Beckmann Sample, near normal, slope_y

Percentage Accurate: 58.0% → 98.4%

Time: 20.6s

Alternatives: 7

Speedup: 2.0×

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

\[\begin{array}{l} \\ \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf((1.0f - u1))) * sinf(((2.0f * ((float) M_PI)) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(-log((single(1.0) - u1))) * sin(((single(2.0) * single(pi)) * u2));
end

Initial Program: 58.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf((1.0f - u1))) * sinf(((2.0f * ((float) M_PI)) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(-log((single(1.0) - u1))) * sin(((single(2.0) * single(pi)) * u2));
end

Alternative 1: 98.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.9%

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

   1. sub-neg 56.9%

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

   2. log1p-def 98.3%

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

   3. associate-*l* 98.3%

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

3. Simplified 98.3%

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

   1. log1p-expm1-u 98.3%

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

5. Applied egg-rr 98.3%

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

6. Final simplification 98.3%

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

Alternative 2: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.9%

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

   1. sub-neg 56.9%

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

   2. log1p-def 98.3%

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

   3. associate-*l* 98.3%

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

3. Simplified 98.3%

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

4. Final simplification 98.3%

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

Alternative 3: 93.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9900000095367432:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 - u1 \cdot \left(u1 \cdot -0.5\right)} \cdot \sin \left(u2 \cdot \left(2 \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9900000095367432)
   (* (sqrt (- (log (- 1.0 u1)))) (* 2.0 (* PI u2)))
   (* (sqrt (- u1 (* u1 (* u1 -0.5)))) (sin (* u2 (* 2.0 PI))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((1.0f - u1) <= 0.9900000095367432f) {
		tmp = sqrtf(-logf((1.0f - u1))) * (2.0f * (((float) M_PI) * u2));
	} else {
		tmp = sqrtf((u1 - (u1 * (u1 * -0.5f)))) * sinf((u2 * (2.0f * ((float) M_PI))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.9900000095367432))
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(sqrt(Float32(u1 - Float32(u1 * Float32(u1 * Float32(-0.5))))) * sin(Float32(u2 * Float32(Float32(2.0) * Float32(pi)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((single(1.0) - u1) <= single(0.9900000095367432))
		tmp = sqrt(-log((single(1.0) - u1))) * (single(2.0) * (single(pi) * u2));
	else
		tmp = sqrt((u1 - (u1 * (u1 * single(-0.5))))) * sin((u2 * (single(2.0) * single(pi))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 1 u1) < 0.99000001

   1. Initial program 96.7%

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

      1. associate-*r* 96.7%

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

      2. add-exp-log 92.5%

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

   3. Applied egg-rr 92.5%

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

   4. Taylor expanded in u2 around 0 77.7%

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

   if 0.99000001 < (-.f32 1 u1)

   1. Initial program 47.5%

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

   2. Taylor expanded in u1 around 0 96.7%

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

      1. +-commutative 96.7%

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

      2. mul-1-neg 96.7%

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

      3. unsub-neg 96.7%

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

      4. *-commutative 96.7%

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

      5. unpow2 96.7%

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

      6. associate-*l* 96.7%

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

   4. Simplified 96.7%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9900000095367432:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 - u1 \cdot \left(u1 \cdot -0.5\right)} \cdot \sin \left(u2 \cdot \left(2 \cdot \pi\right)\right)\\ \end{array} \]

Alternative 4: 85.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(\pi \cdot u2\right)\\ \mathbf{if}\;1 - u1 \leq 0.9990000128746033:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* 2.0 (* PI u2))))
   (if (<= (- 1.0 u1) 0.9990000128746033)
     (* (sqrt (- (log (- 1.0 u1)))) t_0)
     (* (sqrt u1) (sin t_0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = 2.0f * (((float) M_PI) * u2);
	float tmp;
	if ((1.0f - u1) <= 0.9990000128746033f) {
		tmp = sqrtf(-logf((1.0f - u1))) * t_0;
	} else {
		tmp = sqrtf(u1) * sinf(t_0);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(2.0) * Float32(Float32(pi) * u2))
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.9990000128746033))
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * t_0);
	else
		tmp = Float32(sqrt(u1) * sin(t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = single(2.0) * (single(pi) * u2);
	tmp = single(0.0);
	if ((single(1.0) - u1) <= single(0.9990000128746033))
		tmp = sqrt(-log((single(1.0) - u1))) * t_0;
	else
		tmp = sqrt(u1) * sin(t_0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 1 u1) < 0.999000013

