Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.3% → 98.3%

Time: 13.0s

Alternatives: 9

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: 57.3% 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: 98.3% 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 57.0%

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

   1. sub-neg 57.0%

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

   2. log1p-def 98.5%

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

   3. associate-*l* 98.5%

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

3. Simplified 98.5%

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

4. Final simplification 98.5%

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

Alternative 2: 94.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(2 \cdot \pi\right)\\ \mathbf{if}\;t_0 \leq 0.0013000000035390258:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\pi \cdot \left(u2 + u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 - u1 \cdot \left(u1 \cdot -0.5\right)} \cdot \sin t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* 2.0 PI))))
   (if (<= t_0 0.0013000000035390258)
     (* (sqrt (- (log1p (- u1)))) (* PI (+ u2 u2)))
     (* (sqrt (- u1 (* u1 (* u1 -0.5)))) (sin t_0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u2 * (2.0f * ((float) M_PI));
	float tmp;
	if (t_0 <= 0.0013000000035390258f) {
		tmp = sqrtf(-log1pf(-u1)) * (((float) M_PI) * (u2 + u2));
	} else {
		tmp = sqrtf((u1 - (u1 * (u1 * -0.5f)))) * sinf(t_0);
	}
	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(0.0013000000035390258))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(pi) * Float32(u2 + u2)));
	else
		tmp = Float32(sqrt(Float32(u1 - Float32(u1 * Float32(u1 * Float32(-0.5))))) * sin(t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 2 (PI.f32)) u2) < 0.0013

   1. Initial program 58.3%

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

      1. sub-neg 58.3%

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

      2. log1p-def 98.7%

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

      3. associate-*l* 98.7%

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

   3. Simplified 98.7%

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

      1. add-sqr-sqrt 98.1%

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

      2. pow2 98.1%

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

   5. Applied egg-rr 98.1%

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

      1. unpow2 98.1%

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

      2. add-sqr-sqrt 98.7%

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

      3. add-sqr-sqrt 97.8%

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

      4. associate-*r* 97.8%

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

   7. Applied egg-rr 97.8%

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

   8. Taylor expanded in u2 around 0 98.4%

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

      1. *-commutative 98.4%

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

      2. count-2 98.4%

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

      3. distribute-lft-out 98.4%

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

   10. Simplified 98.4%

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

   if 0.0013 < (*.f32 (*.f32 2 (PI.f32)) u2)

   1. Initial program 55.2%

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

   2. Taylor expanded in u1 around 0 90.0%

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

      1. +-commutative 90.0%

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

      2. mul-1-neg 90.0%

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

      3. unsub-neg 90.0%

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

      4. *-commutative 90.0%

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

      5. unpow2 90.0%

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

      6. associate-*l* 90.0%

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

   4. Simplified 90.0%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.0013000000035390258:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\pi \cdot \left(u2 + u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 - u1 \cdot \left(u1 \cdot -0.5\right)} \cdot \sin \left(u2 \cdot \left(2 \cdot \pi\right)\right)\\ \end{array} \]

Alternative 3: 90.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.004600000102072954:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\pi \cdot \left(u2 + u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 (* 2.0 PI)) 0.004600000102072954)
   (* (sqrt (- (log1p (- u1)))) (* PI (+ u2 u2)))
   (* (sqrt u1) (sin (* 2.0 (* PI u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * (2.0f * ((float) M_PI))) <= 0.004600000102072954f) {
		tmp = sqrtf(-log1pf(-u1)) * (((float) M_PI) * (u2 + u2));
	} else {
		tmp = sqrtf(u1) * sinf((2.0f * (((float) M_PI) * u2)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(Float32(2.0) * Float32(pi))) <= Float32(0.004600000102072954))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(pi) * Float32(u2 + u2)));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(Float32(2.0) * Float32(Float32(pi) * u2))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 2 (PI.f32)) u2) < 0.0046000001

   1. Initial program 57.5%

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

      1. sub-neg 57.5%

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

      2. log1p-def 98.6%

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

      3. associate-*l* 98.6%

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

   3. Simplified 98.6%

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

      1. add-sqr-sqrt 98.0%

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

      2. pow2 98.0%

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

   5. Applied egg-rr 98.0%

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

      1. unpow2 98.0%

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

      2. add-sqr-sqrt 98.6%

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

      3. add-sqr-sqrt 97.8%

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

      4. associate-*r* 97.9%

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

   7. Applied egg-rr 97.9%

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

   8. Taylor expanded in u2 around 0 97.0%

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

      1. *-commutative 97.0%

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

      2. count-2 97.0%

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

      3. distribute-lft-out 97.0%

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

   10. Simplified 97.0%

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

   if 0.0046000001 < (*.f32 (*.f32 2 (PI.f32)) u2)

   1. Initial program 55.9%

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

   2. Taylor expanded in u1 around 0 78.5%

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

      1. mul-1-neg 78.5%

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

   4. Simplified 78.5%

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

   5. Taylor expanded in u2 around inf 78.5%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.004600000102072954:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\pi \cdot \left(u2 + u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \end{array} \]

