Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.4% → 99.0%

Time: 15.3s

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: 57.4% 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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\pi \cdot u2\right)\\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) + \mathsf{fma}\left(-t\_0, t\_0, t\_0 \cdot \sin \left(u2 \cdot \left(\left(\pi + 1\right) + -1\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.4%

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

   1. sub-neg 58.4%

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

   2. log1p-define 99.1%

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

3. Simplified 99.1%

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

   1. associate-*l* 99.1%

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

   2. cos-2 98.9%

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

   3. prod-diff 98.9%

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

   4. fma-neg 98.9%

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

   5. cos-2 99.1%

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

6. Applied egg-rr 99.1%

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

   1. associate-*r* 99.1%

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

   2. *-commutative 99.1%

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

   3. *-commutative 99.1%

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

   4. *-commutative 99.1%

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

   5. *-commutative 99.1%

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

8. Simplified 99.1%

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

   1. expm1-log1p-u 99.1%

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

10. Applied egg-rr 99.1%

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

    1. expm1-undefine 99.1%

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

    2. log1p-undefine 99.1%

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

    3. rem-exp-log 99.1%

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

    4. +-commutative 99.1%

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

12. Applied egg-rr 99.1%

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

13. Final simplification 99.1%

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) + \mathsf{fma}\left(-\sin \left(\pi \cdot u2\right), \sin \left(\pi \cdot u2\right), \sin \left(\pi \cdot u2\right) \cdot \sin \left(u2 \cdot \left(\left(\pi + 1\right) + -1\right)\right)\right)\right) \]
14. Add Preprocessing

Alternative 2: 99.0% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* PI u2))))
   (*
    (sqrt (- (log1p (- u1))))
    (+ (cos (* u2 (* 2.0 PI))) (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((u2 * (2.0f * ((float) M_PI)))) + 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(u2 * Float32(Float32(2.0) * Float32(pi)))) + fma(Float32(-t_0), t_0, Float32(t_0 * t_0))))
end

Derivation

1. Initial program 58.4%

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

   1. sub-neg 58.4%

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

   2. log1p-define 99.1%

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

3. Simplified 99.1%

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

   1. associate-*l* 99.1%

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

   2. cos-2 98.9%

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

   3. prod-diff 98.9%

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

   4. fma-neg 98.9%

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

   5. cos-2 99.1%

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

6. Applied egg-rr 99.1%

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

   1. associate-*r* 99.1%

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

   2. *-commutative 99.1%

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

   3. *-commutative 99.1%

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

   4. *-commutative 99.1%

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

   5. *-commutative 99.1%

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

8. Simplified 99.1%

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

9. Final simplification 99.1%

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

Alternative 3: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\pi \cdot u2\right)\\ t_1 := \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(t\_1 + \mathsf{fma}\left(-t\_0, t\_0, 0.5 - t\_1 \cdot 0.5\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* PI u2))) (t_1 (cos (* u2 (* 2.0 PI)))))
   (*
    (sqrt (- (log1p (- u1))))
    (+ t_1 (fma (- t_0) t_0 (- 0.5 (* t_1 0.5)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sinf((((float) M_PI) * u2));
	float t_1 = cosf((u2 * (2.0f * ((float) M_PI))));
	return sqrtf(-log1pf(-u1)) * (t_1 + fmaf(-t_0, t_0, (0.5f - (t_1 * 0.5f))));
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sin(Float32(Float32(pi) * u2))
	t_1 = cos(Float32(u2 * Float32(Float32(2.0) * Float32(pi))))
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(t_1 + fma(Float32(-t_0), t_0, Float32(Float32(0.5) - Float32(t_1 * Float32(0.5))))))
end

Derivation

1. Initial program 58.4%

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

   1. sub-neg 58.4%

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

   2. log1p-define 99.1%

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

3. Simplified 99.1%

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

   1. associate-*l* 99.1%

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

   2. cos-2 98.9%

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

   3. prod-diff 98.9%

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

   4. fma-neg 98.9%

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

   5. cos-2 99.1%

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

6. Applied egg-rr 99.1%

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

   1. associate-*r* 99.1%

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

   2. *-commutative 99.1%

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

   3. *-commutative 99.1%

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

   4. *-commutative 99.1%

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

   5. *-commutative 99.1%

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

8. Simplified 99.1%

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

   1. expm1-log1p-u 99.1%

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

10. Applied egg-rr 99.1%

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

    1. expm1-undefine 99.1%

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

    2. log1p-undefine 99.1%

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

    3. rem-exp-log 99.1%

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

    4. +-commutative 99.1%

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

12. Applied egg-rr 99.1%

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

    1. add-sqr-sqrt 99.1%

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

    2. add-sqr-sqrt 99.1%

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

    3. associate--l+ 99.1%

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

    4. metadata-eval 99.1%

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

    5. +-rgt-identity 99.1%

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

    6. add-sqr-sqrt 99.1%

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

    7. add-sqr-sqrt 99.1%

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

    8. *-commutative 99.1%

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

    9. *-commutative 99.1%

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

    10. sqr-sin-a 99.1%

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

    11. associate-*r* 99.1%

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

    12. *-commutative 99.1%

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

    13. *-commutative 99.1%

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

14. Applied egg-rr 99.1%

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

15. Final simplification 99.1%

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

Alternative 4: 96.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\\ \mathbf{if}\;t\_0 \leq 0.9999995827674866:\\ \;\;\;\;t\_0 \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 - u1 \cdot -0.25\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 (* 2.0 PI)))))
   (if (<= t_0 0.9999995827674866)
     (*
      t_0
      (sqrt
       (*
        u1
        (+ 1.0 (* u1 (+ 0.5 (* u1 (- 0.3333333333333333 (* u1 -0.25)))))))))
     (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 <= 0.9999995827674866f) {
		tmp = t_0 * sqrtf((u1 * (1.0f + (u1 * (0.5f + (u1 * (0.3333333333333333f - (u1 * -0.25f))))))));
	} 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(0.9999995827674866))
		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(-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)) < 0.999999583

