Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.5% → 99.1%

Time: 15.5s

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: 57.5% 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, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 54.4%

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

   1. sub-neg 54.4%

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

   2. log1p-def 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. Step-by-step derivation

   1. add-cbrt-cube 99.0%

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

   2. pow1/3 99.0%

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

   3. pow3 99.0%

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

   4. associate-*l* 99.0%

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

5. Applied egg-rr 99.0%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \color{blue}{\left({\left({\left(2 \cdot \left(\pi \cdot u2\right)\right)}^{3}\right)}^{0.3333333333333333}\right)} \]
6. Step-by-step derivation

   1. unpow1/3 99.0%

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

   2. rem-cbrt-cube 99.0%

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

   3. associate-*r* 99.0%

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

   4. *-commutative 99.0%

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

   5. add-sqr-sqrt 99.0%

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

   6. associate-*r* 99.1%

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

7. Applied egg-rr 99.1%

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

8. Final simplification 99.1%

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

Alternative 2: 90.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\\ \mathbf{if}\;t_0 \leq 0.9999949932098389:\\ \;\;\;\;t_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 (* 2.0 PI)))))
   (if (<= t_0 0.9999949932098389)
     (* 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 <= 0.9999949932098389f) {
		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(0.9999949932098389))
		tmp = Float32(t_0 * sqrt(u1));
	else
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 2 (PI.f32)) u2)) < 0.999994993

   1. Initial program 55.8%

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

      1. sub-neg 55.8%

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

      2. log1p-def 98.0%

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

   3. Simplified 98.0%

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

      1. add-sqr-sqrt 97.5%

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

      2. pow2 97.5%

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

      3. pow1/2 97.5%

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

      4. sqrt-pow1 97.5%

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

      5. add-sqr-sqrt 97.5%

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

      6. sqrt-unprod 97.5%

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

      7. sqr-neg 97.5%

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

      8. sqrt-unprod -0.0%

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

      9. add-sqr-sqrt -0.0%

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

      10. add-sqr-sqrt -0.0%

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

      11. sqrt-unprod 74.9%

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

      12. sqr-neg 74.9%

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

      13. sqrt-unprod 75.0%

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

      14. add-sqr-sqrt 74.9%

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

      15. metadata-eval 74.9%

          \[\leadsto {\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{\color{blue}{0.25}}\right)}^{2} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Applied egg-rr 74.9%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{0.25}\right)}^{2}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   6. Taylor expanded in u1 around 0 77.1%

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

   if 0.999994993 < (cos.f32 (*.f32 (*.f32 2 (PI.f32)) u2))

   1. Initial program 53.6%

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

      1. sub-neg 53.6%

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

      2. log1p-def 99.5%

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

   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. Taylor expanded in u2 around 0 98.1%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \leq 0.9999949932098389:\\ \;\;\;\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]

Alternative 3: 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 54.4%

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

   1. sub-neg 54.4%

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

   2. log1p-def 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. Final simplification 99.0%

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

Alternative 4: 94.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(2 \cdot \pi\right)\\ \mathbf{if}\;t_0 \leq 0.18000000715255737:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(1 + -2 \cdot \left(u2 \cdot \left(\pi \cdot \left(u2 \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos t_0 \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* 2.0 PI))))
   (if (<= t_0 0.18000000715255737)
     (* (sqrt (- (log1p (- u1)))) (+ 1.0 (* -2.0 (* u2 (* PI (* u2 PI))))))
     (* (cos t_0) (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u2 * (2.0f * ((float) M_PI));
	float tmp;
	if (t_0 <= 0.18000000715255737f) {
		tmp = sqrtf(-log1pf(-u1)) * (1.0f + (-2.0f * (u2 * (((float) M_PI) * (u2 * ((float) M_PI))))));
	} else {
		tmp = cosf(t_0) * sqrtf(u1);
	}
	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(0.18000000715255737))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(1.0) + Float32(Float32(-2.0) * Float32(u2 * Float32(Float32(pi) * Float32(u2 * Float32(pi)))))));
	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) < 0.180000007

