Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.2% → 98.4%

Time: 15.0s

Alternatives: 9

Speedup: 2.9×

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: 57.2% 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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.4%

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

   1. sub-neg 60.4%

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

   2. log1p-def 98.2%

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

3. Simplified 98.2%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)} \]
4. Add Preprocessing
5. 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. *-commutative 97.0%

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

   4. associate-*r* 97.0%

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

6. Applied egg-rr 97.0%

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

   1. rem-cube-cbrt 98.2%

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

   2. associate-*l* 98.2%

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

   3. add-sqr-sqrt 98.2%

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

   4. associate-*r* 98.3%

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

   5. *-commutative 98.3%

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

   6. *-commutative 98.3%

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

8. Applied egg-rr 98.3%

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

9. Final simplification 98.3%

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

Alternative 2: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.4%

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

   1. sub-neg 60.4%

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

   2. log1p-def 98.2%

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

3. Simplified 98.2%

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

5. Final simplification 98.2%

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

Alternative 3: 96.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(\pi \cdot 2\right)\\ \mathbf{if}\;t_0 \leq 0.07999999821186066:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \frac{1}{0.6666666666666666 \cdot \left(u2 \cdot \pi\right) + \frac{1}{u2 \cdot \pi}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t_0 \cdot \sqrt{u1 \cdot \left(u1 \cdot \left(--0.5\right) - -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* PI 2.0))))
   (if (<= t_0 0.07999999821186066)
     (*
      (sqrt (- (log1p (- u1))))
      (* 2.0 (/ 1.0 (+ (* 0.6666666666666666 (* u2 PI)) (/ 1.0 (* u2 PI))))))
     (* (sin t_0) (sqrt (* u1 (- (* u1 (- -0.5)) -1.0)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u2 * (((float) M_PI) * 2.0f);
	float tmp;
	if (t_0 <= 0.07999999821186066f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (1.0f / ((0.6666666666666666f * (u2 * ((float) M_PI))) + (1.0f / (u2 * ((float) M_PI))))));
	} else {
		tmp = sinf(t_0) * sqrtf((u1 * ((u1 * -(-0.5f)) - -1.0f)));
	}
	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(0.07999999821186066))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(1.0) / Float32(Float32(Float32(0.6666666666666666) * Float32(u2 * Float32(pi))) + Float32(Float32(1.0) / Float32(u2 * Float32(pi)))))));
	else
		tmp = Float32(sin(t_0) * sqrt(Float32(u1 * Float32(Float32(u1 * Float32(-Float32(-0.5))) - Float32(-1.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 2 (PI.f32)) u2) < 0.0799999982

   1. Initial program 60.0%

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

      1. sub-neg 60.0%

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

      2. log1p-def 98.4%

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

   3. Simplified 98.4%

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

      1. associate-*l* 98.4%

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

      2. sin-2 98.4%

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

   6. Applied egg-rr 98.4%

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

      1. sin-cos-mult 98.4%

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

      2. clear-num 98.2%

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

      3. +-commutative 98.2%

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

      4. count-2 98.2%

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

      5. *-commutative 98.2%

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

      6. *-commutative 98.2%

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

      7. associate-*r* 98.2%

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

      8. +-inverses 98.2%

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \frac{1}{\frac{2}{\sin \left(u2 \cdot \left(\pi \cdot 2\right)\right) + \sin \color{blue}{0}}}\right) \]

   8. Applied egg-rr 98.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \color{blue}{\frac{1}{\frac{2}{\sin \left(u2 \cdot \left(\pi \cdot 2\right)\right) + \sin 0}}}\right) \]

   9. Taylor expanded in u2 around 0 98.0%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \frac{1}{\color{blue}{0.6666666666666666 \cdot \left(u2 \cdot \pi\right) + \frac{1}{u2 \cdot \pi}}}\right) \]

   if 0.0799999982 < (*.f32 (*.f32 2 (PI.f32)) u2)

   1. Initial program 62.0%

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

   3. Taylor expanded in u1 around 0 86.4%

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

      1. unpow2 86.4%

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

      2. associate-*r* 86.4%

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

      3. distribute-rgt-out 86.2%

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

   5. Simplified 86.2%

      \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(-1 + -0.5 \cdot u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.07999999821186066:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \frac{1}{0.6666666666666666 \cdot \left(u2 \cdot \pi\right) + \frac{1}{u2 \cdot \pi}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{u1 \cdot \left(u1 \cdot \left(--0.5\right) - -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(\pi \cdot 2\right)\\ \mathbf{if}\;t_0 \leq 0.001500000013038516:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(u2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t_0 \cdot \sqrt{u1 \cdot \left(u1 \cdot \left(--0.5\right) - -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* PI 2.0))))
   (if (<= t_0 0.001500000013038516)
     (* (sqrt (- (log1p (- u1)))) (* 2.0 (* u2 PI)))
     (* (sin t_0) (sqrt (* u1 (- (* u1 (- -0.5)) -1.0)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u2 * (((float) M_PI) * 2.0f);
	float tmp;
	if (t_0 <= 0.001500000013038516f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (u2 * ((float) M_PI)));
	} else {
		tmp = sinf(t_0) * sqrtf((u1 * ((u1 * -(-0.5f)) - -1.0f)));
	}
	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(0.001500000013038516))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(u2 * Float32(pi))));
	else
		tmp = Float32(sin(t_0) * sqrt(Float32(u1 * Float32(Float32(u1 * Float32(-Float32(-0.5))) - Float32(-1.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 2 (PI.f32)) u2) < 0.00150000001

