Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.3% → 99.0%

Time: 17.3s

Alternatives: 7

Speedup: 3.1×

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(2 \cdot \pi\right)\\ t_1 := 2 + t_0\\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(\frac{t_0 \cdot t_1}{t_1}\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* 2.0 PI))) (t_1 (+ 2.0 t_0)))
   (* (sqrt (- (log1p (- u1)))) (cos (/ (* t_0 t_1) t_1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u2 * (2.0f * ((float) M_PI));
	float t_1 = 2.0f + t_0;
	return sqrtf(-log1pf(-u1)) * cosf(((t_0 * t_1) / t_1));
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u2 * Float32(Float32(2.0) * Float32(pi)))
	t_1 = Float32(Float32(2.0) + t_0)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * cos(Float32(Float32(t_0 * t_1) / t_1)))
end

Derivation

1. Initial program 58.7%

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

   1. sub-neg 58.7%

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

   2. log1p-def 99.3%

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

3. Simplified 99.3%

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

   1. expm1-log1p-u 99.3%

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

   2. associate-*l* 99.3%

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

6. Applied egg-rr 99.3%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right)} \]
7. Step-by-step derivation

   1. expm1-udef 99.3%

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

   2. flip-- 99.2%

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

8. Applied egg-rr 99.2%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \color{blue}{\left(\frac{\left(u2 \cdot \left(\pi \cdot 2\right) + 1\right) \cdot \left(u2 \cdot \left(\pi \cdot 2\right) + 1\right) - 1}{\left(u2 \cdot \left(\pi \cdot 2\right) + 1\right) + 1}\right)} \]
9. Step-by-step derivation

   1. difference-of-sqr-1 99.3%

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

   2. +-commutative 99.3%

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

   3. +-commutative 99.3%

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

   4. associate-+r+ 99.3%

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

   5. metadata-eval 99.3%

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

   6. associate--l+ 99.3%

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

   7. metadata-eval 99.3%

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

   8. +-rgt-identity 99.3%

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

   9. +-commutative 99.3%

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

   10. +-commutative 99.3%

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

   11. associate-+r+ 99.3%

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

   12. metadata-eval 99.3%

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

10. Simplified 99.3%

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

11. Final simplification 99.3%

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(\frac{\left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \left(2 + u2 \cdot \left(2 \cdot \pi\right)\right)}{2 + u2 \cdot \left(2 \cdot \pi\right)}\right) \]
12. Add Preprocessing

Alternative 2: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(2 \cdot \pi\right)\\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(\left(\left(t_0 \cdot t_0\right) \cdot \frac{0.5}{u2 \cdot \pi} + 1\right) + -1\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* 2.0 PI))))
   (*
    (sqrt (- (log1p (- u1))))
    (cos (+ (+ (* (* t_0 t_0) (/ 0.5 (* u2 PI))) 1.0) -1.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u2 * (2.0f * ((float) M_PI));
	return sqrtf(-log1pf(-u1)) * cosf(((((t_0 * t_0) * (0.5f / (u2 * ((float) M_PI)))) + 1.0f) + -1.0f));
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u2 * Float32(Float32(2.0) * Float32(pi)))
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * cos(Float32(Float32(Float32(Float32(t_0 * t_0) * Float32(Float32(0.5) / Float32(u2 * Float32(pi)))) + Float32(1.0)) + Float32(-1.0))))
end

Derivation

1. Initial program 58.7%

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

   1. sub-neg 58.7%

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

   2. log1p-def 99.3%

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

3. Simplified 99.3%

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

   1. expm1-log1p-u 99.3%

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

   2. associate-*l* 99.3%

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

6. Applied egg-rr 99.3%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right)} \]
7. Step-by-step derivation

   1. expm1-udef 99.3%

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

   2. log1p-udef 99.2%

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

   3. rem-exp-log 99.3%

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

   4. +-commutative 99.3%

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

   5. associate-*r* 99.3%

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

   6. *-commutative 99.3%

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

   7. *-commutative 99.3%

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

8. Applied egg-rr 99.3%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \color{blue}{\left(\left(u2 \cdot \left(\pi \cdot 2\right) + 1\right) - 1\right)} \]
9. Step-by-step derivation

