Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.4% → 99.1%

Time: 16.7s

Alternatives: 12

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: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(t\_0 + \left(\left(0.5 + t\_0 \cdot -0.5\right) - \sin \left(u2 \cdot \pi\right) \cdot \sin \left(\sqrt[3]{{\pi}^{3} \cdot {u2}^{3}}\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 (* 2.0 PI)))))
   (*
    (sqrt (- (log1p (- u1))))
    (+
     t_0
     (-
      (+ 0.5 (* t_0 -0.5))
      (* (sin (* u2 PI)) (sin (cbrt (* (pow PI 3.0) (pow u2 3.0))))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((u2 * (2.0f * ((float) M_PI))));
	return sqrtf(-log1pf(-u1)) * (t_0 + ((0.5f + (t_0 * -0.5f)) - (sinf((u2 * ((float) M_PI))) * sinf(cbrtf((powf(((float) M_PI), 3.0f) * powf(u2, 3.0f)))))));
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(u2 * Float32(Float32(2.0) * Float32(pi))))
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(t_0 + Float32(Float32(Float32(0.5) + Float32(t_0 * Float32(-0.5))) - Float32(sin(Float32(u2 * Float32(pi))) * sin(cbrt(Float32((Float32(pi) ^ Float32(3.0)) * (u2 ^ Float32(3.0)))))))))
end

Derivation

1. Initial program 56.7%

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

   1. sub-neg 56.7%

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

   2. log1p-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 99.0%

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

   3. prod-diff 99.1%

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

   4. fma-neg 99.0%

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

   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. associate-*l* 99.1%

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

   7. *-commutative 99.1%

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

   8. associate-*l* 99.1%

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

   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. *-commutative 99.1%

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

   2. *-commutative 99.1%

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

   3. associate-*l* 99.1%

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

   4. fma-undefine 99.1%

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

   5. +-commutative 99.1%

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

   6. distribute-lft-neg-out 99.1%

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

   7. unsub-neg 99.1%

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

8. Simplified 99.1%

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

   1. *-commutative 99.1%

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

   2. add-cbrt-cube 99.1%

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

   3. add-cbrt-cube 99.1%

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

   4. cbrt-unprod 99.2%

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

   5. pow3 99.2%

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

   6. pow3 99.2%

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

10. Applied egg-rr 99.2%

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

11. Final simplification 99.2%

    \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) + \left(\left(0.5 + \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot -0.5\right) - \sin \left(u2 \cdot \pi\right) \cdot \sin \left(\sqrt[3]{{\pi}^{3} \cdot {u2}^{3}}\right)\right)\right) \]
12. Add Preprocessing

Alternative 2: 99.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.7%

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

   1. sub-neg 56.7%

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

   2. log1p-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 99.0%

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

   3. prod-diff 99.1%

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

   4. fma-neg 99.0%

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

   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. associate-*l* 99.1%

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

   7. *-commutative 99.1%

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

   8. associate-*l* 99.1%

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

   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. *-commutative 99.1%

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

   2. *-commutative 99.1%

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

   3. associate-*l* 99.1%

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

   4. fma-undefine 99.1%

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

   5. +-commutative 99.1%

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

   6. distribute-lft-neg-out 99.1%

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

   7. unsub-neg 99.1%

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

8. Simplified 99.1%

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

9. Final simplification 99.1%

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

Alternative 3: 90.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\\ \mathbf{if}\;t\_0 \leq 0.9999589920043945:\\ \;\;\;\;t\_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 (* 2.0 PI)))))
   (if (<= t_0 0.9999589920043945)
     (* t_0 (sqrt u1))
     (sqrt (- (log1p (- u1)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((u2 * (2.0f * ((float) M_PI))));
	float tmp;
	if (t_0 <= 0.9999589920043945f) {
		tmp = t_0 * sqrtf(u1);
	} else {
		tmp = sqrtf(-log1pf(-u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(u2 * Float32(Float32(2.0) * Float32(pi))))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999589920043945))
		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)) < 0.99995899

   1. Initial program 54.2%

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

      1. sub-neg 54.2%

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

      2. log1p-define 97.8%

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

   3. Simplified 97.8%

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

      1. add-cbrt-cube 97.8%

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

      2. pow1/3 94.9%

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

   6. Applied egg-rr 76.7%

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

   7. Taylor expanded in u1 around 0 80.4%

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

   if 0.99995899 < (cos.f32 (*.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. Step-by-step derivation

      1. sub-neg 57.6%

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

      2. log1p-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 96.6%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \leq 0.9999589920043945:\\ \;\;\;\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% 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 56.7%