   1. Initial program 59.9%

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

      1. sub-neg 59.9%

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

      2. log1p-def 98.4%

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

   3. Simplified 98.4%

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

      1. add-exp-log 93.8%

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

      2. *-commutative 93.8%

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

      3. associate-*r* 93.8%

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

   6. Applied egg-rr 93.8%

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

   7. 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 0.00150000001 < (*.f32 (*.f32 2 (PI.f32)) u2)

   1. Initial program 61.1%

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

   3. Taylor expanded in u1 around 0 85.5%

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

      1. unpow2 85.5%

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

      2. associate-*r* 85.5%

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

      3. distribute-rgt-out 85.4%

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

   5. Simplified 85.4%

      \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(-1 + -0.5 \cdot u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.001500000013038516:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(u2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \sqrt{u1 \cdot \left(u1 \cdot \left(--0.5\right) - -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(u2 \cdot \pi\right)\\ \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.023499999195337296:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* 2.0 (* u2 PI))))
   (if (<= (* u2 (* PI 2.0)) 0.023499999195337296)
     (* (sqrt (- (log1p (- u1)))) t_0)
     (* (sqrt u1) (sin t_0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = 2.0f * (u2 * ((float) M_PI));
	float tmp;
	if ((u2 * (((float) M_PI) * 2.0f)) <= 0.023499999195337296f) {
		tmp = sqrtf(-log1pf(-u1)) * t_0;
	} else {
		tmp = sqrtf(u1) * sinf(t_0);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(2.0) * Float32(u2 * Float32(pi)))
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(Float32(pi) * Float32(2.0))) <= Float32(0.023499999195337296))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * t_0);
	else
		tmp = Float32(sqrt(u1) * sin(t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 2 (PI.f32)) u2) < 0.0234999992

   1. Initial program 60.1%

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

      1. sub-neg 60.1%

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

      2. log1p-def 98.5%

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

   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. Add Preprocessing
   5. Step-by-step derivation

      1. add-exp-log 94.1%

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

      2. *-commutative 94.1%

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

      3. associate-*r* 94.1%

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

   6. Applied egg-rr 94.1%

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

   7. Taylor expanded in u2 around 0 95.7%

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

   if 0.0234999992 < (*.f32 (*.f32 2 (PI.f32)) u2)

   1. Initial program 61.2%

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

   3. Taylor expanded in u1 around 0 73.9%

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

      1. mul-1-neg 73.9%

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

   5. Simplified 73.9%

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

   6. Taylor expanded in u2 around inf 73.9%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.023499999195337296:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(u2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.4%

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

3. Taylor expanded in u1 around 0 74.2%

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

   1. mul-1-neg 74.2%

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

5. Simplified 74.2%

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

6. Taylor expanded in u2 around inf 74.2%

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

7. Final simplification 74.2%

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

Alternative 7: 66.2% accurate, 2.9× 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 60.4%

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

3. Taylor expanded in u1 around 0 74.2%

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

   1. mul-1-neg 74.2%

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

5. Simplified 74.2%

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

6. Taylor expanded in u2 around 0 65.7%

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

8. Simplified 65.7%

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

9. Final simplification 65.7%

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

Alternative 8: 66.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.4%

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

3. Taylor expanded in u1 around 0 74.2%

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

   1. mul-1-neg 74.2%

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

5. Simplified 74.2%

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

6. Taylor expanded in u2 around 0 65.7%

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

8. Simplified 65.7%

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

   1. expm1-log1p-u 65.7%

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

   2. expm1-udef 28.6%

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

   3. associate-*r* 28.6%

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

   4. *-commutative 28.6%

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

   5. associate-*l* 28.6%

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

10. Applied egg-rr 28.6%

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

    1. expm1-def 65.8%

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

    2. expm1-log1p 65.8%

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

    3. associate-*r* 65.8%

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

    4. *-commutative 65.8%

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

12. Simplified 65.8%

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

13. Final simplification 65.8%

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

Alternative 9: 7.1% accurate, 310.0× 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 60.4%

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

   1. sub-neg 60.4%

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

   2. log1p-def 98.2%

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

3. Simplified 98.2%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)} \]
4. Add Preprocessing
5. 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. *-commutative 97.0%

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

   4. associate-*r* 97.0%

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

6. Applied egg-rr 97.0%

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

7. Taylor expanded in u2 around 0 7.1%

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

8. Final simplification 7.1%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024019 
(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))))