Beckmann Sample, near normal, slope_x

Percentage Accurate: 56.9% → 99.1%

Time: 12.6s

Alternatives: 13

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: 56.9% 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: 99.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\pi \cdot u2\right)\\ t_1 := \mathsf{fma}\left(-t\_0, t\_0, t\_0 \cdot t\_0\right)\\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(t\_1 + \left(\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + t\_1\right)\right) \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.3%

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

   1. sub-neg 57.3%

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

   2. log1p-define 98.9%

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

3. Simplified 98.9%

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

   1. associate-*l* 98.9%

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

   2. cos-2 98.7%

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

   3. prod-diff 98.7%

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

   4. fmm-def 98.7%

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

   5. cos-2 98.9%

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

   6. associate-*l* 98.9%

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

   7. *-commutative 98.9%

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

   8. associate-*l* 98.9%

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

6. Applied egg-rr 98.9%

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

   1. associate-*r* 98.9%

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

   2. *-commutative 98.9%

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

   3. *-commutative 98.9%

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

   4. *-commutative 98.9%

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

   5. *-commutative 98.9%

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

   6. *-commutative 98.9%

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

8. Simplified 98.9%

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

   1. associate-*l* 98.9%

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

   2. cos-2 98.7%

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

   3. prod-diff 98.7%

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

   4. fmm-def 98.7%

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

   5. cos-2 98.9%

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

   6. associate-*l* 98.9%

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

   7. *-commutative 98.9%

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

   8. associate-*l* 98.9%

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

10. Applied egg-rr 99.0%

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

    1. associate-*r* 98.9%

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

    2. *-commutative 98.9%

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

    3. *-commutative 98.9%

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

    4. *-commutative 98.9%

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

    5. *-commutative 98.9%

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

    6. *-commutative 98.9%

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

12. Simplified 99.0%

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

13. Final simplification 99.0%

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

Alternative 2: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\pi \cdot u2\right)\\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + \mathsf{fma}\left(-t\_0, t\_0, t\_0 \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.3%

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

   1. sub-neg 57.3%

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

   2. log1p-define 98.9%

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

3. Simplified 98.9%

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

   1. associate-*l* 98.9%

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

   2. cos-2 98.7%

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

   3. prod-diff 98.7%

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

   4. fmm-def 98.7%

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

   5. cos-2 98.9%

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

   6. associate-*l* 98.9%

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

   7. *-commutative 98.9%

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

   8. associate-*l* 98.9%

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

6. Applied egg-rr 98.9%

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

   1. associate-*r* 98.9%

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

   2. *-commutative 98.9%

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

   3. *-commutative 98.9%

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

   4. *-commutative 98.9%

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

   5. *-commutative 98.9%

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

   6. *-commutative 98.9%

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

8. Simplified 98.9%

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

9. Final simplification 98.9%

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

Alternative 3: 96.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq 0.9999998211860657:\\ \;\;\;\;t\_0 \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 + u1 \cdot 0.25\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(\frac{u1 \cdot u1}{-u1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 PI) u2))))
   (if (<= t_0 0.9999998211860657)
     (*
      t_0
      (sqrt
       (*
        u1
        (+ 1.0 (* u1 (+ 0.5 (* u1 (+ 0.3333333333333333 (* u1 0.25)))))))))
     (sqrt (- (log1p (/ (* u1 u1) (- u1))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf(((2.0f * ((float) M_PI)) * u2));
	float tmp;
	if (t_0 <= 0.9999998211860657f) {
		tmp = t_0 * sqrtf((u1 * (1.0f + (u1 * (0.5f + (u1 * (0.3333333333333333f + (u1 * 0.25f))))))));
	} else {
		tmp = sqrtf(-log1pf(((u1 * u1) / -u1)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999998211860657))
		tmp = Float32(t_0 * sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(u1 * Float32(Float32(0.5) + Float32(u1 * Float32(Float32(0.3333333333333333) + Float32(u1 * Float32(0.25))))))))));
	else
		tmp = sqrt(Float32(-log1p(Float32(Float32(u1 * u1) / Float32(-u1)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)) < 0.999999821

   1. Initial program 57.6%

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

   3. Taylor expanded in u1 around 0 92.0%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 + 0.25 \cdot u1\right)\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   4. Step-by-step derivation

