Beckmann Sample, near normal, slope_y

Percentage Accurate: 58.2% → 98.4%

Time: 12.1s

Alternatives: 11

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: 58.2% 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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 54.1%

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

   1. sub-neg 54.1%

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

   2. log1p-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. Step-by-step derivation

   1. add-cube-cbrt 97.9%

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

   2. pow3 97.9%

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

8. Applied egg-rr 97.9%

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

   1. *-commutative 97.9%

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

   2. associate-*r* 97.9%

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

   3. rem-cube-cbrt 98.3%

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

   4. associate-*r* 98.3%

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

   5. sin-2 98.3%

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

   6. add-cbrt-cube 98.3%

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

   7. add-cbrt-cube 98.3%

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

   8. cbrt-unprod 98.1%

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

10. Applied egg-rr 98.4%

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

Alternative 2: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 54.1%

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

   1. sub-neg 54.1%

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

   2. log1p-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%

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

Alternative 3: 95.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.001500000013038516:\\ \;\;\;\;t\_0 \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.001500000013038516)
     (* t_0 (sqrt (- (log1p (- u1)))))
     (*
      (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 <= 0.001500000013038516f) {
		tmp = t_0 * sqrtf(-log1pf(-u1));
	} 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(0.001500000013038516))
		tmp = Float32(t_0 * sqrt(Float32(-log1p(Float32(-u1)))));
	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) < 0.00150000001

   1. Initial program 55.0%

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

      1. sub-neg 55.0%

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

      2. log1p-define 98.4%

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

   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. expm1-log1p-u 98.4%

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

      2. *-commutative 98.4%

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

      3. associate-*r* 98.4%

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

   6. Applied egg-rr 98.4%

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

   7. Taylor expanded in u2 around 0 98.1%

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

      1. associate-*r* 98.1%

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

      2. *-commutative 98.1%

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

      3. associate-*r* 98.1%

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

   9. Simplified 98.1%

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

   if 0.00150000001 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 52.6%

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

   3. Taylor expanded in u1 around 0 94.5%

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

      1. *-commutative 94.5%

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

   5. Simplified 94.5%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.001500000013038516:\\ \;\;\;\;\left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.001500000013038516:\\ \;\;\;\;t\_0 \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.001500000013038516)
     (* t_0 (sqrt (- (log1p (- u1)))))
     (* (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 <= 0.001500000013038516f) {
		tmp = t_0 * sqrtf(-log1pf(-u1));
	} 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(0.001500000013038516))
		tmp = Float32(t_0 * sqrt(Float32(-log1p(Float32(-u1)))));
	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) < 0.00150000001

   1. Initial program 55.0%

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

      1. sub-neg 55.0%

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

      2. log1p-define 98.4%

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

   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. expm1-log1p-u 98.4%

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

      2. *-commutative 98.4%

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

      3. associate-*r* 98.4%

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

   6. Applied egg-rr 98.4%

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

   7. Taylor expanded in u2 around 0 98.1%

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

      1. associate-*r* 98.1%

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

      2. *-commutative 98.1%

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

      3. associate-*r* 98.1%

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

   9. Simplified 98.1%

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

   if 0.00150000001 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 52.6%

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

   3. Taylor expanded in u1 around 0 92.0%

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

      1. *-commutative 92.0%

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

   5. Simplified 92.0%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.001500000013038516:\\ \;\;\;\;\left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.3% 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 54.1%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
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 6: 90.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.005400000140070915:\\ \;\;\;\;t\_0 \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.005400000140070915)
     (* t_0 (sqrt (- (log1p (- u1)))))
     (* (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 <= 0.005400000140070915f) {
		tmp = t_0 * sqrtf(-log1pf(-u1));
	} 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(0.005400000140070915))
		tmp = Float32(t_0 * sqrt(Float32(-log1p(Float32(-u1)))));
	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) < 0.00540000014

   1. Initial program 55.0%

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

      1. sub-neg 55.0%

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

      2. log1p-define 98.4%

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

   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. expm1-log1p-u 98.4%

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

      2. *-commutative 98.4%

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

      3. associate-*r* 98.4%

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

   6. Applied egg-rr 98.4%

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

   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. associate-*r* 97.4%

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

      2. *-commutative 97.4%

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

      3. associate-*r* 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 0.00540000014 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 52.3%

