Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.9% → 98.4%

Time: 13.1s

Alternatives: 9

Speedup: 2.0×

• Unsound rule application detected in e-graph. Results may not be sound.

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: 57.9% 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, 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 56.3%

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

   1. sub-neg 56.3%

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

   2. log1p-def 98.4%

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

   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 2: 90.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.006500000134110451:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\pi \cdot \left(2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 (* 2.0 PI)) 0.006500000134110451)
   (* (sqrt (- (log1p (- u1)))) (* PI (* 2.0 u2)))
   (* (sin (* 2.0 (* PI u2))) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * (2.0f * ((float) M_PI))) <= 0.006500000134110451f) {
		tmp = sqrtf(-log1pf(-u1)) * (((float) M_PI) * (2.0f * u2));
	} else {
		tmp = sinf((2.0f * (((float) M_PI) * u2))) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(Float32(2.0) * Float32(pi))) <= Float32(0.006500000134110451))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(pi) * Float32(Float32(2.0) * u2)));
	else
		tmp = Float32(sin(Float32(Float32(2.0) * Float32(Float32(pi) * u2))) * sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 2 (PI.f32)) u2) < 0.00650000013

   1. Initial program 58.4%

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

      1. sub-neg 58.4%

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

      2. log1p-def 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. Step-by-step derivation

      1. add-cbrt-cube 98.5%

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

      2. add-cbrt-cube 98.5%

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

      3. cbrt-unprod 98.5%

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

      4. pow3 98.5%

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

      5. pow3 98.4%

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

   5. Applied egg-rr 98.4%

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

   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 0.00650000013 < (*.f32 (*.f32 2 (PI.f32)) u2)

   1. Initial program 51.5%

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

      1. sub-neg 51.5%

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

      2. log1p-def 98.0%

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

      3. associate-*l* 98.0%

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

   3. Simplified 98.0%

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

      1. log1p-udef 51.5%

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

      2. sub-neg 51.5%

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

      3. add-sqr-sqrt 51.5%

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

      4. pow2 51.5%

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

   5. Applied egg-rr 78.6%

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

   6. Taylor expanded in u1 around 0 80.5%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.006500000134110451:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\pi \cdot \left(2 \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 3: 93.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(2 \cdot u2\right)\\ \mathbf{if}\;u1 \leq 0.00800000037997961:\\ \;\;\;\;\sin t_0 \cdot \sqrt{u1 + 0.5 \cdot \left(u1 \cdot u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* PI (* 2.0 u2))))
   (if (<= u1 0.00800000037997961)
     (* (sin t_0) (sqrt (+ u1 (* 0.5 (* u1 u1)))))
     (* (sqrt (- (log1p (- u1)))) t_0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = ((float) M_PI) * (2.0f * u2);
	float tmp;
	if (u1 <= 0.00800000037997961f) {
		tmp = sinf(t_0) * sqrtf((u1 + (0.5f * (u1 * u1))));
	} else {
		tmp = sqrtf(-log1pf(-u1)) * t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(pi) * Float32(Float32(2.0) * u2))
	tmp = Float32(0.0)
	if (u1 <= Float32(0.00800000037997961))
		tmp = Float32(sin(t_0) * sqrt(Float32(u1 + Float32(Float32(0.5) * Float32(u1 * u1)))));
	else
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.00800000038

   1. Initial program 45.7%

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

      1. associate-*r* 45.7%

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

      2. add-cube-cbrt 45.7%

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)} \cdot \sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right) \cdot \sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)} \]

      3. pow3 45.7%

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{{\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3}} \]

   3. Applied egg-rr 45.7%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{{\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3}} \]

   4. Taylor expanded in u1 around 0 96.1%

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

      1. +-commutative 96.1%

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

      2. mul-1-neg 96.1%

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

      3. unsub-neg 96.1%

         \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} - u1\right)}} \cdot {\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3} \]

      4. unpow2 96.1%

         \[\leadsto \sqrt{-\left(-0.5 \cdot \color{blue}{\left(u1 \cdot u1\right)} - u1\right)} \cdot {\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3} \]

