Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.6% → 99.0%

Time: 12.8s

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: 57.6% 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

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log1p (- u1)))) (log1p (expm1 (cos (* 2.0 (* PI u2)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * log1pf(expm1f(cosf((2.0f * (((float) M_PI) * u2)))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * log1p(expm1(cos(Float32(Float32(2.0) * Float32(Float32(pi) * u2))))))
end

Derivation

1. Initial program 59.5%

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

   1. sub-neg 59.5%

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

   2. log1p-def 98.9%

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

3. Simplified 98.9%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)} \]
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. associate-*l* 98.9%

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

5. Applied egg-rr 98.9%

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

6. Final simplification 98.9%

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

Alternative 2: 90.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \leq 0.9999966025352478:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\pi \cdot \left(2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (cos (* u2 (* 2.0 PI))) 0.9999966025352478)
   (* (sqrt u1) (cos (* PI (* 2.0 u2))))
   (sqrt (- (log1p (- u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (cosf((u2 * (2.0f * ((float) M_PI)))) <= 0.9999966025352478f) {
		tmp = sqrtf(u1) * cosf((((float) M_PI) * (2.0f * u2)));
	} else {
		tmp = sqrtf(-log1pf(-u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (cos(Float32(u2 * Float32(Float32(2.0) * Float32(pi)))) <= Float32(0.9999966025352478))
		tmp = Float32(sqrt(u1) * cos(Float32(Float32(pi) * Float32(Float32(2.0) * u2))));
	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)) < 0.999996603

   1. Initial program 60.7%

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

      1. sub-neg 60.7%

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

      2. log1p-def 97.6%

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

   3. Simplified 97.6%

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

      1. add-cube-cbrt 96.9%

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

      2. pow3 96.9%

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

      3. associate-*l* 96.9%

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

   5. Applied egg-rr 96.9%

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

   6. Taylor expanded in u1 around 0 74.1%

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

      1. mul-1-neg 74.1%

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

   8. Simplified 74.1%

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

      1. expm1-log1p-u 74.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{-\left(-u1\right)} \cdot \cos \left({\left(\sqrt[3]{2 \cdot \left(\pi \cdot u2\right)}\right)}^{3}\right)\right)\right)} \]

      2. expm1-udef 62.1%

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

      3. *-commutative 62.1%

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

      4. rem-cube-cbrt 62.1%

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

      5. *-commutative 62.1%

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

      6. associate-*l* 62.1%

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

      7. remove-double-neg 62.1%

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

   10. Applied egg-rr 62.1%

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

       1. expm1-def 74.2%

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

       2. expm1-log1p 74.2%

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

       3. *-commutative 74.2%

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

   12. Simplified 74.2%

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

   if 0.999996603 < (cos.f32 (*.f32 (*.f32 2 (PI.f32)) u2))

   1. Initial program 58.8%

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

      1. sub-neg 58.8%

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

      2. log1p-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in u2 around 0 98.4%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \leq 0.9999966025352478:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\pi \cdot \left(2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]

Alternative 3: 94.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(2 \cdot \pi\right)\\ \mathbf{if}\;t_0 \leq 0.002199999988079071:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos t_0 \cdot \sqrt{u1 \cdot \left(\left(--1\right) - u1 \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* 2.0 PI))))
   (if (<= t_0 0.002199999988079071)
     (sqrt (- (log1p (- u1))))
     (* (cos t_0) (sqrt (* u1 (- (- -1.0) (* u1 -0.5))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u2 * (2.0f * ((float) M_PI));
	float tmp;
	if (t_0 <= 0.002199999988079071f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = cosf(t_0) * sqrtf((u1 * (-(-1.0f) - (u1 * -0.5f))));
	}
	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(0.002199999988079071))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(cos(t_0) * sqrt(Float32(u1 * Float32(Float32(-Float32(-1.0)) - Float32(u1 * Float32(-0.5))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 2 (PI.f32)) u2) < 0.0022

   1. Initial program 58.8%

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

      1. sub-neg 58.8%

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

      2. log1p-def 99.6%

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

   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. Taylor expanded in u2 around 0 98.7%

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

   if 0.0022 < (*.f32 (*.f32 2 (PI.f32)) u2)

   1. Initial program 60.7%

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

   2. Taylor expanded in u1 around 0 86.1%

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

      1. *-commutative 86.1%

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

      2. *-commutative 86.1%

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

      3. unpow2 86.1%

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

      4. associate-*l* 86.1%

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

      5. distribute-lft-out 86.1%

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

   4. Simplified 86.1%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.002199999988079071:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{u1 \cdot \left(\left(--1\right) - u1 \cdot -0.5\right)}\\ \end{array} \]

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 59.5%

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

   1. sub-neg 59.5%

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

   2. log1p-def 98.9%

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

3. Simplified 98.9%

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

4. Final simplification 98.9%

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

Alternative 5: 80.2% 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 59.5%

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

   1. sub-neg 59.5%

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

   2. log1p-def 98.9%

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

3. Simplified 98.9%

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

4. Taylor expanded in u2 around 0 78.0%

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

5. Final simplification 78.0%

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

Alternative 6: 72.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1 \cdot \left(\left(--1\right) - u1 \cdot -0.5\right)} \end{array} \]

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(-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 59.5%

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

2. Taylor expanded in u1 around 0 87.1%

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

   1. *-commutative 87.1%

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

   2. *-commutative 87.1%

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

   3. unpow2 87.1%

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

   4. associate-*l* 87.1%

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

   5. distribute-lft-out 87.1%

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

4. Simplified 87.1%

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

5. Taylor expanded in u2 around 0 70.6%

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

6. Final simplification 70.6%

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

Alternative 7: 65.0% 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 59.5%

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

   1. add-exp-log 54.8%

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

   2. *-commutative 54.8%

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

   3. associate-*l* 54.8%

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

   4. add-sqr-sqrt 54.8%

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

   5. sqrt-unprod 54.8%

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

   6. sqr-neg 54.8%

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

   7. sqrt-unprod 1.5%

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

   8. add-sqr-sqrt 1.5%

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

   9. sub-neg 1.5%

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

   10. log1p-udef -0.0%

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

   11. add-sqr-sqrt -0.0%

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

   12. sqrt-unprod 66.5%

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

   13. sqr-neg 66.5%

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

   14. sqrt-unprod 66.4%

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

   15. add-sqr-sqrt 66.5%

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

3. Applied egg-rr 66.5%

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

4. Taylor expanded in u2 around 0 35.4%

   \[\leadsto \color{blue}{\sqrt{\log \left(1 + u1\right)}} \]
5. Step-by-step derivation

   1. log1p-def 60.8%

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

6. Simplified 60.8%

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

7. Taylor expanded in u1 around 0 62.3%

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

8. Final simplification 62.3%

   \[\leadsto \sqrt{u1} \]

Reproduce

herbie shell --seed 2023320 
(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))))