Beckmann Sample, near normal, slope_y

Percentage Accurate: 58.2% → 98.3%

Time: 1.2min

Alternatives: 4

Speedup: 2.9×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 58.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

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(Float32(2.0) * Float32(pi)) * u2)))
end

Derivation

1. Initial program 56.5%

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

   1. sub-neg 56.5%

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

   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%

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

Alternative 2: 90.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.003000000026077032:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.003000000026077032)
     (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))
     (* (sin t_0) (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.003000000026077032f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	} else {
		tmp = sinf(t_0) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(2.0) * Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.003000000026077032))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(sin(t_0) * sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00300000003

   1. Initial program 57.6%

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

      1. sub-neg 57.6%

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

      2. log1p-define 98.5%

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

   3. Simplified 98.5%

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

      1. add-sqr-sqrt 97.7%

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

      2. pow2 97.7%

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

      3. associate-*l* 97.7%

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

   6. Applied egg-rr 97.7%

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

      1. unpow2 97.7%

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

      2. add-sqr-sqrt 98.5%

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

      3. *-commutative 98.5%

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

      4. associate-*r* 98.5%

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

      5. *-commutative 98.5%

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

      6. +-lft-identity 98.5%

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(0 + \sin \left(\pi \cdot \left(2 \cdot u2\right)\right)\right)} \]

      7. sin-0 98.5%

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\color{blue}{\sin 0} + \sin \left(\pi \cdot \left(2 \cdot u2\right)\right)\right) \]

      8. +-commutative 98.5%

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(\sin \left(\pi \cdot \left(2 \cdot u2\right)\right) + \sin 0\right)} \]

      9. *-un-lft-identity 98.5%

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(1 \cdot \left(\sin \left(\pi \cdot \left(2 \cdot u2\right)\right) + \sin 0\right)\right)} \]

      10. metadata-eval 98.5%

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\color{blue}{\left(2 \cdot 0.5\right)} \cdot \left(\sin \left(\pi \cdot \left(2 \cdot u2\right)\right) + \sin 0\right)\right) \]

      11. associate-*r* 98.5%

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(2 \cdot \left(0.5 \cdot \left(\sin \left(\pi \cdot \left(2 \cdot u2\right)\right) + \sin 0\right)\right)\right)} \]

      12. metadata-eval 98.5%

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\color{blue}{\frac{1}{2}} \cdot \left(\sin \left(\pi \cdot \left(2 \cdot u2\right)\right) + \sin 0\right)\right)\right) \]

      13. associate-/r/ 98.5%

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \color{blue}{\frac{1}{\frac{2}{\sin \left(\pi \cdot \left(2 \cdot u2\right)\right) + \sin 0}}}\right) \]

      14. un-div-inv 98.5%

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\frac{2}{\frac{2}{\sin \left(\pi \cdot \left(2 \cdot u2\right)\right) + \sin 0}}} \]

      15. +-commutative 98.5%

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \frac{2}{\frac{2}{\color{blue}{\sin 0 + \sin \left(\pi \cdot \left(2 \cdot u2\right)\right)}}} \]

      16. sin-0 98.5%

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \frac{2}{\frac{2}{\color{blue}{0} + \sin \left(\pi \cdot \left(2 \cdot u2\right)\right)}} \]

      17. +-lft-identity 98.5%

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

      18. *-commutative 98.5%

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

      19. associate-*r* 98.5%

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

      20. *-commutative 98.5%

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

   8. Applied egg-rr 98.5%

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

   9. Taylor expanded in u2 around 0 97.9%

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

   if 0.00300000003 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 54.3%

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

      1. sub-neg 54.3%

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

      2. log1p-define 98.2%

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

   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. Add Preprocessing
   5. Step-by-step derivation

      1. expm1-log1p-u 98.2%

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

      2. expm1-undefine 69.0%

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

   6. Applied egg-rr 54.9%

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

   7. Taylor expanded in u1 around 0 77.2%

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

      1. associate-*r* 77.2%

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

      2. *-commutative 77.2%

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

      3. associate-*r* 77.2%

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

      4. *-commutative 77.2%

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

   9. Simplified 77.2%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.003000000026077032:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.1% accurate, 1.5× speedup

Math

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

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(Float32(2.0) * 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 56.5%

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

   1. sub-neg 56.5%

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

   2. log1p-define 98.4%

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

3. Simplified 98.4%

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

   1. expm1-log1p-u 98.4%

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

   2. expm1-undefine 37.2%

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

6. Applied egg-rr 31.3%

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

7. Taylor expanded in u1 around 0 77.5%

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

   1. associate-*r* 77.5%

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

   2. *-commutative 77.5%

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

   3. associate-*r* 77.5%

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

   4. *-commutative 77.5%

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

9. Simplified 77.5%

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

10. Final simplification 77.5%

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

Alternative 4: 66.2% 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 56.5%

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

   1. sub-neg 56.5%

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

   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. Taylor expanded in u1 around 0 -0.0%

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
6. 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 4.0%

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

   4. *-commutative 4.0%

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

   5. associate-*l* 4.0%

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

   6. *-commutative 4.0%

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

   7. mul-1-neg 4.0%

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

7. Simplified 4.0%

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

   1. *-commutative 4.0%

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

   2. add-sqr-sqrt -0.0%

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

   3. sqrt-unprod 77.5%

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

   4. sqr-neg 77.5%

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

   5. add-sqr-sqrt 77.5%

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

   6. add-exp-log 73.0%

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

   7. *-commutative 73.0%

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

   8. associate-*r* 73.0%

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

   9. *-commutative 73.0%

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

9. Applied egg-rr 73.0%

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

10. Taylor expanded in u2 around 0 67.6%

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

11. Final simplification 67.6%

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

Reproduce

herbie shell --seed 2024115 
(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))))