      5. associate-*r* 96.1%

         \[\leadsto \sqrt{-\left(\color{blue}{\left(-0.5 \cdot u1\right) \cdot u1} - u1\right)} \cdot {\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3} \]

   6. Simplified 96.1%

      \[\leadsto \sqrt{-\color{blue}{\left(\left(-0.5 \cdot u1\right) \cdot u1 - u1\right)}} \cdot {\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3} \]

   7. Taylor expanded in u2 around inf 97.5%

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

      1. sub-neg 97.5%

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

      2. +-commutative 97.5%

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

      3. remove-double-neg 97.5%

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

      4. distribute-neg-in 97.5%

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

      5. *-commutative 97.5%

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

      6. unpow2 97.5%

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

      7. associate-*r* 97.5%

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

      8. sub-neg 97.5%

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

      9. associate-*r* 97.5%

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

      10. *-commutative 97.5%

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

      11. *-commutative 97.5%

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

      12. sub-neg 97.5%

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

      13. associate-*r* 97.5%

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

      14. unpow2 97.5%

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

      15. *-commutative 97.5%

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

      16. distribute-neg-in 97.5%

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

   9. Simplified 97.5%

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

   if 0.00800000038 < u1

   1. Initial program 95.0%

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

      1. sub-neg 95.0%

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

      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. Step-by-step derivation

      1. add-cbrt-cube 98.5%

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

      2. add-cbrt-cube 98.4%

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

      3. cbrt-unprod 98.6%

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

      4. pow3 98.6%

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

      5. pow3 98.5%

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

   5. Applied egg-rr 98.5%

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

   6. Taylor expanded in u2 around 0 87.5%

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

      1. associate-*r* 87.5%

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

      2. *-commutative 87.5%

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

   8. Simplified 87.5%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(\pi \cdot \left(2 \cdot u2\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u1 \leq 0.00800000037997961:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot u2\right)\right) \cdot \sqrt{u1 + 0.5 \cdot \left(u1 \cdot u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\pi \cdot \left(2 \cdot u2\right)\right)\\ \end{array} \]

Alternative 4: 83.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.0010000000474974513:\\ \;\;\;\;\left(\pi \cdot u2\right) \cdot \left(2 \cdot \sqrt{u1 + 0.5 \cdot \left(u1 \cdot u1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.0010000000474974513)
   (* (* PI u2) (* 2.0 (sqrt (+ u1 (* 0.5 (* u1 u1))))))
   (* (sin (* 2.0 (* PI u2))) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.0010000000474974513f) {
		tmp = (((float) M_PI) * u2) * (2.0f * sqrtf((u1 + (0.5f * (u1 * u1)))));
	} else {
		tmp = sinf((2.0f * (((float) M_PI) * u2))) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.0010000000474974513))
		tmp = Float32(Float32(Float32(pi) * u2) * Float32(Float32(2.0) * sqrt(Float32(u1 + Float32(Float32(0.5) * Float32(u1 * u1))))));
	else
		tmp = Float32(sin(Float32(Float32(2.0) * Float32(Float32(pi) * u2))) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (u2 <= single(0.0010000000474974513))
		tmp = (single(pi) * u2) * (single(2.0) * sqrt((u1 + (single(0.5) * (u1 * u1)))));
	else
		tmp = sin((single(2.0) * (single(pi) * u2))) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.00100000005

   1. Initial program 58.4%

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

      1. associate-*r* 58.4%

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

      2. add-cube-cbrt 58.4%

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)} \cdot \sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right) \cdot \sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)} \]

      3. pow3 58.4%

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{{\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3}} \]

   3. Applied egg-rr 58.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{{\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3}} \]

   4. Taylor expanded in u1 around 0 87.0%

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

      1. +-commutative 87.0%

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

      2. mul-1-neg 87.0%

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

      3. unsub-neg 87.0%

         \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} - u1\right)}} \cdot {\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3} \]

      4. unpow2 87.0%

         \[\leadsto \sqrt{-\left(-0.5 \cdot \color{blue}{\left(u1 \cdot u1\right)} - u1\right)} \cdot {\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3} \]