Alternative 4: 76.7% 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 57.0%

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

2. Taylor expanded in u1 around 0 77.3%

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

   1. mul-1-neg 77.3%

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

4. Simplified 77.3%

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

5. Taylor expanded in u2 around inf 77.3%

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

6. Final simplification 77.3%

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

Alternative 5: 66.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.0%

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

2. Taylor expanded in u1 around 0 77.3%

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

   1. mul-1-neg 77.3%

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

4. Simplified 77.3%

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

5. Taylor expanded in u2 around 0 66.3%

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

   1. associate-*r* 66.3%

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

   2. *-commutative 66.3%

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

   3. *-commutative 66.3%

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

7. Simplified 66.3%

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

   1. add-sqr-sqrt 66.2%

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

   2. sqrt-unprod 66.3%

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

   3. *-commutative 66.3%

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

   4. *-commutative 66.3%

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

   5. swap-sqr 66.2%

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

   6. add-sqr-sqrt 66.3%

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

9. Applied egg-rr 66.3%

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

    1. associate-*r* 66.3%

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

    2. *-commutative 66.3%

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

    3. *-commutative 66.3%

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

11. Simplified 66.3%

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

12. Final simplification 66.3%

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

Alternative 6: 66.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.0%

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

2. Taylor expanded in u1 around 0 77.3%

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

   1. mul-1-neg 77.3%

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

4. Simplified 77.3%

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

5. Taylor expanded in u2 around 0 66.3%

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

   1. associate-*r* 66.3%

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

   2. *-commutative 66.3%

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

   3. *-commutative 66.3%

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

7. Simplified 66.3%

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

8. Taylor expanded in u2 around 0 66.3%

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

   1. *-commutative 66.3%

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

   2. associate-*r* 66.3%

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

   3. *-commutative 66.3%

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

   4. *-commutative 66.3%

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

10. Simplified 66.3%

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

11. Final simplification 66.3%

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

Alternative 7: 66.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.0%

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

2. Taylor expanded in u1 around 0 77.3%

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

   1. mul-1-neg 77.3%

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

4. Simplified 77.3%

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

5. Taylor expanded in u2 around 0 66.3%

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

   1. associate-*r* 66.3%

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

   2. *-commutative 66.3%

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

   3. *-commutative 66.3%

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

7. Simplified 66.3%

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

8. Final simplification 66.3%

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

Alternative 8: 66.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.0%

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

2. Taylor expanded in u1 around 0 77.3%

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

   1. mul-1-neg 77.3%

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

4. Simplified 77.3%

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

5. Taylor expanded in u2 around 0 66.3%

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

6. Final simplification 66.3%

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

Alternative 9: 7.1% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1 \cdot 0} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (sqrt (* u1 0.0)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 * 0.0f));
}

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 * 0.0e0))
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(u1 * Float32(0.0)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 * single(0.0)));
end

Derivation

1. Initial program 57.0%

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

2. Taylor expanded in u1 around 0 77.3%

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

   1. mul-1-neg 77.3%

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

4. Simplified 77.3%

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

   1. add-sqr-sqrt 74.5%

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

   2. sqrt-unprod 74.9%

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

   3. associate-*r* 74.9%

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

   4. associate-*r* 74.9%

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

   5. swap-sqr 74.9%

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

   6. add-sqr-sqrt 74.9%

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

   7. remove-double-neg 74.9%

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

   8. pow2 74.9%

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

6. Applied egg-rr 74.9%

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

   1. unpow2 74.9%

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

   2. sqr-sin-a 35.9%

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

   3. *-commutative 35.9%

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

   4. associate-*r* 35.9%

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

   5. *-commutative 35.9%

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

   6. sqr-sin-a 74.9%

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

   7. sin-mult 35.9%

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

8. Applied egg-rr 35.9%

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

   1. div-sub 35.9%

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

   2. +-inverses 35.9%

      \[\leadsto \sqrt{u1 \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(\pi \cdot \left(u2 \cdot 2\right) + \pi \cdot \left(u2 \cdot 2\right)\right)}{2}\right)} \]

   3. cos-0 35.9%

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

   4. metadata-eval 35.9%

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

   5. distribute-lft-out 35.9%

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

   6. distribute-lft-out 35.9%

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

   7. metadata-eval 35.9%

      \[\leadsto \sqrt{u1 \cdot \left(0.5 - \frac{\cos \left(\pi \cdot \left(u2 \cdot \color{blue}{4}\right)\right)}{2}\right)} \]

10. Simplified 35.9%

    \[\leadsto \sqrt{u1 \cdot \color{blue}{\left(0.5 - \frac{\cos \left(\pi \cdot \left(u2 \cdot 4\right)\right)}{2}\right)}} \]

11. Taylor expanded in u2 around 0 7.1%

    \[\leadsto \sqrt{u1 \cdot \left(0.5 - \frac{\color{blue}{1}}{2}\right)} \]

12. Final simplification 7.1%

    \[\leadsto \sqrt{u1 \cdot 0} \]

Reproduce

herbie shell --seed 2023293 
(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))))