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

   3. Taylor expanded in u1 around 0 80.3%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.005400000140070915:\\ \;\;\;\;\left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.005400000140070915:\\ \;\;\;\;u2 \cdot \left(2 \cdot \left(\pi \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot 0.5\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.005400000140070915)
     (* u2 (* 2.0 (* PI (sqrt (* u1 (+ 1.0 (* u1 0.5)))))))
     (* (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 <= 0.005400000140070915f) {
		tmp = u2 * (2.0f * (((float) M_PI) * sqrtf((u1 * (1.0f + (u1 * 0.5f))))));
	} 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(0.005400000140070915))
		tmp = Float32(u2 * Float32(Float32(2.0) * Float32(Float32(pi) * sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(u1 * Float32(0.5))))))));
	else
		tmp = Float32(sin(t_0) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = (single(2.0) * single(pi)) * u2;
	tmp = single(0.0);
	if (t_0 <= single(0.005400000140070915))
		tmp = u2 * (single(2.0) * (single(pi) * sqrt((u1 * (single(1.0) + (u1 * single(0.5)))))));
	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) < 0.00540000014

   1. Initial program 55.0%

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

      1. sub-neg 55.0%

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

      2. log1p-define 98.4%

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

   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 88.2%

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

   8. Taylor expanded in u2 around 0 88.3%

      \[\leadsto \color{blue}{u2 \cdot \left(2 \cdot \left(\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \pi\right) + 2 \cdot \left(\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left({u2}^{2} \cdot \left(-0.5 \cdot {\pi}^{3} + -0.16666666666666666 \cdot {\pi}^{3}\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 88.3%

         \[\leadsto u2 \cdot \left(\color{blue}{\left(2 \cdot \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)}\right) \cdot \pi} + 2 \cdot \left(\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left({u2}^{2} \cdot \left(-0.5 \cdot {\pi}^{3} + -0.16666666666666666 \cdot {\pi}^{3}\right)\right)\right)\right) \]

      2. associate-*r* 88.3%

         \[\leadsto u2 \cdot \left(\left(2 \cdot \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)}\right) \cdot \pi + \color{blue}{\left(2 \cdot \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)}\right) \cdot \left({u2}^{2} \cdot \left(-0.5 \cdot {\pi}^{3} + -0.16666666666666666 \cdot {\pi}^{3}\right)\right)}\right) \]

      3. distribute-lft-out 88.3%

         \[\leadsto u2 \cdot \color{blue}{\left(\left(2 \cdot \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)}\right) \cdot \left(\pi + {u2}^{2} \cdot \left(-0.5 \cdot {\pi}^{3} + -0.16666666666666666 \cdot {\pi}^{3}\right)\right)\right)} \]

      4. +-commutative 88.3%

         \[\leadsto u2 \cdot \left(\left(2 \cdot \sqrt{u1 \cdot \color{blue}{\left(0.5 \cdot u1 + 1\right)}}\right) \cdot \left(\pi + {u2}^{2} \cdot \left(-0.5 \cdot {\pi}^{3} + -0.16666666666666666 \cdot {\pi}^{3}\right)\right)\right) \]

      5. *-commutative 88.3%

         \[\leadsto u2 \cdot \left(\left(2 \cdot \sqrt{u1 \cdot \left(\color{blue}{u1 \cdot 0.5} + 1\right)}\right) \cdot \left(\pi + {u2}^{2} \cdot \left(-0.5 \cdot {\pi}^{3} + -0.16666666666666666 \cdot {\pi}^{3}\right)\right)\right) \]

      6. fma-undefine 88.3%

         \[\leadsto u2 \cdot \left(\left(2 \cdot \sqrt{u1 \cdot \color{blue}{\mathsf{fma}\left(u1, 0.5, 1\right)}}\right) \cdot \left(\pi + {u2}^{2} \cdot \left(-0.5 \cdot {\pi}^{3} + -0.16666666666666666 \cdot {\pi}^{3}\right)\right)\right) \]

   10. Simplified 88.3%

       \[\leadsto \color{blue}{u2 \cdot \left(\left(2 \cdot \sqrt{u1 \cdot \mathsf{fma}\left(u1, 0.5, 1\right)}\right) \cdot \left(\pi + {u2}^{2} \cdot \left({\pi}^{3} \cdot -0.6666666666666666\right)\right)\right)} \]

   11. Taylor expanded in u2 around 0 87.6%

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

   if 0.00540000014 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 52.3%

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

   3. Taylor expanded in u1 around 0 80.3%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.005400000140070915:\\ \;\;\;\;u2 \cdot \left(2 \cdot \left(\pi \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot 0.5\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.2% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 54.1%

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

   1. sub-neg 54.1%

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

   2. log1p-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 89.2%

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

8. Taylor expanded in u2 around 0 81.3%

   \[\leadsto \color{blue}{u2 \cdot \left(2 \cdot \left(\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \pi\right) + 2 \cdot \left(\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left({u2}^{2} \cdot \left(-0.5 \cdot {\pi}^{3} + -0.16666666666666666 \cdot {\pi}^{3}\right)\right)\right)\right)} \]
9. Step-by-step derivation

