Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.6% → 98.4%

Time: 12.4s

Alternatives: 8

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: 57.6% 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, 0.6× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((2.0f * ((float) M_PI)));
	return sqrtf(-log1pf(-u1)) * sinf((t_0 * (u2 * t_0)));
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(Float32(2.0) * Float32(pi)))
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * sin(Float32(t_0 * Float32(u2 * t_0))))
end

Derivation

1. Initial program 53.7%

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

   1. sub-neg 53.7%

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

   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. Simplified 98.3%

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

   1. add-cube-cbrt 97.0%

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

   2. pow3 97.0%

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

   3. associate-*l* 97.0%

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

5. Applied egg-rr 97.0%

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

   1. rem-cube-cbrt 98.3%

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

   2. associate-*r* 98.3%

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

   3. *-commutative 98.3%

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

   4. add-sqr-sqrt 98.3%

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

   5. associate-*r* 98.4%

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

7. Applied egg-rr 98.4%

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

8. Final simplification 98.4%

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

Alternative 2: 98.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 53.7%

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

   1. sub-neg 53.7%

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

   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. Simplified 98.3%

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

   1. add-cbrt-cube 98.3%

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

   2. add-cbrt-cube 98.3%

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

   3. cbrt-unprod 98.4%

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

   4. pow3 98.4%

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

   5. pow3 98.3%

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

5. Applied egg-rr 98.3%

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

6. Final simplification 98.3%

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

Alternative 3: 91.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(2 \cdot \pi\right)\\ \mathbf{if}\;t_0 \leq 0.006000000052154064:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\pi \cdot \left(u2 \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t_0 \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* 2.0 PI))))
   (if (<= t_0 0.006000000052154064)
     (* (sqrt (- (log1p (- u1)))) (* PI (* u2 2.0)))
     (* (sin t_0) (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u2 * (2.0f * ((float) M_PI));
	float tmp;
	if (t_0 <= 0.006000000052154064f) {
		tmp = sqrtf(-log1pf(-u1)) * (((float) M_PI) * (u2 * 2.0f));
	} else {
		tmp = sinf(t_0) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u2 * Float32(Float32(2.0) * Float32(pi)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.006000000052154064))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(pi) * Float32(u2 * Float32(2.0))));
	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) < 0.00600000005

   1. Initial program 54.8%

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

      1. sub-neg 54.8%

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

      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. Simplified 98.5%

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

      1. add-cbrt-cube 98.5%

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

      2. add-cbrt-cube 98.4%

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

      3. cbrt-unprod 98.6%

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

      4. pow3 98.6%

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

      5. pow3 98.5%

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

   5. Applied egg-rr 98.5%

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

   6. Taylor expanded in u2 around 0 97.3%

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

      1. associate-*r* 97.3%

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

      2. *-commutative 97.3%

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

   8. Simplified 97.3%

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

   if 0.00600000005 < (*.f32 (*.f32 2 (PI.f32)) u2)

   1. Initial program 51.4%

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

      1. sub-neg 51.4%

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

      2. log1p-def 97.9%

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

   3. Simplified 97.9%

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

      1. neg-mul-1 97.9%

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

      2. log1p-udef 51.4%

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

      3. sub-neg 51.4%

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

      4. neg-mul-1 51.4%

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

      5. add-sqr-sqrt 51.3%

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

      6. pow2 51.3%

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

   5. Applied egg-rr 78.5%

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

   6. Taylor expanded in u1 around 0 80.3%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.006000000052154064:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\pi \cdot \left(u2 \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 53.7%

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

   1. sub-neg 53.7%

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

   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. Simplified 98.3%

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

4. Final simplification 98.3%

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

Alternative 5: 76.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 53.7%

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

   1. sub-neg 53.7%

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

   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. Simplified 98.3%

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

   1. neg-mul-1 98.3%

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

   2. log1p-udef 53.7%

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

   3. sub-neg 53.7%

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

   4. neg-mul-1 53.7%

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

   5. add-sqr-sqrt 53.7%

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

   6. pow2 53.7%

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

5. Applied egg-rr 77.3%

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

6. Taylor expanded in u1 around 0 79.4%

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

7. Final simplification 79.4%

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

Alternative 6: 66.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \pi \cdot \left(u2 \cdot \sqrt{u1 \cdot 4}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (* PI (* u2 (sqrt (* u1 4.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return ((float) M_PI) * (u2 * sqrtf((u1 * 4.0f)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(pi) * Float32(u2 * sqrt(Float32(u1 * Float32(4.0)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(pi) * (u2 * sqrt((u1 * single(4.0))));
end

Derivation

1. Initial program 53.7%

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

2. Taylor expanded in u1 around 0 79.4%

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

   1. mul-1-neg 79.4%

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

4. Simplified 79.4%

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

5. Taylor expanded in u2 around 0 67.7%

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

   1. associate-*r* 67.7%

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

   2. *-commutative 67.7%

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

   3. *-commutative 67.7%

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

   4. *-commutative 67.7%

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

7. Simplified 67.7%

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

   1. expm1-log1p-u 67.7%

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

   2. expm1-udef 25.8%

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

   3. *-commutative 25.8%

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

   4. associate-*l* 25.8%

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

   5. add-sqr-sqrt 25.8%

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

   6. sqrt-unprod 25.8%

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

   7. *-commutative 25.8%

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

   8. *-commutative 25.8%

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

   9. swap-sqr 25.8%

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

   10. add-sqr-sqrt 25.8%

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

   11. metadata-eval 25.8%

       \[\leadsto e^{\mathsf{log1p}\left(\pi \cdot \left(u2 \cdot \sqrt{u1 \cdot \color{blue}{4}}\right)\right)} - 1 \]

9. Applied egg-rr 25.8%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\pi \cdot \left(u2 \cdot \sqrt{u1 \cdot 4}\right)\right)} - 1} \]
10. Step-by-step derivation

    1. expm1-def 67.6%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(u2 \cdot \sqrt{u1 \cdot 4}\right)\right)\right)} \]

    2. expm1-log1p 67.6%

       \[\leadsto \color{blue}{\pi \cdot \left(u2 \cdot \sqrt{u1 \cdot 4}\right)} \]

11. Simplified 67.6%

    \[\leadsto \color{blue}{\pi \cdot \left(u2 \cdot \sqrt{u1 \cdot 4}\right)} \]

12. Final simplification 67.6%

    \[\leadsto \pi \cdot \left(u2 \cdot \sqrt{u1 \cdot 4}\right) \]

Alternative 7: 66.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 53.7%

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

2. Taylor expanded in u1 around 0 79.4%

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

   1. mul-1-neg 79.4%

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

4. Simplified 79.4%

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

5. Taylor expanded in u2 around 0 67.7%

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

   1. associate-*r* 67.7%

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

   2. *-commutative 67.7%

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

   3. *-commutative 67.7%

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

   4. *-commutative 67.7%

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

7. Simplified 67.7%

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

8. Final simplification 67.7%

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

Alternative 8: 7.1% accurate, 409.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return 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 = 0.0e0
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 53.7%

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

   1. sub-neg 53.7%

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

   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. Simplified 98.3%

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

   1. add-cube-cbrt 97.0%

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

   2. pow3 97.0%

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

   3. associate-*l* 97.0%

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

5. Applied egg-rr 97.0%

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

6. Taylor expanded in u2 around 0 7.2%

   \[\leadsto \color{blue}{0} \]

7. Final simplification 7.2%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023305 
(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))))