Beckmann Sample, near normal, slope_x

Percentage Accurate: 58.0% → 99.0%

Time: 15.3s

Alternatives: 6

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: 58.0% 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

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* PI u2))) (t_1 (cos (* PI (* 2.0 u2)))))
   (*
    (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((((float) M_PI) * (2.0f * u2)));
	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(Float32(pi) * Float32(Float32(2.0) * u2)))
	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 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 99.0%

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

3. Simplified 99.0%

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

   1. associate-*l* 99.0%

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

   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 99.0%

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

   6. associate-*l* 99.0%

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

   7. *-commutative 99.0%

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

   8. associate-*l* 99.0%

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

   9. sqr-sin-a 99.1%

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

   10. associate-*l* 99.1%

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

6. Applied egg-rr 99.1%

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

7. Final simplification 99.1%

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

Alternative 2: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + {u2}^{6} \cdot \mathsf{fma}\left({\pi}^{6}, -0.044444444444444446, {\pi}^{6} \cdot 0.044444444444444446\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (- (log1p (- u1))))
  (+
   (cos (* PI (* 2.0 u2)))
   (*
    (pow u2 6.0)
    (fma
     (pow PI 6.0)
     -0.044444444444444446
     (* (pow PI 6.0) 0.044444444444444446))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * (cosf((((float) M_PI) * (2.0f * u2))) + (powf(u2, 6.0f) * fmaf(powf(((float) M_PI), 6.0f), -0.044444444444444446f, (powf(((float) M_PI), 6.0f) * 0.044444444444444446f))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(cos(Float32(Float32(pi) * Float32(Float32(2.0) * u2))) + Float32((u2 ^ Float32(6.0)) * fma((Float32(pi) ^ Float32(6.0)), Float32(-0.044444444444444446), Float32((Float32(pi) ^ Float32(6.0)) * Float32(0.044444444444444446))))))
end

Derivation

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 99.0%

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

3. Simplified 99.0%

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

   1. associate-*l* 99.0%

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

   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 99.0%

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

   6. associate-*l* 99.0%

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

   7. *-commutative 99.0%

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

   8. associate-*l* 99.0%

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

   9. sqr-sin-a 99.1%

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

   10. associate-*l* 99.1%

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

6. Applied egg-rr 99.1%

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

   1. associate-*r* 99.1%

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

   2. *-commutative 99.1%

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

   3. add-cbrt-cube 99.1%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + \mathsf{fma}\left(-\sin \left(\pi \cdot u2\right), \sin \left(\pi \cdot u2\right), 0.5 - 0.5 \cdot \cos \color{blue}{\left(\sqrt[3]{\left(\left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \left(\left(2 \cdot \pi\right) \cdot u2\right)\right) \cdot \left(\left(2 \cdot \pi\right) \cdot u2\right)}\right)}\right)\right) \]

   4. pow1/3 99.1%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + \mathsf{fma}\left(-\sin \left(\pi \cdot u2\right), \sin \left(\pi \cdot u2\right), 0.5 - 0.5 \cdot \cos \color{blue}{\left({\left(\left(\left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \left(\left(2 \cdot \pi\right) \cdot u2\right)\right) \cdot \left(\left(2 \cdot \pi\right) \cdot u2\right)\right)}^{0.3333333333333333}\right)}\right)\right) \]

   5. pow3 99.0%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + \mathsf{fma}\left(-\sin \left(\pi \cdot u2\right), \sin \left(\pi \cdot u2\right), 0.5 - 0.5 \cdot \cos \left({\color{blue}{\left({\left(\left(2 \cdot \pi\right) \cdot u2\right)}^{3}\right)}}^{0.3333333333333333}\right)\right)\right) \]

   6. *-commutative 99.0%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + \mathsf{fma}\left(-\sin \left(\pi \cdot u2\right), \sin \left(\pi \cdot u2\right), 0.5 - 0.5 \cdot \cos \left({\left({\color{blue}{\left(u2 \cdot \left(2 \cdot \pi\right)\right)}}^{3}\right)}^{0.3333333333333333}\right)\right)\right) \]