      1. *-commutative 92.0%

         \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 + \color{blue}{u1 \cdot 0.25}\right)\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Simplified 92.0%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 + u1 \cdot 0.25\right)\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   if 0.999999821 < (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   1. Initial program 57.1%

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

      1. sub-neg 57.1%

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

      2. log1p-define 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 99.6%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. neg-sub0 99.6%

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

      2. flip-- 99.6%

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

      3. metadata-eval 99.6%

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

      4. pow2 99.6%

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

      5. add-sqr-sqrt 98.8%

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

      6. sqrt-unprod 99.6%

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

      7. sqr-neg 99.6%

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

      8. sqrt-unprod -0.0%

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

      9. add-sqr-sqrt -0.0%

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

      10. sub-neg -0.0%

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

      11. neg-sub0 -0.0%

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

      12. add-sqr-sqrt -0.0%

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

      13. sqrt-unprod 99.6%

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

      14. sqr-neg 99.6%

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

      15. sqrt-unprod 98.8%

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

      16. add-sqr-sqrt 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. unpow2 99.6%

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

   9. Applied egg-rr 99.6%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(\frac{0 - \color{blue}{u1 \cdot u1}}{u1}\right)} \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.9999998211860657:\\ \;\;\;\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 + u1 \cdot 0.25\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(\frac{u1 \cdot u1}{-u1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq 0.9999499917030334:\\ \;\;\;\;t\_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(\frac{u1 \cdot u1}{-u1}\right)}\\ \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)) < 0.999949992

   1. Initial program 58.0%

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

   3. Taylor expanded in u1 around 0 75.6%

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

   if 0.999949992 < (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   1. Initial program 57.0%

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

      1. sub-neg 57.0%

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

      2. log1p-define 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in u2 around 0 96.6%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. neg-sub0 96.6%

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

      2. flip-- 96.6%

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

      3. metadata-eval 96.6%

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

      4. pow2 96.6%

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

      5. add-sqr-sqrt 96.0%

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

      6. sqrt-unprod 96.6%

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

      7. sqr-neg 96.6%

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

      8. sqrt-unprod -0.0%

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

      9. add-sqr-sqrt -0.0%

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

      10. sub-neg -0.0%

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

      11. neg-sub0 -0.0%

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

      12. add-sqr-sqrt -0.0%

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

      13. sqrt-unprod 96.6%

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

      14. sqr-neg 96.6%

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

      15. sqrt-unprod 96.0%

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

      16. add-sqr-sqrt 96.6%

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

   7. Applied egg-rr 96.6%

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

      1. unpow2 96.6%

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

   9. Applied egg-rr 96.6%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(\frac{0 - \color{blue}{u1 \cdot u1}}{u1}\right)} \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.0%

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

Alternative 5: 90.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq 0.9999499917030334:\\ \;\;\;\;t\_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 PI) u2))))
   (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(((2.0f * ((float) M_PI)) * u2));
	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(Float32(Float32(2.0) * Float32(pi)) * u2))
	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 #s(literal 2 binary32) (PI.f32)) u2)) < 0.999949992

   1. Initial program 58.0%

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

   3. Taylor expanded in u1 around 0 75.6%

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

   if 0.999949992 < (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   1. Initial program 57.0%

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

      1. sub-neg 57.0%

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

      2. log1p-define 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in u2 around 0 96.6%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. neg-sub0 96.6%

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

      2. flip-- 96.6%

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

      3. metadata-eval 96.6%

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

      4. pow2 96.6%

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

      5. add-sqr-sqrt 96.0%

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

      6. sqrt-unprod 96.6%

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

      7. sqr-neg 96.6%

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

      8. sqrt-unprod -0.0%

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

      9. add-sqr-sqrt -0.0%

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

      10. sub-neg -0.0%

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

      11. neg-sub0 -0.0%

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

      12. add-sqr-sqrt -0.0%

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

      13. sqrt-unprod 96.6%

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

      14. sqr-neg 96.6%

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

      15. sqrt-unprod 96.0%

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

      16. add-sqr-sqrt 96.6%

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

   7. Applied egg-rr 96.6%

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

      1. *-rgt-identity 96.6%

         \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(\frac{0 - {u1}^{2}}{u1}\right)}} \]

      2. pow1/2 96.6%

         \[\leadsto \color{blue}{{\left(-\mathsf{log1p}\left(\frac{0 - {u1}^{2}}{u1}\right)\right)}^{0.5}} \]