   1. +-rgt-identity 99.3%

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

   2. flip-+ 99.3%

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

   3. pow2 99.3%

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

   4. *-commutative 99.3%

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

   5. *-commutative 99.3%

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

   6. associate-*l* 99.3%

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

   7. metadata-eval 99.3%

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

   8. *-commutative 99.3%

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

   9. *-commutative 99.3%

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

   10. associate-*l* 99.3%

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

10. Applied egg-rr 99.3%

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(\left(\color{blue}{\frac{{\left(2 \cdot \left(\pi \cdot u2\right)\right)}^{2} - 0}{2 \cdot \left(\pi \cdot u2\right) - 0}} + 1\right) - 1\right) \]
11. Step-by-step derivation

    1. --rgt-identity 99.3%

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

    2. --rgt-identity 99.3%

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

    3. remove-double-div 99.3%

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

    4. associate-/r/ 99.3%

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

    5. unpow2 99.3%

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

    6. associate-*r/ 99.3%

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

    7. /-rgt-identity 99.3%

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

    8. unpow2 99.3%

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

    9. *-commutative 99.3%

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

    10. *-commutative 99.3%

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

    11. associate-*l* 99.3%

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

    12. associate-/r* 99.3%

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

    13. metadata-eval 99.3%

        \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(\left({\left(u2 \cdot \left(\pi \cdot 2\right)\right)}^{2} \cdot \frac{\color{blue}{0.5}}{\pi \cdot u2} + 1\right) - 1\right) \]

    14. *-commutative 99.3%

        \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(\left({\left(u2 \cdot \left(\pi \cdot 2\right)\right)}^{2} \cdot \frac{0.5}{\color{blue}{u2 \cdot \pi}} + 1\right) - 1\right) \]

12. Simplified 99.3%

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(\left(\color{blue}{{\left(u2 \cdot \left(\pi \cdot 2\right)\right)}^{2} \cdot \frac{0.5}{u2 \cdot \pi}} + 1\right) - 1\right) \]
13. Step-by-step derivation

    1. unpow2 99.3%

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(\left(\color{blue}{\left(\left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right)} \cdot \frac{0.5}{u2 \cdot \pi} + 1\right) - 1\right) \]

14. Applied egg-rr 99.3%

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(\left(\color{blue}{\left(\left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right)} \cdot \frac{0.5}{u2 \cdot \pi} + 1\right) - 1\right) \]

15. Final simplification 99.3%

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(\left(\left(\left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \left(u2 \cdot \left(2 \cdot \pi\right)\right)\right) \cdot \frac{0.5}{u2 \cdot \pi} + 1\right) + -1\right) \]
16. Add Preprocessing

Alternative 3: 90.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\\ \mathbf{if}\;t_0 \leq 0.9999499917030334:\\ \;\;\;\;t_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 (* 2.0 PI)))))
   (if (<= t_0 0.9999499917030334)
     (* t_0 (sqrt u1))
     (sqrt (- (log1p (- u1)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((u2 * (2.0f * ((float) M_PI))));
	float tmp;
	if (t_0 <= 0.9999499917030334f) {
		tmp = t_0 * sqrtf(u1);
	} else {
		tmp = sqrtf(-log1pf(-u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(u2 * Float32(Float32(2.0) * Float32(pi))))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999499917030334))
		tmp = Float32(t_0 * sqrt(u1));
	else
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 2 (PI.f32)) u2)) < 0.999949992

   1. Initial program 59.7%

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

      1. add-cube-cbrt 59.6%

         \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\sqrt{-\log \left(1 - u1\right)}} \cdot \sqrt[3]{\sqrt{-\log \left(1 - u1\right)}}\right) \cdot \sqrt[3]{\sqrt{-\log \left(1 - u1\right)}}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. pow3 59.6%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{-\log \left(1 - u1\right)}}\right)}^{3}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. add-sqr-sqrt 59.6%

         \[\leadsto {\left(\sqrt[3]{\sqrt{\color{blue}{\sqrt{-\log \left(1 - u1\right)} \cdot \sqrt{-\log \left(1 - u1\right)}}}}\right)}^{3} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. sqrt-unprod 59.6%

         \[\leadsto {\left(\sqrt[3]{\sqrt{\color{blue}{\sqrt{\left(-\log \left(1 - u1\right)\right) \cdot \left(-\log \left(1 - u1\right)\right)}}}}\right)}^{3} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. sqr-neg 59.6%

         \[\leadsto {\left(\sqrt[3]{\sqrt{\sqrt{\color{blue}{\log \left(1 - u1\right) \cdot \log \left(1 - u1\right)}}}}\right)}^{3} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. sqrt-unprod 1.3%

         \[\leadsto {\left(\sqrt[3]{\sqrt{\color{blue}{\sqrt{\log \left(1 - u1\right)} \cdot \sqrt{\log \left(1 - u1\right)}}}}\right)}^{3} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      7. add-sqr-sqrt 1.3%

         \[\leadsto {\left(\sqrt[3]{\sqrt{\color{blue}{\log \left(1 - u1\right)}}}\right)}^{3} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      8. sub-neg 1.3%

         \[\leadsto {\left(\sqrt[3]{\sqrt{\log \color{blue}{\left(1 + \left(-u1\right)\right)}}}\right)}^{3} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      9. log1p-udef -0.0%