   1. Initial program 92.7%

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

      1. associate-*r* 92.7%

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

      2. add-exp-log 90.1%

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

   3. Applied egg-rr 90.1%

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

   4. Taylor expanded in u2 around 0 75.3%

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

   if 0.999000013 < (-.f32 1 u1)

   1. Initial program 41.8%

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

   2. Taylor expanded in u1 around 0 88.9%

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

      1. mul-1-neg 88.9%

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

   4. Simplified 88.9%

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

   5. Taylor expanded in u2 around inf 88.9%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9990000128746033:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \end{array} \]

Alternative 5: 76.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1} \cdot \sin \left(2 \cdot \left(\pi \cdot u2\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt u1) (sin (* 2.0 (* PI u2)))))

C

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

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(u1) * sin(Float32(Float32(2.0) * Float32(Float32(pi) * u2))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(u1) * sin((single(2.0) * (single(pi) * u2)));
end

Derivation

1. Initial program 56.9%

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

2. Taylor expanded in u1 around 0 76.4%

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

   1. mul-1-neg 76.4%

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

4. Simplified 76.4%

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

5. Taylor expanded in u2 around inf 76.4%

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

6. Final simplification 76.4%

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

Alternative 6: 66.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(pi) * (single(2.0) * sqrt((u2 * (u1 * u2))));
end

Derivation

1. Initial program 56.9%

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

2. Taylor expanded in u1 around 0 76.4%

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

   1. mul-1-neg 76.4%

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

4. Simplified 76.4%

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

5. Taylor expanded in u2 around 0 65.5%

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

   1. associate-*r* 65.5%

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

   2. *-commutative 65.5%

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

7. Simplified 65.5%

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

8. Taylor expanded in u2 around 0 65.5%

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

   1. *-commutative 65.5%

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

   2. *-commutative 65.5%

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

   3. *-commutative 65.5%

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

   4. associate-*l* 65.6%

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

   5. associate-*l* 65.6%

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

10. Simplified 65.6%

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

    1. add-sqr-sqrt 65.4%

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

    2. sqrt-unprod 65.6%

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

    3. *-commutative 65.6%

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

    4. *-commutative 65.6%

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

    5. swap-sqr 65.5%

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

    6. add-sqr-sqrt 65.6%

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

12. Applied egg-rr 65.6%

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

    1. *-commutative 65.6%

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

    2. associate-*l* 65.6%

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

14. Simplified 65.6%

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

15. Final simplification 65.6%

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

Alternative 7: 66.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(pi) * (single(2.0) * (u2 * sqrt(u1)));
end

Derivation

1. Initial program 56.9%

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

2. Taylor expanded in u1 around 0 76.4%

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

   1. mul-1-neg 76.4%

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

4. Simplified 76.4%

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

5. Taylor expanded in u2 around 0 65.5%

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

   1. associate-*r* 65.5%

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

   2. *-commutative 65.5%

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

7. Simplified 65.5%

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

8. Taylor expanded in u2 around 0 65.5%

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

   1. *-commutative 65.5%

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

   2. *-commutative 65.5%

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

   3. *-commutative 65.5%

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

   4. associate-*l* 65.6%

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

   5. associate-*l* 65.6%

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

10. Simplified 65.6%

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

11. Final simplification 65.6%

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

Reproduce

herbie shell --seed 2023283 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_y"
  :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)))) (sin (* (* 2.0 PI) u2))))