      5. associate-*r* 87.0%

         \[\leadsto \sqrt{-\left(\color{blue}{\left(-0.5 \cdot u1\right) \cdot u1} - u1\right)} \cdot {\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3} \]

   6. Simplified 87.0%

      \[\leadsto \sqrt{-\color{blue}{\left(\left(-0.5 \cdot u1\right) \cdot u1 - u1\right)}} \cdot {\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3} \]

   7. Taylor expanded in u2 around 0 87.4%

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

      1. associate-*r* 87.4%

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

      2. cancel-sign-sub-inv 87.4%

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

      3. metadata-eval 87.4%

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

      4. unpow2 87.4%

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

      5. *-commutative 87.4%

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

   9. Simplified 87.4%

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

   if 0.00100000005 < u2

   1. Initial program 51.5%

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

      1. sub-neg 51.5%

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

      2. log1p-def 98.0%

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

      3. associate-*l* 98.0%

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

   3. Simplified 98.0%

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

      1. log1p-udef 51.5%

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

      2. sub-neg 51.5%

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

      3. add-sqr-sqrt 51.5%

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

      4. pow2 51.5%

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

   5. Applied egg-rr 78.6%

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

   6. Taylor expanded in u1 around 0 80.5%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \leq 0.0010000000474974513:\\ \;\;\;\;\left(\pi \cdot u2\right) \cdot \left(2 \cdot \sqrt{u1 + 0.5 \cdot \left(u1 \cdot u1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{u1}\\ \end{array} \]

Alternative 5: 74.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\pi \cdot \left(u2 \cdot \sqrt{u1 + 0.5 \cdot \left(u1 \cdot u1\right)}\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* 2.0 (* PI (* u2 (sqrt (+ u1 (* 0.5 (* u1 u1))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return 2.0f * (((float) M_PI) * (u2 * sqrtf((u1 + (0.5f * (u1 * u1))))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(2.0) * Float32(Float32(pi) * Float32(u2 * sqrt(Float32(u1 + Float32(Float32(0.5) * Float32(u1 * u1)))))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(2.0) * (single(pi) * (u2 * sqrt((u1 + (single(0.5) * (u1 * u1))))));
end

Derivation

1. Initial program 56.3%

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

   1. associate-*r* 56.3%

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

   2. add-cube-cbrt 56.3%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)} \cdot \sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right) \cdot \sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)} \]

   3. pow3 56.3%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{{\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3}} \]

3. Applied egg-rr 56.3%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{{\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3}} \]

4. Taylor expanded in u1 around 0 87.7%

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

   1. +-commutative 87.7%

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

   2. mul-1-neg 87.7%

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

   3. unsub-neg 87.7%

      \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} - u1\right)}} \cdot {\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3} \]

   4. unpow2 87.7%

      \[\leadsto \sqrt{-\left(-0.5 \cdot \color{blue}{\left(u1 \cdot u1\right)} - u1\right)} \cdot {\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3} \]

   5. associate-*r* 87.7%

      \[\leadsto \sqrt{-\left(\color{blue}{\left(-0.5 \cdot u1\right) \cdot u1} - u1\right)} \cdot {\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3} \]

6. Simplified 87.7%

   \[\leadsto \sqrt{-\color{blue}{\left(\left(-0.5 \cdot u1\right) \cdot u1 - u1\right)}} \cdot {\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3} \]

7. Taylor expanded in u2 around 0 74.5%

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

   1. *-commutative 74.5%

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

   2. *-commutative 74.5%

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

   3. associate-*l* 74.5%

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

   4. cancel-sign-sub-inv 74.5%

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

   5. metadata-eval 74.5%

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

   6. unpow2 74.5%

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

9. Simplified 74.5%

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

10. Final simplification 74.5%

    \[\leadsto 2 \cdot \left(\pi \cdot \left(u2 \cdot \sqrt{u1 + 0.5 \cdot \left(u1 \cdot u1\right)}\right)\right) \]

