Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.5% → 98.5%

Time: 14.9s

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: 57.5% 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: 98.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.8%

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

   1. sub-neg 56.8%

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

   2. log1p-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%

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

   2. add-cbrt-cube 98.3%

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

   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 98.4%

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

6. Applied egg-rr 98.4%

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

   1. add-cube-cbrt 98.4%

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

   2. unpow-prod-down 98.6%

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

   3. pow2 98.6%

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

   4. *-commutative 98.6%

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

   5. pow3 98.6%

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

   6. add-cube-cbrt 98.6%

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

   7. *-commutative 98.6%

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

8. Applied egg-rr 98.6%

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

9. Final simplification 98.6%

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

Alternative 2: 98.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.8%

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

   1. sub-neg 56.8%

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

   2. log1p-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%

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

   2. add-cbrt-cube 98.3%

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

   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 98.4%

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

6. Applied egg-rr 98.4%

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

7. Final simplification 98.4%

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

Alternative 3: 98.3% 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 56.8%

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

   1. sub-neg 56.8%

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

   2. log1p-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. Final simplification 98.4%

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

Alternative 4: 96.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(\pi \cdot 2\right)\\ \mathbf{if}\;t\_0 \leq 0.0013500000350177288:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 - u1 \cdot -0.25\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* PI 2.0))))
   (if (<= t_0 0.0013500000350177288)
     (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))
     (*
      (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 <= 0.0013500000350177288f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	} 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(0.0013500000350177288))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	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) < 0.00135000004

   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 98.7%

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

   3. Simplified 98.7%

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

      1. add-cbrt-cube 98.7%

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

      2. pow1/3 92.4%

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

      3. pow3 92.4%

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

      4. *-commutative 92.4%

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

      5. associate-*r* 92.4%

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

   6. Applied egg-rr 92.4%

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

   7. Taylor expanded in u2 around 0 98.6%

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

   if 0.00135000004 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 56.1%

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

   3. Taylor expanded in u1 around 0 93.1%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.0013500000350177288:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 - u1 \cdot -0.25\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(\pi \cdot 2\right)\\ \mathbf{if}\;t\_0 \leq 0.0013500000350177288:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 - u1 \cdot -0.3333333333333333\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* PI 2.0))))
   (if (<= t_0 0.0013500000350177288)
     (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))
     (*
      (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 <= 0.0013500000350177288f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	} 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(0.0013500000350177288))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	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) < 0.00135000004

   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 98.7%

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

   3. Simplified 98.7%

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

      1. add-cbrt-cube 98.7%

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

      2. pow1/3 92.4%

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

      3. pow3 92.4%

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

      4. *-commutative 92.4%

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

      5. associate-*r* 92.4%

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

   6. Applied egg-rr 92.4%

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

   7. Taylor expanded in u2 around 0 98.6%

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

   if 0.00135000004 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 56.1%

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

   3. Taylor expanded in u1 around 0 91.4%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.0013500000350177288:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 - u1 \cdot -0.3333333333333333\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(\pi \cdot 2\right)\\ \mathbf{if}\;t\_0 \leq 0.007499999832361937:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{u1 \cdot \left(1 - u1 \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* PI 2.0))))
   (if (<= t_0 0.007499999832361937)
     (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))
     (* (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 <= 0.007499999832361937f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	} 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(0.007499999832361937))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	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) < 0.00749999983

   1. Initial program 57.3%

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

      1. sub-neg 57.3%

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

      2. log1p-define 98.6%

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

   3. Simplified 98.6%

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

      1. add-cbrt-cube 98.6%

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

      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 97.3%

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

   if 0.00749999983 < (*.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 88.9%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.007499999832361937:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{u1 \cdot \left(1 - u1 \cdot -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(\pi \cdot u2\right)\\ \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.04500000178813934:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* 2.0 (* PI u2))))
   (if (<= (* u2 (* PI 2.0)) 0.04500000178813934)
     (* (sqrt (- (log1p (- u1)))) t_0)
     (* (sqrt u1) (sin t_0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = 2.0f * (((float) M_PI) * u2);
	float tmp;
	if ((u2 * (((float) M_PI) * 2.0f)) <= 0.04500000178813934f) {
		tmp = sqrtf(-log1pf(-u1)) * t_0;
	} else {
		tmp = sqrtf(u1) * sinf(t_0);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(2.0) * Float32(Float32(pi) * u2))
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(Float32(pi) * Float32(2.0))) <= Float32(0.04500000178813934))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * t_0);
	else
		tmp = Float32(sqrt(u1) * sin(t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.0450000018

   1. Initial program 58.4%

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

      1. sub-neg 58.4%

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

      2. log1p-define 98.6%

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

   3. Simplified 98.6%

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

      1. add-cbrt-cube 98.6%

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

      2. pow1/3 93.1%

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

      3. pow3 93.1%

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

      4. *-commutative 93.1%

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

      5. associate-*r* 93.1%

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

   6. Applied egg-rr 93.1%

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

   7. Taylor expanded in u2 around 0 95.5%

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

   if 0.0450000018 < (*.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 -0.0%

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

      1. associate-*r* -0.0%

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

      2. unpow2 -0.0%

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

      3. rem-square-sqrt 3.6%

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

      4. *-commutative 3.6%

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

      5. associate-*l* 3.6%

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

      6. *-commutative 3.6%

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

      7. mul-1-neg 3.6%

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

   5. Simplified 3.6%

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

      1. expm1-log1p-u 3.6%

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

      2. expm1-undefine 3.6%

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

      3. *-commutative 3.6%

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

      4. add-sqr-sqrt -0.0%

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

      5. sqrt-unprod 61.7%

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

      6. sqr-neg 61.7%

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

      7. add-sqr-sqrt 61.7%

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

      8. associate-*r* 61.7%

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

      9. *-commutative 61.7%

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

      10. *-commutative 61.7%

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

      11. *-commutative 61.7%

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

   7. Applied egg-rr 61.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{u1} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right)\right)} - 1} \]
   8. Step-by-step derivation

