Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.3% → 99.0%

Time: 11.7s

Alternatives: 7

Speedup: 4.0×

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: 57.3% 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, 0.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot {\left({\left(\pi \cdot u2\right)}^{3}\right)}^{0.3333333333333333}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (- (log1p (- u1))))
  (cos (* 2.0 (pow (pow (* PI u2) 3.0) 0.3333333333333333)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * cosf((2.0f * powf(powf((((float) M_PI) * u2), 3.0f), 0.3333333333333333f)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * cos(Float32(Float32(2.0) * ((Float32(Float32(pi) * u2) ^ Float32(3.0)) ^ Float32(0.3333333333333333)))))
end

Derivation

1. Initial program 55.3%

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

   1. sub-neg 55.3%

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

   2. log1p-def 99.1%

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

   3. associate-*l* 99.1%

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

3. Simplified 99.1%

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

   1. add-cbrt-cube 99.1%

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

   2. pow1/3 99.1%

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

   3. pow3 99.1%

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

5. Applied egg-rr 99.1%

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

6. Final simplification 99.1%

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

Alternative 2: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(2 \cdot \pi\right)\\ \mathbf{if}\;t_0 \leq 0.0013500000350177288:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos t_0 \cdot \sqrt{u1 + 0.5 \cdot \left(u1 \cdot u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* 2.0 PI))))
   (if (<= t_0 0.0013500000350177288)
     (sqrt (- (log1p (- u1))))
     (* (cos t_0) (sqrt (+ u1 (* 0.5 (* u1 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 <= 0.0013500000350177288f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = cosf(t_0) * sqrtf((u1 + (0.5f * (u1 * 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(0.0013500000350177288))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(cos(t_0) * sqrt(Float32(u1 + Float32(Float32(0.5) * Float32(u1 * u1)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 2 (PI.f32)) u2) < 0.00135000004

   1. Initial program 52.5%

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

      1. sub-neg 52.5%

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

      2. log1p-def 99.6%

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

      3. associate-*l* 99.6%

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

   3. Simplified 99.6%

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

      1. add-cbrt-cube 99.6%

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

      2. pow1/3 99.6%

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

      3. pow3 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. unpow1/3 99.6%

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

      2. rem-cbrt-cube 99.6%

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

      3. *-commutative 99.6%

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

      4. add-cube-cbrt 99.6%

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

      5. associate-*l* 99.6%

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

      6. pow2 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. add-log-exp 99.6%

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

      2. associate-*r* 99.6%

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

      3. unpow2 99.6%

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

      4. add-cube-cbrt 99.6%

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

   9. Applied egg-rr 99.6%

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

   10. Taylor expanded in u2 around 0 98.9%

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

   if 0.00135000004 < (*.f32 (*.f32 2 (PI.f32)) u2)

   1. Initial program 58.4%

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

   2. Taylor expanded in u1 around 0 85.9%

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

      1. +-commutative 85.9%

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

      2. mul-1-neg 85.9%

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

      3. unsub-neg 85.9%

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

      4. unpow2 85.9%

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

      5. associate-*r* 85.9%

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

   4. Simplified 85.9%

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

      1. associate-*r* 85.9%

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

      2. add-exp-log 85.9%

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

   6. Applied egg-rr 85.9%

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

   7. Taylor expanded in u2 around inf 85.9%

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

      1. *-commutative 85.9%

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

      2. *-commutative 85.9%

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

      3. associate-*l* 85.9%

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

      4. unpow2 85.9%

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

      5. cancel-sign-sub-inv 85.9%

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

      6. metadata-eval 85.9%

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

   9. Simplified 85.9%

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

4. Final simplification 92.8%

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

Alternative 3: 90.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.009534000419080257:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 (* 2.0 PI)) 0.009534000419080257)
   (sqrt (- (log1p (- u1))))
   (* (cos (* PI (* 2.0 u2))) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * (2.0f * ((float) M_PI))) <= 0.009534000419080257f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = cosf((((float) M_PI) * (2.0f * u2))) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(Float32(2.0) * Float32(pi))) <= Float32(0.009534000419080257))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(cos(Float32(Float32(pi) * Float32(Float32(2.0) * u2))) * sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 2 (PI.f32)) u2) < 0.00953400042

