrt (* u1 (+ 1.0 (* u1 (+ 0.5 (* u1 0.3333333333333333))))))
       (* PI u2)))
     (* (sin t_0) (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u2 * (((float) M_PI) * 2.0f);
	float tmp;
	if (t_0 <= 0.0031999999191612005f) {
		tmp = 2.0f * (sqrtf((u1 * (1.0f + (u1 * (0.5f + (u1 * 0.3333333333333333f)))))) * (((float) M_PI) * u2));
	} else {
		tmp = sinf(t_0) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u2 * Float32(Float32(pi) * Float32(2.0)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0031999999191612005))
		tmp = Float32(Float32(2.0) * Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(u1 * Float32(Float32(0.5) + Float32(u1 * Float32(0.3333333333333333))))))) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(sin(t_0) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = u2 * (single(pi) * single(2.0));
	tmp = single(0.0);
	if (t_0 <= single(0.0031999999191612005))
		tmp = single(2.0) * (sqrt((u1 * (single(1.0) + (u1 * (single(0.5) + (u1 * single(0.3333333333333333))))))) * (single(pi) * u2));
	else
		tmp = sin(t_0) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00319999992

   1. Initial program 58.7%

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

      1. sub-neg 58.7%

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

      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. associate-*l* 98.4%

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

      2. sin-2 98.4%

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

   6. Applied egg-rr 98.4%

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

   7. Taylor expanded in u1 around 0 91.3%

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

   8. Taylor expanded in u2 around 0 90.7%

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

   if 0.00319999992 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 55.6%

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

   3. Taylor expanded in u1 around 0 78.8%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.0031999999191612005:\\ \;\;\;\;2 \cdot \left(\sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right)\right)} \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.1% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.7%

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

   1. sub-neg 57.7%

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

   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. associate-*l* 98.3%

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

   2. sin-2 98.3%

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

6. Applied egg-rr 98.3%

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

7. Taylor expanded in u1 around 0 92.5%

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

8. Taylor expanded in u2 around 0 76.8%

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

9. Final simplification 76.8%

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

Alternative 9: 66.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.7%

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

3. Taylor expanded in u1 around 0 76.1%

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

4. Taylor expanded in u2 around 0 65.6%

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

5. Taylor expanded in u1 around -inf -0.0%

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

   1. mul-1-neg -0.0%

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

   2. associate-*r* -0.0%

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

   3. distribute-rgt-neg-in -0.0%

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

   4. unpow2 -0.0%

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

   5. rem-square-sqrt 65.6%

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

7. Simplified 65.6%

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

8. Taylor expanded in u1 around 0 65.6%

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

   1. *-commutative 65.6%

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

   2. *-commutative 65.6%

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

   3. associate-*r* 65.6%

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

   4. associate-*r* 65.6%

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

   5. *-commutative 65.6%

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

10. Simplified 65.6%

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

11. Final simplification 65.6%

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

Alternative 10: 66.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.7%

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

3. Taylor expanded in u1 around 0 76.1%

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

4. Taylor expanded in u2 around 0 65.6%

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

5. Final simplification 65.6%

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

Reproduce

herbie shell --seed 2024132 
(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))))