Alternative 6: 74.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(\pi \cdot u2\right) \cdot \left(2 \cdot \sqrt{u1 + 0.5 \cdot \left(u1 \cdot u1\right)}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (* PI u2) (* 2.0 (sqrt (+ u1 (* 0.5 (* u1 u1)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return (((float) M_PI) * u2) * (2.0f * sqrtf((u1 + (0.5f * (u1 * u1)))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(Float32(pi) * u2) * Float32(Float32(2.0) * sqrt(Float32(u1 + Float32(Float32(0.5) * Float32(u1 * u1))))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (single(pi) * u2) * (single(2.0) * sqrt((u1 + (single(0.5) * (u1 * u1)))));
end

Derivation

1. Initial program 56.3%

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

   1. associate-*r* 56.3%

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

   2. add-cube-cbrt 56.3%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)} \cdot \sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right) \cdot \sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)} \]

   3. pow3 56.3%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{{\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3}} \]

3. Applied egg-rr 56.3%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{{\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3}} \]

4. Taylor expanded in u1 around 0 87.7%

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

   1. +-commutative 87.7%

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

   2. mul-1-neg 87.7%

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

   3. unsub-neg 87.7%

      \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} - u1\right)}} \cdot {\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3} \]

   4. unpow2 87.7%

      \[\leadsto \sqrt{-\left(-0.5 \cdot \color{blue}{\left(u1 \cdot u1\right)} - u1\right)} \cdot {\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3} \]

   5. associate-*r* 87.7%

      \[\leadsto \sqrt{-\left(\color{blue}{\left(-0.5 \cdot u1\right) \cdot u1} - u1\right)} \cdot {\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3} \]

6. Simplified 87.7%

   \[\leadsto \sqrt{-\color{blue}{\left(\left(-0.5 \cdot u1\right) \cdot u1 - u1\right)}} \cdot {\left(\sqrt[3]{\sin \left(2 \cdot \left(\pi \cdot u2\right)\right)}\right)}^{3} \]

7. Taylor expanded in u2 around 0 74.5%

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

   1. associate-*r* 74.5%

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

   2. cancel-sign-sub-inv 74.5%

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

   3. metadata-eval 74.5%

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

   4. unpow2 74.5%

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

   5. *-commutative 74.5%

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

9. Simplified 74.5%

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

10. Final simplification 74.5%

    \[\leadsto \left(\pi \cdot u2\right) \cdot \left(2 \cdot \sqrt{u1 + 0.5 \cdot \left(u1 \cdot u1\right)}\right) \]

Alternative 7: 66.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.3%

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

2. Taylor expanded in u1 around 0 77.8%

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

   1. mul-1-neg 77.8%

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

4. Simplified 77.8%

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

5. Taylor expanded in u2 around 0 66.6%

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

   1. associate-*l* 66.7%

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

7. Simplified 66.7%

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

8. Taylor expanded in u2 around 0 66.6%

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

   1. *-commutative 66.6%

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

   2. associate-*l* 66.6%

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

10. Simplified 66.6%

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

    1. add-sqr-sqrt 66.6%

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

    2. sqrt-unprod 66.6%

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

    3. *-commutative 66.6%

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

    4. *-commutative 66.6%

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

    5. swap-sqr 66.6%

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

    6. add-sqr-sqrt 66.7%

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

12. Applied egg-rr 66.7%

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

13. Final simplification 66.7%

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

Alternative 8: 66.3% 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 56.3%

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

2. Taylor expanded in u1 around 0 77.8%

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

   1. mul-1-neg 77.8%

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

4. Simplified 77.8%

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

5. Taylor expanded in u2 around 0 66.6%

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

   1. associate-*l* 66.7%

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

7. Simplified 66.7%

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

8. Final simplification 66.7%

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

Alternative 9: 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 56.3%

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

2. Taylor expanded in u1 around 0 77.8%

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

   1. mul-1-neg 77.8%

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

4. Simplified 77.8%

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

   1. associate-*r* 77.8%

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

   2. add-cube-cbrt 77.3%

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

   3. pow3 77.4%

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

   4. add-exp-log 76.6%

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

   5. pow-exp 76.3%

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

6. Applied egg-rr 76.3%

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

7. Taylor expanded in u2 around 0 7.2%

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

8. Final simplification 7.2%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023178 
(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))))