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

   1. sub-neg 56.7%

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

   2. log1p-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 5: 94.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.0013500000350177288:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(u1 \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right) \cdot \sqrt{0.5 + \frac{1}{u1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 (* 2.0 PI)) 0.0013500000350177288)
   (sqrt (- (log1p (- u1))))
   (* (* u1 (cos (* 2.0 (* u2 PI)))) (sqrt (+ 0.5 (/ 1.0 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * (2.0f * ((float) M_PI))) <= 0.0013500000350177288f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = (u1 * cosf((2.0f * (u2 * ((float) M_PI))))) * sqrtf((0.5f + (1.0f / u1)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(Float32(2.0) * Float32(pi))) <= Float32(0.0013500000350177288))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(Float32(u1 * cos(Float32(Float32(2.0) * Float32(u2 * Float32(pi))))) * sqrt(Float32(Float32(0.5) + Float32(Float32(1.0) / u1))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00135000004

   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 99.2%

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

   if 0.00135000004 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 56.1%

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

      1. sub-neg 56.1%

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

      2. log1p-define 98.2%

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

   3. Simplified 98.2%

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

      1. add-cube-cbrt 97.3%

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

      2. pow3 97.2%

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

      3. *-commutative 97.2%

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

      4. associate-*l* 97.2%

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

   6. Applied egg-rr 97.2%

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

   7. Taylor expanded in u1 around 0 87.9%

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

   8. Taylor expanded in u1 around inf 87.9%

      \[\leadsto \sqrt{-\color{blue}{-1 \cdot \left({u1}^{2} \cdot \left(0.5 + \frac{1}{u1}\right)\right)}} \cdot {\left(\sqrt[3]{\cos \left(\pi \cdot \left(2 \cdot u2\right)\right)}\right)}^{3} \]
   9. Step-by-step derivation

      1. mul-1-neg 87.9%

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

      2. distribute-rgt-neg-in 87.9%

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

      3. +-commutative 87.9%

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

   10. Simplified 87.9%

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

   11. Taylor expanded in u2 around inf 88.3%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.0013500000350177288:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(u1 \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right) \cdot \sqrt{0.5 + \frac{1}{u1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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 56.7%

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

   1. sub-neg 56.7%

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

   2. log1p-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 81.3%

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

6. Final simplification 81.3%

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

Alternative 7: 76.4% 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 56.7%

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

   1. sub-neg 56.7%

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

   2. log1p-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 81.3%

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

6. Taylor expanded in u1 around 0 77.5%

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

7. Final simplification 77.5%

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

Alternative 8: 75.1% 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 56.7%

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

   1. sub-neg 56.7%

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

   2. log1p-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 81.3%

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

6. Taylor expanded in u1 around 0 76.1%

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

7. Final simplification 76.1%

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

Alternative 9: 72.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ u1 \cdot \sqrt{0.5 + \frac{1}{u1}} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return u1 * sqrtf((0.5f + (1.0f / 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 = u1 * sqrt((0.5e0 + (1.0e0 / u1)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 56.7%

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

   1. sub-neg 56.7%

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

   2. log1p-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. add-cube-cbrt 98.7%

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

   2. pow3 98.7%

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

   3. *-commutative 98.7%

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

   4. associate-*l* 98.7%

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

6. Applied egg-rr 98.7%

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

7. Taylor expanded in u1 around 0 88.7%

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

8. Taylor expanded in u1 around inf 88.6%

   \[\leadsto \sqrt{-\color{blue}{-1 \cdot \left({u1}^{2} \cdot \left(0.5 + \frac{1}{u1}\right)\right)}} \cdot {\left(\sqrt[3]{\cos \left(\pi \cdot \left(2 \cdot u2\right)\right)}\right)}^{3} \]
9. Step-by-step derivation

   1. mul-1-neg 88.6%

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

   2. distribute-rgt-neg-in 88.6%

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

   3. +-commutative 88.6%

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

10. Simplified 88.6%

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

11. Taylor expanded in u2 around 0 73.8%

    \[\leadsto \color{blue}{u1 \cdot \sqrt{0.5 + \frac{1}{u1}}} \]

12. Final simplification 73.8%

    \[\leadsto u1 \cdot \sqrt{0.5 + \frac{1}{u1}} \]
13. Add Preprocessing

Alternative 10: 72.5% 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 56.7%

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

   1. sub-neg 56.7%

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

   2. log1p-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 81.3%

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

6. Taylor expanded in u1 around 0 74.0%

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

7. Final simplification 74.0%

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

Alternative 11: 5.0% accurate, 3.0× 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 Float32(-sqrt(u1))
end

MATLAB

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

Derivation

1. Initial program 56.7%

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

   1. sub-neg 56.7%

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

   2. log1p-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 81.3%

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

6. Taylor expanded in u1 around 0 -0.0%

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

   1. *-commutative -0.0%

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

   2. unpow2 -0.0%

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

   3. rem-square-sqrt 5.1%

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

   4. mul-1-neg 5.1%

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

8. Simplified 5.1%

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

9. Final simplification 5.1%

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

Alternative 12: 64.6% 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 56.7%

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

   1. sub-neg 56.7%

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

   2. log1p-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 81.3%

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

6. Taylor expanded in u1 around 0 66.0%

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

8. Simplified 66.0%

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

9. Final simplification 66.0%

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

Reproduce

herbie shell --seed 2024078 
(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))))