Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.7% → 98.4%

Time: 13.3s

Alternatives: 7

Speedup: 2.0×

Specification

\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf((1.0f - u1))) * sinf(((2.0f * ((float) M_PI)) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(-log((single(1.0) - u1))) * sin(((single(2.0) * single(pi)) * u2));
end

Initial Program: 57.7% 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

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

FPCore

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

C

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

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(Float32(2.0) * Float32(pi)))
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * sin(Float32(t_0 * Float32(u2 * t_0))))
end

Derivation

1. Initial program 54.4%

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

   1. sub-neg 54.4%

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

   2. log1p-def 98.2%

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

3. Simplified 98.2%

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

   1. add-exp-log 95.0%

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

   2. associate-*l* 95.0%

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

5. Applied egg-rr 95.0%

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

   1. add-exp-log 98.2%

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

   2. associate-*r* 98.2%

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

   3. *-commutative 98.2%

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

   4. add-sqr-sqrt 98.2%

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

   5. associate-*r* 98.5%

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

7. Applied egg-rr 98.5%

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

8. Final simplification 98.5%

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

Alternative 2: 98.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * sinf(cbrtf((powf(u2, 3.0f) * powf((2.0f * ((float) M_PI)), 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(2.0) * Float32(pi)) ^ Float32(3.0))))))
end

Derivation

1. Initial program 54.4%

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

   1. sub-neg 54.4%

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

   2. log1p-def 98.2%

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

3. Simplified 98.2%

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

   1. add-cbrt-cube 98.2%

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

   2. add-cbrt-cube 98.2%

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

   3. cbrt-unprod 98.3%

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

   4. *-commutative 98.3%

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

   5. pow3 98.3%

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

   6. pow3 98.3%

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

5. Applied egg-rr 98.3%

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

6. Final simplification 98.3%

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

Alternative 3: 98.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 54.4%

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

   1. sub-neg 54.4%

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

   2. log1p-def 98.2%

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

3. Simplified 98.2%

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

   1. expm1-log1p-u 98.2%

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

   2. associate-*l* 98.2%

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

5. Applied egg-rr 98.2%

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

6. Final simplification 98.2%

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

Alternative 4: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 54.4%

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

   1. sub-neg 54.4%

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

   2. log1p-def 98.2%

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

3. Simplified 98.2%

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

4. Final simplification 98.2%

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

Alternative 5: 76.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 54.4%

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

   1. sub-neg 54.4%

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

   2. log1p-def 98.2%

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

3. Simplified 98.2%

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

   1. add-cube-cbrt 97.2%

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

   2. pow3 97.1%

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

5. Applied egg-rr 76.0%

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

6. Taylor expanded in u1 around 0 78.4%

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

7. Final simplification 78.4%

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

Alternative 6: 66.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 54.4%

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

2. Taylor expanded in u1 around 0 78.4%

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

   1. mul-1-neg 78.4%

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

4. Simplified 78.4%

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

   1. add-exp-log 73.7%

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

   2. *-commutative 73.7%

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

   3. associate-*r* 73.7%

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

   4. remove-double-neg 73.7%

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

6. Applied egg-rr 73.7%

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

7. Taylor expanded in u2 around 0 67.6%

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

   1. *-commutative 67.6%

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

   2. associate-*r* 67.6%

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

9. Simplified 67.6%

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

10. Final simplification 67.6%

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

Alternative 7: 66.6% accurate, 2.0× 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 54.4%

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

2. Taylor expanded in u1 around 0 78.4%

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

   1. mul-1-neg 78.4%

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

4. Simplified 78.4%

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

5. Taylor expanded in u2 around 0 67.6%

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

   1. associate-*r* 67.7%

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

7. Simplified 67.7%

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

8. Final simplification 67.7%

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

Reproduce

herbie shell --seed 2023336 
(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))))