   1. associate-*r* 81.3%

      \[\leadsto u2 \cdot \left(\color{blue}{\left(2 \cdot \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)}\right) \cdot \pi} + 2 \cdot \left(\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left({u2}^{2} \cdot \left(-0.5 \cdot {\pi}^{3} + -0.16666666666666666 \cdot {\pi}^{3}\right)\right)\right)\right) \]

   2. associate-*r* 81.3%

      \[\leadsto u2 \cdot \left(\left(2 \cdot \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)}\right) \cdot \pi + \color{blue}{\left(2 \cdot \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)}\right) \cdot \left({u2}^{2} \cdot \left(-0.5 \cdot {\pi}^{3} + -0.16666666666666666 \cdot {\pi}^{3}\right)\right)}\right) \]

   3. distribute-lft-out 81.3%

      \[\leadsto u2 \cdot \color{blue}{\left(\left(2 \cdot \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)}\right) \cdot \left(\pi + {u2}^{2} \cdot \left(-0.5 \cdot {\pi}^{3} + -0.16666666666666666 \cdot {\pi}^{3}\right)\right)\right)} \]

   4. +-commutative 81.3%

      \[\leadsto u2 \cdot \left(\left(2 \cdot \sqrt{u1 \cdot \color{blue}{\left(0.5 \cdot u1 + 1\right)}}\right) \cdot \left(\pi + {u2}^{2} \cdot \left(-0.5 \cdot {\pi}^{3} + -0.16666666666666666 \cdot {\pi}^{3}\right)\right)\right) \]

   5. *-commutative 81.3%

      \[\leadsto u2 \cdot \left(\left(2 \cdot \sqrt{u1 \cdot \left(\color{blue}{u1 \cdot 0.5} + 1\right)}\right) \cdot \left(\pi + {u2}^{2} \cdot \left(-0.5 \cdot {\pi}^{3} + -0.16666666666666666 \cdot {\pi}^{3}\right)\right)\right) \]

   6. fma-undefine 81.3%

      \[\leadsto u2 \cdot \left(\left(2 \cdot \sqrt{u1 \cdot \color{blue}{\mathsf{fma}\left(u1, 0.5, 1\right)}}\right) \cdot \left(\pi + {u2}^{2} \cdot \left(-0.5 \cdot {\pi}^{3} + -0.16666666666666666 \cdot {\pi}^{3}\right)\right)\right) \]

10. Simplified 81.3%

    \[\leadsto \color{blue}{u2 \cdot \left(\left(2 \cdot \sqrt{u1 \cdot \mathsf{fma}\left(u1, 0.5, 1\right)}\right) \cdot \left(\pi + {u2}^{2} \cdot \left({\pi}^{3} \cdot -0.6666666666666666\right)\right)\right)} \]

11. Taylor expanded in u2 around 0 74.5%

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

12. Final simplification 74.5%

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

Alternative 9: 74.2% accurate, 2.7× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return 2.0f * (sqrtf((u1 * (1.0f + (u1 * 0.5f)))) * (((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(0.5))))) * 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))))) * (single(pi) * u2));
end

Derivation

1. Initial program 54.1%

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

   1. sub-neg 54.1%

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

   2. log1p-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 89.2%

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

8. Taylor expanded in u2 around 0 74.4%

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

9. Final simplification 74.4%

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

Alternative 10: 66.1% 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 54.1%

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

3. Taylor expanded in u1 around 0 78.8%

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

4. Taylor expanded in u2 around 0 67.8%

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

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 67.8%

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

7. Simplified 67.8%

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

   1. neg-sub0 67.8%

      \[\leadsto 2 \cdot \left(\left(\sqrt{u1} \cdot u2\right) \cdot \color{blue}{\left(0 - \pi \cdot -1\right)}\right) \]

9. Applied egg-rr 67.8%

   \[\leadsto 2 \cdot \left(\left(\sqrt{u1} \cdot u2\right) \cdot \color{blue}{\left(0 - \pi \cdot -1\right)}\right) \]
10. Step-by-step derivation

    1. neg-sub0 67.8%

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

    2. distribute-rgt-neg-in 67.8%

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

    3. metadata-eval 67.8%

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

    4. *-rgt-identity 67.8%

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

11. Simplified 67.8%

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

12. Final simplification 67.8%

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

Alternative 11: 66.1% 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 54.1%

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

3. Taylor expanded in u1 around 0 78.8%

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

4. Taylor expanded in u2 around 0 67.8%

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

5. Final simplification 67.8%

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

Reproduce

herbie shell --seed 2024129 
(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))))