Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.2% → 98.4%

Time: 11.8s

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: 57.2% 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

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

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 56.2%

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

   1. sub-neg 56.2%

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

   2. log1p-def 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. Step-by-step derivation

   1. add-sqr-sqrt 97.5%

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

   2. pow2 97.5%

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

   3. *-commutative 97.5%

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

   4. associate-*r* 97.5%

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

5. Applied egg-rr 97.5%

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

   1. unpow2 97.5%

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

   2. add-sqr-sqrt 98.3%

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

   3. associate-*l* 98.3%

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

   4. add-sqr-sqrt 98.3%

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

   5. associate-*r* 98.4%

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

7. Applied egg-rr 98.4%

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

8. Final simplification 98.4%

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

Alternative 2: 98.4% 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 56.2%

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

   1. sub-neg 56.2%

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

   2. log1p-def 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. Final simplification 98.3%

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

Alternative 3: 88.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.2%

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

2. Taylor expanded in u1 around 0 88.4%

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

   1. *-commutative 88.4%

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

   2. *-commutative 88.4%

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

   3. unpow2 88.4%

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

   4. associate-*l* 88.4%

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

   5. distribute-lft-out 88.4%

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

4. Simplified 88.4%

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

5. Final simplification 88.4%

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

Alternative 4: 76.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.2%

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

2. Taylor expanded in u1 around 0 77.4%

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

   1. mul-1-neg 77.4%

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

4. Simplified 77.4%

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

5. Taylor expanded in u2 around inf 77.4%

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

6. Final simplification 77.4%

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

Alternative 5: 66.4% 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 56.2%

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

2. Taylor expanded in u1 around 0 77.4%

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

   1. mul-1-neg 77.4%

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

4. Simplified 77.4%

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

5. Taylor expanded in u2 around 0 67.7%

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

   1. *-commutative 67.7%

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

7. Simplified 67.7%

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

8. Taylor expanded in u2 around 0 67.7%

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

   1. associate-*r* 67.9%

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

   2. *-commutative 67.9%

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

   3. *-commutative 67.9%

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

10. Simplified 67.9%

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

11. Taylor expanded in u2 around 0 67.7%

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

    1. associate-*r* 67.9%

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

    2. *-commutative 67.9%

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

    3. associate-*l* 67.9%

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

13. Simplified 67.9%

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

14. Final simplification 67.9%

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

Alternative 6: 66.4% 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 56.2%

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

2. Taylor expanded in u1 around 0 77.4%

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

   1. mul-1-neg 77.4%

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

4. Simplified 77.4%

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

5. Taylor expanded in u2 around 0 67.7%

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

   1. *-commutative 67.7%

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

7. Simplified 67.7%

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

8. Taylor expanded in u2 around 0 67.7%

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

   1. associate-*r* 67.9%

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

   2. *-commutative 67.9%

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

   3. *-commutative 67.9%

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

10. Simplified 67.9%

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

11. Final simplification 67.9%

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

Alternative 7: 7.1% accurate, 409.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 0.0)

C

float code(float cosTheta_i, float u1, float u2) {
	return 0.0f;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = 0.0e0
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(0.0)
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
end

Derivation

1. Initial program 56.2%

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

   1. sub-neg 56.2%

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

   2. log1p-def 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. Step-by-step derivation

   1. add-cube-cbrt 97.2%

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

   2. pow3 97.3%

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

   3. *-commutative 97.3%

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

   4. associate-*r* 97.3%

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

5. Applied egg-rr 97.3%

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

6. Taylor expanded in u2 around 0 7.2%

   \[\leadsto \color{blue}{0} \]

7. Final simplification 7.2%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023321 
(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))))