   1. Initial program 53.8%

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

      1. sub-neg 53.8%

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

      2. log1p-def 99.4%

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

   3. Simplified 99.4%

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

      1. expm1-log1p-u 99.1%

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

      2. associate-*l* 99.1%

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

   5. Applied egg-rr 99.1%

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

   6. Taylor expanded in u2 around 0 98.0%

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

      1. *-commutative 98.0%

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

      2. unpow2 98.0%

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

      3. unpow2 98.0%

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

      4. swap-sqr 98.0%

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

      5. unpow2 98.0%

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

      6. *-commutative 98.0%

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

   8. Simplified 98.0%

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

      1. *-commutative 98.0%

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

      2. unpow2 98.0%

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

      3. associate-*r* 98.0%

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

      4. *-commutative 98.0%

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

   10. Applied egg-rr 98.0%

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

   if 0.180000007 < (*.f32 (*.f32 2 (PI.f32)) u2)

   1. Initial program 56.9%

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

      1. sub-neg 56.9%

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

      2. log1p-def 97.5%

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

   3. Simplified 97.5%

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

      1. add-sqr-sqrt 97.2%

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

      2. pow2 97.2%

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

      3. pow1/2 97.2%

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

      4. sqrt-pow1 97.0%

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

      5. add-sqr-sqrt 97.2%

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

      6. sqrt-unprod 97.0%

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

      7. sqr-neg 97.0%

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

      8. sqrt-unprod -0.0%

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

      9. add-sqr-sqrt -0.0%

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

      10. add-sqr-sqrt -0.0%

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

      11. sqrt-unprod 73.4%

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

      12. sqr-neg 73.4%

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

      13. sqrt-unprod 73.5%

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

      14. add-sqr-sqrt 73.4%

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

      15. metadata-eval 73.4%

          \[\leadsto {\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{\color{blue}{0.25}}\right)}^{2} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Applied egg-rr 73.4%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{0.25}\right)}^{2}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   6. Taylor expanded in u1 around 0 75.4%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.18000000715255737:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(1 + -2 \cdot \left(u2 \cdot \left(\pi \cdot \left(u2 \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 5: 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 54.4%

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

   1. sub-neg 54.4%

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

   2. log1p-def 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. Taylor expanded in u2 around 0 79.5%

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

5. Final simplification 79.5%

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

Alternative 6: 65.2% 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 54.4%

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

   1. sub-neg 54.4%

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

   2. log1p-def 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. Step-by-step derivation

   1. log1p-expm1-u 98.9%

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

   2. log1p-udef 97.3%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\log \left(1 + \mathsf{expm1}\left(\cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\right)\right)} \]

   3. associate-*l* 97.3%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \log \left(1 + \mathsf{expm1}\left(\cos \color{blue}{\left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)\right) \]

5. Applied egg-rr 97.3%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\log \left(1 + \mathsf{expm1}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right)} \]

6. Taylor expanded in u1 around 0 77.5%

   \[\leadsto \sqrt{-\color{blue}{-1 \cdot u1}} \cdot \log \left(1 + \mathsf{expm1}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right) \]
7. Step-by-step derivation

   1. mul-1-neg 77.5%

      \[\leadsto \sqrt{-\color{blue}{\left(-u1\right)}} \cdot \log \left(1 + \mathsf{expm1}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right) \]

8. Simplified 77.5%

   \[\leadsto \sqrt{-\color{blue}{\left(-u1\right)}} \cdot \log \left(1 + \mathsf{expm1}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right) \]

9. Taylor expanded in u2 around 0 66.3%

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

10. Final simplification 66.3%

    \[\leadsto \sqrt{u1} \]

Reproduce

herbie shell --seed 2023336 
(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))))