Beckmann Sample, near normal, slope_y

Percentage Accurate: 58.1% → 98.3%

Time: 11.5s

Alternatives: 6

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: 58.1% 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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.6%

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

   1. sub-neg 59.6%

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

   2. log1p-def 98.4%

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

   3. associate-*l* 98.4%

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

3. Simplified 98.4%

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

   1. log1p-expm1-u 98.4%

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

5. Applied egg-rr 98.4%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(2 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\pi \cdot u2\right)\right)}\right) \]
6. Step-by-step derivation

   1. log1p-expm1-u 98.4%

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

   2. add-sqr-sqrt 97.7%

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

   3. associate-*r* 97.7%

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

7. Applied egg-rr 97.7%

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

   1. pow1 97.7%

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

   2. metadata-eval 97.7%

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

   3. sqrt-pow1 94.8%

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

   4. associate-*l* 94.9%

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

   5. add-sqr-sqrt 95.4%

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

   6. *-commutative 95.4%

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

   7. associate-*l* 95.4%

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

   8. expm1-log1p-u 95.5%

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

   9. associate-*l* 95.5%

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

   10. *-commutative 95.5%

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

   11. add-sqr-sqrt 94.9%

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

   12. associate-*l* 94.8%

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

9. Applied egg-rr 98.4%

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

10. Final simplification 98.4%

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

Alternative 2: 94.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(\pi \cdot 2\right)\\ \mathbf{if}\;t_0 \leq 0.0020000000949949026:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 - u1 \cdot \left(u1 \cdot -0.5\right)} \cdot \sin t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* PI 2.0))))
   (if (<= t_0 0.0020000000949949026)
     (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))
     (* (sqrt (- u1 (* u1 (* u1 -0.5)))) (sin t_0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u2 * (((float) M_PI) * 2.0f);
	float tmp;
	if (t_0 <= 0.0020000000949949026f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * 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(pi) * Float32(2.0)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0020000000949949026))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * 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) < 0.00200000009

   1. Initial program 59.1%

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

      1. sub-neg 59.1%

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

      2. log1p-def 98.4%

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

      3. associate-*l* 98.4%

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

   3. Simplified 98.4%

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

      1. log1p-expm1-u 98.4%

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

   5. Applied egg-rr 98.4%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(2 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\pi \cdot u2\right)\right)}\right) \]
   6. Step-by-step derivation

      1. log1p-expm1-u 98.4%

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

      2. add-sqr-sqrt 97.9%

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

      3. 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 98.0%

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

   if 0.00200000009 < (*.f32 (*.f32 2 (PI.f32)) u2)

   1. Initial program 60.3%

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

   2. Taylor expanded in u1 around 0 85.9%

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

      1. +-commutative 85.9%

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

      2. mul-1-neg 85.9%

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

      3. unsub-neg 85.9%

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

      4. unpow2 85.9%

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

      5. associate-*r* 85.9%

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

   4. Simplified 85.9%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.0020000000949949026:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 - u1 \cdot \left(u1 \cdot -0.5\right)} \cdot \sin \left(u2 \cdot \left(\pi \cdot 2\right)\right)\\ \end{array} \]

Alternative 3: 90.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(\pi \cdot u2\right)\\ \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.019999999552965164:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\sin t_0 \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* 2.0 (* PI u2))))
   (if (<= (* u2 (* PI 2.0)) 0.019999999552965164)
     (* (sqrt (- (log1p (- u1)))) t_0)
     (* (sin t_0) (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = 2.0f * (((float) M_PI) * u2);
	float tmp;
	if ((u2 * (((float) M_PI) * 2.0f)) <= 0.019999999552965164f) {
		tmp = sqrtf(-log1pf(-u1)) * t_0;
	} else {
		tmp = sinf(t_0) * sqrtf(u1);
	}
	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(u2 * Float32(Float32(pi) * Float32(2.0))) <= Float32(0.019999999552965164))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * t_0);
	else
		tmp = Float32(sin(t_0) * sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 2 (PI.f32)) u2) < 0.0199999996

   1. Initial program 60.6%

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

      1. sub-neg 60.6%

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

      2. log1p-def 98.4%

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

      3. associate-*l* 98.4%

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

   3. Simplified 98.4%

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

      1. log1p-expm1-u 98.4%

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

   5. Applied egg-rr 98.4%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(2 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\pi \cdot u2\right)\right)}\right) \]
   6. Step-by-step derivation

      1. log1p-expm1-u 98.4%

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

      2. add-sqr-sqrt 97.8%

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

      3. 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 94.9%

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

   if 0.0199999996 < (*.f32 (*.f32 2 (PI.f32)) u2)

   1. Initial program 56.6%

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

      1. sub-neg 56.6%

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

      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-udef 56.6%

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

      2. sub-neg 56.6%

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

      3. add-sqr-sqrt 56.5%

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

      4. pow2 56.5%

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

   5. Applied egg-rr 72.4%

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

   6. Taylor expanded in u1 around 0 74.7%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.019999999552965164:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 4: 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 59.6%

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

   1. sub-neg 59.6%

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

   2. log1p-def 98.4%

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

   3. associate-*l* 98.4%

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

3. Simplified 98.4%

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

4. Final simplification 98.4%

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

Alternative 5: 76.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.6%

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

   1. sub-neg 59.6%

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

   2. log1p-def 98.4%

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

   3. associate-*l* 98.4%

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

3. Simplified 98.4%

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

   1. log1p-udef 59.6%

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

   2. sub-neg 59.6%

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

   3. add-sqr-sqrt 59.4%

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

   4. pow2 59.4%

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

5. Applied egg-rr 73.0%

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

6. Taylor expanded in u1 around 0 75.3%

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

7. Final simplification 75.3%

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

Alternative 6: 66.4% 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 59.6%

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

2. Taylor expanded in u1 around 0 75.3%

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

   1. mul-1-neg 75.3%

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

4. Simplified 75.3%

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

5. Taylor expanded in u2 around 0 66.1%

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

   1. associate-*r* 66.1%

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

   2. associate-*r* 66.1%

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

7. Simplified 66.1%

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

8. Taylor expanded in u2 around 0 66.1%

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

   1. associate-*l* 66.2%

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

10. Simplified 66.2%

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

11. Final simplification 66.2%

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

Reproduce

herbie shell --seed 2023182 
(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))))