   7. *-commutative 99.0%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + \mathsf{fma}\left(-\sin \left(\pi \cdot u2\right), \sin \left(\pi \cdot u2\right), 0.5 - 0.5 \cdot \cos \left({\left({\left(u2 \cdot \color{blue}{\left(\pi \cdot 2\right)}\right)}^{3}\right)}^{0.3333333333333333}\right)\right)\right) \]

8. Applied egg-rr 99.0%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + \mathsf{fma}\left(-\sin \left(\pi \cdot u2\right), \sin \left(\pi \cdot u2\right), 0.5 - 0.5 \cdot \cos \color{blue}{\left({\left({\left(u2 \cdot \left(\pi \cdot 2\right)\right)}^{3}\right)}^{0.3333333333333333}\right)}\right)\right) \]

9. Taylor expanded in u2 around inf 99.0%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + \color{blue}{\left(\left(0.5 + -1 \cdot {\sin \left(u2 \cdot \pi\right)}^{2}\right) - 0.5 \cdot \cos \left({\left({\left(2 \cdot \left(u2 \cdot \pi\right)\right)}^{3}\right)}^{0.3333333333333333}\right)\right)}\right) \]

10. Taylor expanded in u2 around 0 99.0%

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + \color{blue}{{u2}^{6} \cdot \left(-1 \cdot \left(0.016666666666666666 \cdot {\pi}^{6} + 0.027777777777777776 \cdot {\pi}^{6}\right) - -0.044444444444444446 \cdot {\pi}^{6}\right)}\right) \]
11. Step-by-step derivation

    1. mul-1-neg 99.0%

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + {u2}^{6} \cdot \left(\color{blue}{\left(-\left(0.016666666666666666 \cdot {\pi}^{6} + 0.027777777777777776 \cdot {\pi}^{6}\right)\right)} - -0.044444444444444446 \cdot {\pi}^{6}\right)\right) \]

    2. distribute-rgt-out 99.0%

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + {u2}^{6} \cdot \left(\left(-\color{blue}{{\pi}^{6} \cdot \left(0.016666666666666666 + 0.027777777777777776\right)}\right) - -0.044444444444444446 \cdot {\pi}^{6}\right)\right) \]

    3. metadata-eval 99.0%

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + {u2}^{6} \cdot \left(\left(-{\pi}^{6} \cdot \color{blue}{0.044444444444444446}\right) - -0.044444444444444446 \cdot {\pi}^{6}\right)\right) \]

    4. metadata-eval 99.0%

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + {u2}^{6} \cdot \left(\left(-{\pi}^{6} \cdot \color{blue}{\left(--0.044444444444444446\right)}\right) - -0.044444444444444446 \cdot {\pi}^{6}\right)\right) \]

    5. distribute-rgt-neg-in 99.0%

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + {u2}^{6} \cdot \left(\color{blue}{{\pi}^{6} \cdot \left(-\left(--0.044444444444444446\right)\right)} - -0.044444444444444446 \cdot {\pi}^{6}\right)\right) \]

    6. metadata-eval 99.0%

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + {u2}^{6} \cdot \left({\pi}^{6} \cdot \left(-\color{blue}{0.044444444444444446}\right) - -0.044444444444444446 \cdot {\pi}^{6}\right)\right) \]

    7. metadata-eval 99.0%

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + {u2}^{6} \cdot \left({\pi}^{6} \cdot \color{blue}{-0.044444444444444446} - -0.044444444444444446 \cdot {\pi}^{6}\right)\right) \]

    8. *-commutative 99.0%

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + {u2}^{6} \cdot \left({\pi}^{6} \cdot -0.044444444444444446 - \color{blue}{{\pi}^{6} \cdot -0.044444444444444446}\right)\right) \]

    9. metadata-eval 99.0%

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + {u2}^{6} \cdot \left({\pi}^{6} \cdot -0.044444444444444446 - {\pi}^{6} \cdot \color{blue}{\left(-0.044444444444444446\right)}\right)\right) \]

    10. metadata-eval 99.0%

        \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + {u2}^{6} \cdot \left({\pi}^{6} \cdot -0.044444444444444446 - {\pi}^{6} \cdot \left(-\color{blue}{\left(--0.044444444444444446\right)}\right)\right)\right) \]