      3. div-sub 96.6%

         \[\leadsto {\left(-\mathsf{log1p}\left(\color{blue}{\frac{0}{u1} - \frac{{u1}^{2}}{u1}}\right)\right)}^{0.5} \]

      4. pow1 96.6%

         \[\leadsto {\left(-\mathsf{log1p}\left(\frac{0}{u1} - \frac{{u1}^{2}}{\color{blue}{{u1}^{1}}}\right)\right)}^{0.5} \]

      5. pow-div 96.6%

         \[\leadsto {\left(-\mathsf{log1p}\left(\frac{0}{u1} - \color{blue}{{u1}^{\left(2 - 1\right)}}\right)\right)}^{0.5} \]

      6. metadata-eval 96.6%

         \[\leadsto {\left(-\mathsf{log1p}\left(\frac{0}{u1} - {u1}^{\color{blue}{1}}\right)\right)}^{0.5} \]

      7. pow1 96.6%

         \[\leadsto {\left(-\mathsf{log1p}\left(\frac{0}{u1} - \color{blue}{u1}\right)\right)}^{0.5} \]

   9. Applied egg-rr 96.6%

      \[\leadsto \color{blue}{{\left(-\mathsf{log1p}\left(\frac{0}{u1} - u1\right)\right)}^{0.5}} \]
   10. Step-by-step derivation

       1. unpow1/2 96.6%

          \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(\frac{0}{u1} - u1\right)}} \]

       2. log1p-define 56.0%

          \[\leadsto \sqrt{-\color{blue}{\log \left(1 + \left(\frac{0}{u1} - u1\right)\right)}} \]

       3. div0 56.0%

          \[\leadsto \sqrt{-\log \left(1 + \left(\color{blue}{0} - u1\right)\right)} \]

       4. neg-sub0 56.0%

          \[\leadsto \sqrt{-\log \left(1 + \color{blue}{\left(-u1\right)}\right)} \]

       5. log1p-undefine 96.6%

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

   11. Simplified 96.6%

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

4. Final simplification 89.9%

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

Alternative 6: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.3%

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

   1. sub-neg 57.3%

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

   2. log1p-define 98.9%

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

3. Simplified 98.9%

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

Alternative 7: 96.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.0006000000284984708:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(\frac{u1 \cdot u1}{-u1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos t\_0 \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.0006000000284984708)
     (sqrt (- (log1p (/ (* u1 u1) (- u1)))))
     (*
      (cos t_0)
      (sqrt (* u1 (+ 1.0 (* u1 (+ 0.5 (* u1 0.3333333333333333))))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.0006000000284984708f) {
		tmp = sqrtf(-log1pf(((u1 * u1) / -u1)));
	} else {
		tmp = cosf(t_0) * sqrtf((u1 * (1.0f + (u1 * (0.5f + (u1 * 0.3333333333333333f))))));
	}
	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(0.0006000000284984708))
		tmp = sqrt(Float32(-log1p(Float32(Float32(u1 * u1) / Float32(-u1)))));
	else
		tmp = Float32(cos(t_0) * sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(u1 * Float32(Float32(0.5) + Float32(u1 * Float32(0.3333333333333333))))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 6.00000028e-4

   1. Initial program 57.1%

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

      1. sub-neg 57.1%

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

      2. log1p-define 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 99.6%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. neg-sub0 99.6%

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

      2. flip-- 99.6%

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

      3. metadata-eval 99.6%

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

      4. pow2 99.6%

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

      5. add-sqr-sqrt 98.8%

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

      6. sqrt-unprod 99.6%

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

      7. sqr-neg 99.6%

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

      8. sqrt-unprod -0.0%

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

      9. add-sqr-sqrt -0.0%

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

      10. sub-neg -0.0%

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

      11. neg-sub0 -0.0%

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

      12. add-sqr-sqrt -0.0%

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

      13. sqrt-unprod 99.6%

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

      14. sqr-neg 99.6%

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

      15. sqrt-unprod 98.8%

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

      16. add-sqr-sqrt 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. unpow2 99.6%

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

   9. Applied egg-rr 99.6%

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

   if 6.00000028e-4 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.6%

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

   3. Taylor expanded in u1 around 0 90.1%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(0.5 + 0.3333333333333333 \cdot u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   4. Step-by-step derivation

