Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.9% → 98.4%

Time: 11.0s

Alternatives: 5

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: 57.9% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 53.4%

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

   1. sub-neg 53.4%

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

   2. log1p-def 98.6%

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

   3. associate-*l* 98.6%

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

3. Simplified 98.6%

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

4. Final simplification 98.6%

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

Alternative 2: 90.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.013000000268220901:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\pi \cdot \left(2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 (* 2.0 PI)) 0.013000000268220901)
   (* (sqrt (- (log1p (- u1)))) (* PI (* 2.0 u2)))
   (* (sin (* 2.0 (* PI u2))) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * (2.0f * ((float) M_PI))) <= 0.013000000268220901f) {
		tmp = sqrtf(-log1pf(-u1)) * (((float) M_PI) * (2.0f * u2));
	} else {
		tmp = sinf((2.0f * (((float) M_PI) * u2))) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(Float32(2.0) * Float32(pi))) <= Float32(0.013000000268220901))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(pi) * Float32(Float32(2.0) * u2)));
	else
		tmp = Float32(sin(Float32(Float32(2.0) * Float32(Float32(pi) * u2))) * sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 2 (PI.f32)) u2) < 0.0130000003

   1. Initial program 54.9%

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

      1. sub-neg 54.9%

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

      2. log1p-def 98.6%

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

      3. associate-*l* 98.6%

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

   3. Simplified 98.6%

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

      1. add-cbrt-cube 98.6%

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

      2. add-cbrt-cube 98.6%

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

      3. cbrt-unprod 98.5%

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

      4. pow3 98.5%

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

      5. pow3 98.5%

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

   5. Applied egg-rr 98.5%

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

   6. Taylor expanded in u2 around 0 96.2%

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

      1. associate-*r* 96.2%

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

      2. *-commutative 96.2%

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

   8. Simplified 96.2%

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

   if 0.0130000003 < (*.f32 (*.f32 2 (PI.f32)) u2)

   1. Initial program 50.3%

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

      1. sub-neg 50.3%

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

      2. log1p-def 98.5%

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

      3. associate-*l* 98.5%

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

   3. Simplified 98.5%

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

      1. neg-mul-1 98.5%

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

      2. log1p-udef 50.3%

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

      3. sub-neg 50.3%

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

      4. neg-mul-1 50.3%

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

      5. add-sqr-sqrt 50.4%

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

      6. pow2 50.4%

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

   5. Applied egg-rr 78.4%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{0.25}\right)}^{2}} \cdot \sin \left(2 \cdot \left(\pi \cdot u2\right)\right) \]

   6. Taylor expanded in u1 around 0 80.5%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.013000000268220901:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\pi \cdot \left(2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 3: 76.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 53.4%

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

   1. sub-neg 53.4%

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

   2. log1p-def 98.6%

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

   3. associate-*l* 98.6%

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

3. Simplified 98.6%

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

   1. neg-mul-1 98.6%

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

   2. log1p-udef 53.4%

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

   3. sub-neg 53.4%

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

   4. neg-mul-1 53.4%

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

   5. add-sqr-sqrt 53.4%

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

   6. pow2 53.4%

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

5. Applied egg-rr 77.0%

   \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{0.25}\right)}^{2}} \cdot \sin \left(2 \cdot \left(\pi \cdot u2\right)\right) \]

6. Taylor expanded in u1 around 0 79.1%

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

7. Final simplification 79.1%

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

Alternative 4: 66.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 53.4%

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

2. Taylor expanded in u1 around 0 79.1%

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

   1. mul-1-neg 79.1%

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

4. Simplified 79.1%

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

5. Taylor expanded in u2 around 0 67.5%

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

   1. associate-*r* 67.5%

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

   2. associate-*r* 67.5%

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

   3. *-commutative 67.5%

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

7. Simplified 67.5%

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

8. Final simplification 67.5%

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

Alternative 5: 66.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 53.4%

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

2. Taylor expanded in u1 around 0 79.1%

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

   1. mul-1-neg 79.1%

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

4. Simplified 79.1%

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

5. Taylor expanded in u2 around 0 67.5%

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

   1. associate-*r* 67.5%

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

   2. associate-*r* 67.5%

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

   3. *-commutative 67.5%

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

7. Simplified 67.5%

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

   1. add-cube-cbrt 67.3%

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

   2. pow3 67.3%

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

9. Applied egg-rr 67.3%

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

    1. add-log-exp 27.4%

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

    2. rem-cube-cbrt 27.4%

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

    3. associate-*l* 27.4%

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

    4. exp-prod 27.4%

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

11. Applied egg-rr 27.4%

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

    1. log-pow 67.5%

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

    2. *-commutative 67.5%

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

    3. rem-log-exp 67.5%

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

13. Simplified 67.5%

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

14. Final simplification 67.5%

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

Reproduce

herbie shell --seed 2023240 
(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))))