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

      10. add-sqr-sqrt -0.0%

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

      11. sqrt-unprod 74.0%

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

      12. sqr-neg 74.0%

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

      13. sqrt-unprod 74.0%

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

      14. add-sqr-sqrt 74.0%

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

   4. Applied egg-rr 74.0%

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

   5. Taylor expanded in u1 around 0 76.3%

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

   if 0.999949992 < (cos.f32 (*.f32 (*.f32 2 (PI.f32)) u2))

   1. Initial program 58.3%

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

      1. sub-neg 58.3%

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

      2. log1p-def 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in u2 around 0 96.5%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \leq 0.9999499917030334:\\ \;\;\;\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.7%

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

   1. sub-neg 58.7%

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

   2. log1p-def 99.3%

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

3. Simplified 99.3%

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

5. Final simplification 99.3%

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

Alternative 5: 94.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(2 \cdot \pi\right)\\ \mathbf{if}\;t_0 \leq 0.0013800000306218863:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos t_0 \cdot \sqrt{u1 \cdot \left(\left(--1\right) - u1 \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* 2.0 PI))))
   (if (<= t_0 0.0013800000306218863)
     (sqrt (- (log1p (- u1))))
     (* (cos t_0) (sqrt (* u1 (- (- -1.0) (* u1 -0.5))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u2 * (2.0f * ((float) M_PI));
	float tmp;
	if (t_0 <= 0.0013800000306218863f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = cosf(t_0) * sqrtf((u1 * (-(-1.0f) - (u1 * -0.5f))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u2 * Float32(Float32(2.0) * Float32(pi)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0013800000306218863))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(cos(t_0) * sqrt(Float32(u1 * Float32(Float32(-Float32(-1.0)) - Float32(u1 * Float32(-0.5))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 2 (PI.f32)) u2) < 0.00138000003

   1. Initial program 57.6%

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

      1. sub-neg 57.6%

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

      2. log1p-def 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in u2 around 0 99.0%

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

   if 0.00138000003 < (*.f32 (*.f32 2 (PI.f32)) u2)

   1. Initial program 60.4%

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

   3. Taylor expanded in u1 around 0 88.3%

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

      1. unpow2 88.3%

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

      2. associate-*r* 88.3%

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

      3. distribute-rgt-out 88.2%

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

   5. Simplified 88.2%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.0013800000306218863:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{u1 \cdot \left(\left(--1\right) - u1 \cdot -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (sqrt (- (log1p (- u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1));
}

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(-log1p(Float32(-u1))))
end

Derivation

1. Initial program 58.7%

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

   1. sub-neg 58.7%

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

   2. log1p-def 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in u2 around 0 81.5%

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

6. Final simplification 81.5%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \]
7. Add Preprocessing

Alternative 7: 65.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1} \end{array} \]

FPCore

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

C

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

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 = sqrt(u1)
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(u1)
end

MATLAB

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

Derivation

1. Initial program 58.7%

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

   1. add-sqr-sqrt 55.9%

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

   2. sqrt-unprod 56.2%

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

   3. swap-sqr 56.2%

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

4. Applied egg-rr 71.0%

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

   1. associate-*r* 71.0%

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

   2. *-commutative 71.0%

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

   3. associate-*l* 71.0%

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

6. Simplified 71.0%

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

7. Taylor expanded in u2 around 0 37.0%

   \[\leadsto \sqrt{\color{blue}{\log \left(1 + u1\right)}} \]
8. Step-by-step derivation

   1. log1p-def 64.1%

      \[\leadsto \sqrt{\color{blue}{\mathsf{log1p}\left(u1\right)}} \]

9. Simplified 64.1%

   \[\leadsto \sqrt{\color{blue}{\mathsf{log1p}\left(u1\right)}} \]

10. Taylor expanded in u1 around 0 65.6%

    \[\leadsto \sqrt{\color{blue}{u1}} \]

11. Final simplification 65.6%

    \[\leadsto \sqrt{u1} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024018 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_x"
  :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)))) (cos (* (* 2.0 PI) u2))))