    11. distribute-rgt-neg-in 99.0%

        \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + {u2}^{6} \cdot \left({\pi}^{6} \cdot -0.044444444444444446 - \color{blue}{\left(-{\pi}^{6} \cdot \left(--0.044444444444444446\right)\right)}\right)\right) \]

    12. metadata-eval 99.0%

        \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + {u2}^{6} \cdot \left({\pi}^{6} \cdot -0.044444444444444446 - \left(-{\pi}^{6} \cdot \color{blue}{0.044444444444444446}\right)\right)\right) \]

    13. metadata-eval 99.0%

        \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + {u2}^{6} \cdot \left({\pi}^{6} \cdot -0.044444444444444446 - \left(-{\pi}^{6} \cdot \color{blue}{\left(0.016666666666666666 + 0.027777777777777776\right)}\right)\right)\right) \]

    14. distribute-rgt-out 99.0%

        \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + {u2}^{6} \cdot \left({\pi}^{6} \cdot -0.044444444444444446 - \left(-\color{blue}{\left(0.016666666666666666 \cdot {\pi}^{6} + 0.027777777777777776 \cdot {\pi}^{6}\right)}\right)\right)\right) \]

    15. mul-1-neg 99.0%

        \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + {u2}^{6} \cdot \left({\pi}^{6} \cdot -0.044444444444444446 - \color{blue}{-1 \cdot \left(0.016666666666666666 \cdot {\pi}^{6} + 0.027777777777777776 \cdot {\pi}^{6}\right)}\right)\right) \]

12. Simplified 99.0%

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + \color{blue}{{u2}^{6} \cdot \mathsf{fma}\left({\pi}^{6}, -0.044444444444444446, {\pi}^{6} \cdot 0.044444444444444446\right)}\right) \]

13. Final simplification 99.0%

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) + {u2}^{6} \cdot \mathsf{fma}\left({\pi}^{6}, -0.044444444444444446, {\pi}^{6} \cdot 0.044444444444444446\right)\right) \]
14. Add Preprocessing

Alternative 3: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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 99.0%

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

3. Simplified 99.0%

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

5. Final simplification 99.0%

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

Alternative 4: 90.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(\pi \cdot 2\right)\\ \mathbf{if}\;t\_0 \leq 0.00800000037997961:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos t\_0 \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* PI 2.0))))
   (if (<= t_0 0.00800000037997961)
     (sqrt (- (log1p (- u1))))
     (* (cos t_0) (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u2 * (((float) M_PI) * 2.0f);
	float tmp;
	if (t_0 <= 0.00800000037997961f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = cosf(t_0) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u2 * Float32(Float32(pi) * Float32(2.0)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.00800000037997961))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(cos(t_0) * sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 2 (PI.f32)) u2) < 0.00800000038

   1. Initial program 55.6%

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

      1. sub-neg 55.6%

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

      2. log1p-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 97.1%

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

   if 0.00800000038 < (*.f32 (*.f32 2 (PI.f32)) u2)

   1. Initial program 58.0%

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

      1. sub-neg 58.0%

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

      2. log1p-define 98.1%

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

   3. Simplified 98.1%

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

      1. add-cbrt-cube 41.2%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}}} \cdot 1 \]

      2. pow1/3 41.2%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right)}^{0.3333333333333333}} \cdot 1 \]

   6. Applied egg-rr 75.2%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   7. Taylor expanded in u1 around 0 78.8%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.00800000037997961:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.8% 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 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 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in u2 around 0 79.6%

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

6. Final simplification 79.6%

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

Alternative 6: 64.5% 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 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 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in u2 around 0 79.6%

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

   1. add-cbrt-cube 79.6%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}}} \cdot 1 \]

   2. pow1/3 77.4%

      \[\leadsto \color{blue}{{\left(\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right)}^{0.3333333333333333}} \cdot 1 \]

7. Applied egg-rr 63.6%

   \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot 1 \]

8. Taylor expanded in u1 around 0 66.4%

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

9. Final simplification 66.4%

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

Reproduce

herbie shell --seed 2024043 
(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))))