      1. *-commutative 90.1%

         \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + \color{blue}{u1 \cdot 0.3333333333333333}\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Simplified 90.1%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.0006000000284984708:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(\frac{u1 \cdot u1}{-u1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.0012000000569969416:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(\frac{u1 \cdot u1}{-u1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos t\_0 \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.0012000000569969416)
     (sqrt (- (log1p (/ (* u1 u1) (- u1)))))
     (* (cos t_0) (sqrt (* u1 (+ 1.0 (* u1 0.5))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.0012000000569969416f) {
		tmp = sqrtf(-log1pf(((u1 * u1) / -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(Float32(Float32(2.0) * Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0012000000569969416))
		tmp = sqrt(Float32(-log1p(Float32(Float32(u1 * u1) / Float32(-u1)))));
	else
		tmp = Float32(cos(t_0) * sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(u1 * Float32(0.5))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00120000006

   1. Initial program 56.4%

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

      1. sub-neg 56.4%

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

      2. log1p-define 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 99.1%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. neg-sub0 99.1%

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

      2. flip-- 99.2%

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

      3. metadata-eval 99.2%

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

      4. pow2 99.1%

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

      5. add-sqr-sqrt 98.5%

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

      6. sqrt-unprod 99.1%

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

      7. sqr-neg 99.1%

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

      8. sqrt-unprod -0.0%

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

      9. add-sqr-sqrt -0.0%

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

      10. sub-neg -0.0%

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

      11. neg-sub0 -0.0%

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

      12. add-sqr-sqrt -0.0%

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

      13. sqrt-unprod 99.1%

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

      14. sqr-neg 99.1%

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

      15. sqrt-unprod 98.5%

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

      16. add-sqr-sqrt 99.1%

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

   7. Applied egg-rr 99.1%

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

      1. unpow2 99.2%

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

   9. Applied egg-rr 99.2%

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

   if 0.00120000006 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 58.6%

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

   3. Taylor expanded in u1 around 0 86.1%

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

      1. *-commutative 86.1%

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

   5. Simplified 86.1%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.0012000000569969416:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(\frac{u1 \cdot u1}{-u1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.0% 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 57.3%

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

   1. sub-neg 57.3%

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

   2. log1p-define 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in u2 around 0 79.1%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]
6. Step-by-step derivation

   1. neg-sub0 79.1%

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

   2. flip-- 79.1%

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

   3. metadata-eval 79.1%

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

   4. pow2 79.1%

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

   5. add-sqr-sqrt 78.7%

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

   6. sqrt-unprod 79.1%

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

   7. sqr-neg 79.1%

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

   8. sqrt-unprod -0.0%

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

   9. add-sqr-sqrt -0.0%

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

   10. sub-neg -0.0%

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

   11. neg-sub0 -0.0%

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

   12. add-sqr-sqrt -0.0%

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

   13. sqrt-unprod 79.1%

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

   14. sqr-neg 79.1%

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

   15. sqrt-unprod 78.7%

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

   16. add-sqr-sqrt 79.1%

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

7. Applied egg-rr 79.1%

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

   1. *-rgt-identity 79.1%

      \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(\frac{0 - {u1}^{2}}{u1}\right)}} \]

   2. pow1/2 79.1%

      \[\leadsto \color{blue}{{\left(-\mathsf{log1p}\left(\frac{0 - {u1}^{2}}{u1}\right)\right)}^{0.5}} \]

   3. div-sub 79.1%

      \[\leadsto {\left(-\mathsf{log1p}\left(\color{blue}{\frac{0}{u1} - \frac{{u1}^{2}}{u1}}\right)\right)}^{0.5} \]

   4. pow1 79.1%

      \[\leadsto {\left(-\mathsf{log1p}\left(\frac{0}{u1} - \frac{{u1}^{2}}{\color{blue}{{u1}^{1}}}\right)\right)}^{0.5} \]

   5. pow-div 79.1%

      \[\leadsto {\left(-\mathsf{log1p}\left(\frac{0}{u1} - \color{blue}{{u1}^{\left(2 - 1\right)}}\right)\right)}^{0.5} \]

   6. metadata-eval 79.1%

      \[\leadsto {\left(-\mathsf{log1p}\left(\frac{0}{u1} - {u1}^{\color{blue}{1}}\right)\right)}^{0.5} \]

   7. pow1 79.1%

      \[\leadsto {\left(-\mathsf{log1p}\left(\frac{0}{u1} - \color{blue}{u1}\right)\right)}^{0.5} \]