   1. Initial program 54.8%

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

      1. sub-neg 54.8%

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

      2. log1p-def 99.5%

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

      3. associate-*l* 99.5%

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

   3. Simplified 99.5%

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

      1. add-cbrt-cube 99.5%

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

      2. pow1/3 99.5%

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

      3. pow3 99.5%

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

   5. Applied egg-rr 99.5%

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

      1. unpow1/3 99.5%

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

      2. rem-cbrt-cube 99.5%

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

      3. *-commutative 99.5%

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

      4. add-cube-cbrt 99.5%

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

      5. associate-*l* 99.5%

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

      6. pow2 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. add-log-exp 99.5%

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

      2. associate-*r* 99.5%

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

      3. unpow2 99.5%

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

      4. add-cube-cbrt 99.5%

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

   9. Applied egg-rr 99.5%

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

   10. Taylor expanded in u2 around 0 96.0%

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

   if 0.00953400042 < (*.f32 (*.f32 2 (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. add-sqr-sqrt 48.5%

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

      2. pow2 48.5%

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

   3. Applied egg-rr 62.6%

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

   4. Taylor expanded in u1 around 0 75.2%

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

      1. associate-*r* 75.2%

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

      2. *-commutative 75.2%

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

      3. *-commutative 75.2%

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

   6. Simplified 75.2%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.009534000419080257:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 4: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.3%

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

   1. sub-neg 55.3%

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

   2. log1p-def 99.1%

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

   3. associate-*l* 99.1%

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

3. Simplified 99.1%

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

4. Final simplification 99.1%

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

Alternative 5: 80.1% accurate, 2.0× 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 55.3%

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

   1. sub-neg 55.3%

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

   2. log1p-def 99.1%

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

   3. associate-*l* 99.1%

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

3. Simplified 99.1%

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

   1. add-cbrt-cube 99.1%

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

   2. pow1/3 99.1%

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

   3. pow3 99.1%

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

5. Applied egg-rr 99.1%

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

   1. unpow1/3 99.1%

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

   2. rem-cbrt-cube 99.1%

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

   3. *-commutative 99.1%

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

   4. add-cube-cbrt 99.0%

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

   5. associate-*l* 99.0%

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

   6. pow2 99.0%

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

7. Applied egg-rr 99.0%

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

   1. add-log-exp 98.9%

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

   2. associate-*r* 98.9%

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

   3. unpow2 98.9%

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

   4. add-cube-cbrt 99.1%

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

9. Applied egg-rr 99.1%

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

10. Taylor expanded in u2 around 0 77.6%

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

11. Final simplification 77.6%

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

Alternative 6: 72.7% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.3%

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

2. Taylor expanded in u1 around 0 89.1%

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

   1. +-commutative 89.1%

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

   2. mul-1-neg 89.1%

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

   3. unsub-neg 89.1%

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

   4. unpow2 89.1%

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

   5. associate-*r* 89.1%

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

4. Simplified 89.1%

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

   1. associate-*r* 89.1%

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

   2. add-exp-log 89.1%

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

6. Applied egg-rr 89.1%

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

7. Taylor expanded in u2 around 0 72.7%

   \[\leadsto \color{blue}{\sqrt{u1 - -0.5 \cdot {u1}^{2}}} \]
8. Step-by-step derivation

   1. unpow2 72.7%

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

   2. cancel-sign-sub-inv 72.7%

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

   3. metadata-eval 72.7%

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

9. Simplified 72.7%

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

10. Final simplification 72.7%

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

Alternative 7: 64.8% accurate, 4.0× 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 sqrt(u1)
end

MATLAB

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

Derivation

1. Initial program 55.3%

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

2. Taylor expanded in u2 around 0 46.2%

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

3. Taylor expanded in u1 around 0 65.2%

   \[\leadsto \sqrt{-\color{blue}{-1 \cdot u1}} \cdot 1 \]
4. Step-by-step derivation

   1. mul-1-neg 65.2%

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

5. Simplified 65.2%

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

6. Final simplification 65.2%

   \[\leadsto \sqrt{u1} \]

Reproduce

herbie shell --seed 2023215 
(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))))