      1. sub-neg 61.7%

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

      2. metadata-eval 61.7%

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

      3. +-commutative 61.7%

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

      4. log1p-undefine 61.7%

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

      5. rem-exp-log 61.7%

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

      6. associate-+r+ 82.4%

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

      7. metadata-eval 82.4%

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

      8. mul0-lft 82.4%

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

      9. *-commutative 82.4%

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

      10. *-commutative 82.4%

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

      11. associate-*r* 82.4%

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

      12. distribute-rgt-in 82.4%

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

      13. +-lft-identity 82.4%

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

   9. Simplified 82.4%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.04500000178813934:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.8%

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

3. Taylor expanded in u1 around 0 -0.0%

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

   1. associate-*r* -0.0%

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

   2. unpow2 -0.0%

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

   3. rem-square-sqrt 4.1%

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

   4. *-commutative 4.1%

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

   5. associate-*l* 4.1%

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

   6. *-commutative 4.1%

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

   7. mul-1-neg 4.1%

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

5. Simplified 4.1%

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

   1. expm1-log1p-u 4.1%

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

   2. expm1-undefine 5.3%

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

   3. *-commutative 5.3%

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

   4. add-sqr-sqrt -0.0%

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

   5. sqrt-unprod 31.8%

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

   6. sqr-neg 31.8%

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

   7. add-sqr-sqrt 31.8%

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

   8. associate-*r* 31.8%

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

   9. *-commutative 31.8%

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

   10. *-commutative 31.8%

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

   11. *-commutative 31.8%

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

7. Applied egg-rr 31.8%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{u1} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right)\right)} - 1} \]
8. Step-by-step derivation

   1. sub-neg 31.8%

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

   2. metadata-eval 31.8%

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

   3. +-commutative 31.8%

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

   4. log1p-undefine 31.8%

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

   5. rem-exp-log 31.8%

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

   6. associate-+r+ 78.1%

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

   7. metadata-eval 78.1%

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

   8. mul0-lft 78.1%

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

   9. *-commutative 78.1%

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

   10. *-commutative 78.1%

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

   11. associate-*r* 78.1%

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

   12. distribute-rgt-in 78.1%

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

   13. +-lft-identity 78.1%

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

9. Simplified 78.1%

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

10. Final simplification 78.1%

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

Alternative 9: 4.7% accurate, 2.9× speedup

Math

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

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(Float32(pi) * 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 56.8%

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

3. Taylor expanded in u1 around 0 -0.0%

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

   1. associate-*r* -0.0%

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

   2. unpow2 -0.0%

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

   3. rem-square-sqrt 4.1%

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

   4. *-commutative 4.1%

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

   5. associate-*l* 4.1%

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

   6. *-commutative 4.1%

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

   7. mul-1-neg 4.1%

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

5. Simplified 4.1%

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

6. Taylor expanded in u2 around 0 4.8%

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

7. Final simplification 4.8%

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

Alternative 10: 65.9% accurate, 2.9× speedup

Math

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

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(Float32(2.0) * Float32(Float32(pi) * 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 56.8%

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

3. Taylor expanded in u1 around 0 -0.0%

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

   1. associate-*r* -0.0%

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

   2. unpow2 -0.0%

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

   3. rem-square-sqrt 4.1%

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

   4. *-commutative 4.1%

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

   5. associate-*l* 4.1%

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

   6. *-commutative 4.1%

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

   7. mul-1-neg 4.1%

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

5. Simplified 4.1%

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

   1. expm1-log1p-u 4.1%

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

   2. expm1-undefine 5.3%

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

   3. *-commutative 5.3%

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

   4. add-sqr-sqrt -0.0%

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

   5. sqrt-unprod 31.8%

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

   6. sqr-neg 31.8%

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

   7. add-sqr-sqrt 31.8%

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

   8. associate-*r* 31.8%

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

   9. *-commutative 31.8%

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

   10. *-commutative 31.8%

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

   11. *-commutative 31.8%

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

7. Applied egg-rr 31.8%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{u1} \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right)\right)} - 1} \]
8. Step-by-step derivation

   1. sub-neg 31.8%

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

   2. metadata-eval 31.8%

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

   3. +-commutative 31.8%

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

   4. log1p-undefine 31.8%

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

   5. rem-exp-log 31.8%

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

   6. associate-+r+ 78.1%

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

   7. metadata-eval 78.1%

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

   8. mul0-lft 78.1%

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

   9. *-commutative 78.1%

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

   10. *-commutative 78.1%

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

   11. associate-*r* 78.1%

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

   12. distribute-rgt-in 78.1%

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

   13. +-lft-identity 78.1%

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

9. Simplified 78.1%

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

10. Taylor expanded in u2 around 0 67.3%

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

11. Final simplification 67.3%

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

Reproduce

herbie shell --seed 2024078 
(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))))