   1. Initial program 56.7%

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

   3. Taylor expanded in u1 around 0 93.1%

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

   if 0.999999583 < (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   1. Initial program 59.8%

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

      1. sub-neg 59.8%

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

      2. log1p-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in u2 around 0 99.6%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \leq 0.9999995827674866:\\ \;\;\;\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 - u1 \cdot -0.25\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\\ \mathbf{if}\;t\_0 \leq 0.9999986886978149:\\ \;\;\;\;t\_0 \cdot \sqrt{u1 \cdot \left(1 - u1 \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 (* 2.0 PI)))))
   (if (<= t_0 0.9999986886978149)
     (* t_0 (sqrt (* u1 (- 1.0 (* u1 -0.5)))))
     (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 <= 0.9999986886978149f) {
		tmp = t_0 * sqrtf((u1 * (1.0f - (u1 * -0.5f))));
	} 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(0.9999986886978149))
		tmp = Float32(t_0 * sqrt(Float32(u1 * Float32(Float32(1.0) - Float32(u1 * Float32(-0.5))))));
	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)) < 0.999998689

   1. Initial program 57.3%

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

   3. Taylor expanded in u1 around 0 88.4%

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

   if 0.999998689 < (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   1. Initial program 59.1%

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

      1. sub-neg 59.1%

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

      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 98.9%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \leq 0.9999986886978149:\\ \;\;\;\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{u1 \cdot \left(1 - u1 \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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 58.4%

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

   1. sub-neg 58.4%

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

   2. log1p-define 99.1%

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

3. Simplified 99.1%

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

5. Final simplification 99.1%

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

Alternative 7: 96.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(2 \cdot \pi\right)\\ \mathbf{if}\;t\_0 \leq 0.0013650000328198075:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos t\_0 \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 - u1 \cdot -0.3333333333333333\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* 2.0 PI))))
   (if (<= t_0 0.0013650000328198075)
     (sqrt (- (log1p (- 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 = u2 * (2.0f * ((float) M_PI));
	float tmp;
	if (t_0 <= 0.0013650000328198075f) {
		tmp = sqrtf(-log1pf(-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(u2 * Float32(Float32(2.0) * Float32(pi)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0013650000328198075))
		tmp = sqrt(Float32(-log1p(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) < 0.00136500003

   1. Initial program 59.2%

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

      1. sub-neg 59.2%

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

      2. log1p-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in u2 around 0 99.2%

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

   if 0.00136500003 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.3%

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

   3. Taylor expanded in u1 around 0 91.5%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.0013650000328198075:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 - u1 \cdot -0.3333333333333333\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.011500000022351742:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 (* 2.0 PI)) 0.011500000022351742)
   (sqrt (- (log1p (- u1))))
   (* (sqrt u1) (cos (* 2.0 (* PI u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * (2.0f * ((float) M_PI))) <= 0.011500000022351742f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = sqrtf(u1) * cosf((2.0f * (((float) M_PI) * u2)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(Float32(2.0) * Float32(pi))) <= Float32(0.011500000022351742))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(sqrt(u1) * cos(Float32(Float32(2.0) * Float32(Float32(pi) * u2))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.0115

   1. Initial program 59.4%

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

      1. sub-neg 59.4%

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

      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 95.9%

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

   if 0.0115 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 56.3%

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

   3. Taylor expanded in u1 around 0 77.6%

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

      1. mul-1-neg 77.6%

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

   5. Simplified 77.6%

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

   6. Taylor expanded in u1 around 0 77.6%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.011500000022351742:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.1% 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 58.4%

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

   1. sub-neg 58.4%

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

   2. log1p-define 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in u2 around 0 77.9%

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

6. Final simplification 77.9%

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

Alternative 10: 76.5% 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 58.4%

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

   1. sub-neg 58.4%

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

   2. log1p-define 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in u2 around 0 77.9%

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

6. Taylor expanded in u1 around 0 74.0%

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

7. Final simplification 74.0%

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

Alternative 11: 75.3% 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 58.4%

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

   1. sub-neg 58.4%

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

   2. log1p-define 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in u2 around 0 77.9%

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

6. Taylor expanded in u1 around 0 72.7%

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

7. Final simplification 72.7%

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

Alternative 12: 72.8% 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 58.4%

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

   1. sub-neg 58.4%

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

   2. log1p-define 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in u2 around 0 77.9%

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

6. Taylor expanded in u1 around 0 70.2%

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

7. Final simplification 70.2%

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

Alternative 13: 64.9% 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 58.4%

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

3. Taylor expanded in u1 around 0 75.8%

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

   1. mul-1-neg 75.8%

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

5. Simplified 75.8%

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

6. Taylor expanded in u2 around 0 62.9%

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

7. Final simplification 62.9%

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

Reproduce

herbie shell --seed 2024067 
(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))))