9. Applied egg-rr 79.1%

   \[\leadsto \color{blue}{{\left(-\mathsf{log1p}\left(\frac{0}{u1} - u1\right)\right)}^{0.5}} \]
10. Step-by-step derivation

    1. unpow1/2 79.1%

       \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(\frac{0}{u1} - u1\right)}} \]

    2. log1p-define 48.1%

       \[\leadsto \sqrt{-\color{blue}{\log \left(1 + \left(\frac{0}{u1} - u1\right)\right)}} \]

    3. div0 48.1%

       \[\leadsto \sqrt{-\log \left(1 + \left(\color{blue}{0} - u1\right)\right)} \]

    4. neg-sub0 48.1%

       \[\leadsto \sqrt{-\log \left(1 + \color{blue}{\left(-u1\right)}\right)} \]

    5. log1p-undefine 79.1%

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

11. Simplified 79.1%

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

Alternative 10: 76.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 + u1 \cdot 0.25\right)\right)\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (sqrt
  (* u1 (+ 1.0 (* u1 (+ 0.5 (* u1 (+ 0.3333333333333333 (* u1 0.25)))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 * (1.0f + (u1 * (0.5f + (u1 * (0.3333333333333333f + (u1 * 0.25f))))))));
}

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 * (1.0e0 + (u1 * (0.5e0 + (u1 * (0.3333333333333333e0 + (u1 * 0.25e0))))))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(u1 * Float32(Float32(0.5) + Float32(u1 * Float32(Float32(0.3333333333333333) + Float32(u1 * Float32(0.25)))))))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 * (single(1.0) + (u1 * (single(0.5) + (u1 * (single(0.3333333333333333) + (u1 * single(0.25)))))))));
end

Derivation

1. Initial program 57.3%

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

   1. sub-neg 57.3%

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

   2. log1p-define 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in u2 around 0 79.1%

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

6. Taylor expanded in u1 around 0 75.0%

   \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 + 0.25 \cdot u1\right)\right)\right)}} \cdot 1 \]

7. Final simplification 75.0%

   \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 + u1 \cdot 0.25\right)\right)\right)} \]
8. Add Preprocessing

Alternative 11: 75.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right)\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (sqrt (* u1 (+ 1.0 (* u1 (+ 0.5 (* u1 0.3333333333333333)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 * (1.0f + (u1 * (0.5f + (u1 * 0.3333333333333333f))))));
}

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 * (1.0e0 + (u1 * (0.5e0 + (u1 * 0.3333333333333333e0))))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(u1 * Float32(Float32(0.5) + Float32(u1 * Float32(0.3333333333333333)))))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 * (single(1.0) + (u1 * (single(0.5) + (u1 * single(0.3333333333333333)))))));
end

Derivation

1. Initial program 57.3%

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

   1. sub-neg 57.3%

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

   2. log1p-define 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in u2 around 0 79.1%

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

6. Taylor expanded in u1 around 0 73.8%

   \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(0.5 + 0.3333333333333333 \cdot u1\right)\right)}} \cdot 1 \]

7. Final simplification 73.8%

   \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot 0.3333333333333333\right)\right)} \]
8. Add Preprocessing

Alternative 12: 72.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1 \cdot \left(1 + u1 \cdot 0.5\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (sqrt (* u1 (+ 1.0 (* u1 0.5)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 * (1.0f + (u1 * 0.5f))));
}

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 * (1.0e0 + (u1 * 0.5e0))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(u1 * Float32(0.5)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 * (single(1.0) + (u1 * single(0.5)))));
end

Derivation

1. Initial program 57.3%

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

   1. sub-neg 57.3%

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

   2. log1p-define 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in u2 around 0 79.1%

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

6. Taylor expanded in u1 around 0 71.5%

   \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + 0.5 \cdot u1\right)}} \cdot 1 \]

7. Final simplification 71.5%

   \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot 0.5\right)} \]
8. Add Preprocessing

Alternative 13: 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 57.3%

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

   1. sub-neg 57.3%

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

   2. log1p-define 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in u2 around 0 79.1%

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

6. Taylor expanded in u1 around 0 63.9%

   \[\leadsto \sqrt{\color{blue}{u1}} \cdot 1 \]

7. Taylor expanded in u1 around 0 63.9%

   \[\leadsto \color{blue}{\sqrt{u1}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024163 
(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))))