Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.4% → 98.3%

Time: 5.1s

Alternatives: 10

Speedup: 4.7×

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: 57.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf((1.0f - u1))) * sinf(((2.0f * ((float) M_PI)) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(-log((single(1.0) - u1))) * sin(((single(2.0) * single(pi)) * u2));
end

Alternative 1: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\left|\pi\right|}\\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\mathsf{fma}\left(\left(\sqrt[3]{\pi} \cdot u2\right) \cdot t\_0, t\_0, u2 \cdot \pi\right)\right) \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.4%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-flip N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 98.3

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

3. Applied rewrites 98.3%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. associate-*l* N/A

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

   6. lift-PI.f32 N/A

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

   7. add-cube-cbrt N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-cbrt.f32 N/A

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

   10. associate-*l* N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-cbrt.f32 N/A

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

   13. lift-PI.f32 N/A

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

   14. lift-cbrt.f32 N/A

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

   15. lift-*.f32 N/A

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

   16. associate-*r* N/A

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

   17. lift-*.f32 N/A

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

5. Applied rewrites 98.3%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\left(\sqrt[3]{\pi} \cdot u2\right) \cdot \sqrt[3]{\left|\pi\right|}, \sqrt[3]{\left|\pi\right|}, u2 \cdot \pi\right)\right)} \]
6. Add Preprocessing

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.4%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-flip N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 98.3

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

3. Applied rewrites 98.3%

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

   1. lift-*.f32 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f32 98.3

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

5. Applied rewrites 98.3%

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

Alternative 3: 97.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ t_1 := \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq -0.003700000001117587:\\ \;\;\;\;\sqrt{-t\_0} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))) (t_1 (sin (* (+ PI PI) u2))))
   (if (<= t_0 -0.003700000001117587)
     (* (sqrt (- t_0)) t_1)
     (* (sqrt (* u1 (+ 1.0 (* 0.5 u1)))) t_1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float t_1 = sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	float tmp;
	if (t_0 <= -0.003700000001117587f) {
		tmp = sqrtf(-t_0) * t_1;
	} else {
		tmp = sqrtf((u1 * (1.0f + (0.5f * u1)))) * t_1;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = log(Float32(Float32(1.0) - u1))
	t_1 = sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.003700000001117587))
		tmp = Float32(sqrt(Float32(-t_0)) * t_1);
	else
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * u1)))) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = log((single(1.0) - u1));
	t_1 = sin(((single(pi) + single(pi)) * u2));
	tmp = single(0.0);
	if (t_0 <= single(-0.003700000001117587))
		tmp = sqrt(-t_0) * t_1;
	else
		tmp = sqrt((u1 * (single(1.0) + (single(0.5) * u1)))) * t_1;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0037

   1. Initial program 57.4%

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

      1. lift-*.f32 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f32 57.4

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

   3. Applied rewrites 57.4%

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

   if -0.0037 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 57.4%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.3

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

   3. Applied rewrites 98.3%

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

      1. lift-*.f32 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f32 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 87.9

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

   8. Applied rewrites 87.9%

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

Alternative 4: 94.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.00014000000373926014:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(u2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.00014000000373926014)
   (* (sqrt (- (log1p (- u1)))) (* 2.0 (* u2 PI)))
   (* (sqrt (* u1 (+ 1.0 (* 0.5 u1)))) (sin (* (+ PI PI) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.00014000000373926014f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (u2 * ((float) M_PI)));
	} else {
		tmp = sqrtf((u1 * (1.0f + (0.5f * u1)))) * sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.00014000000373926014))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(u2 * Float32(pi))));
	else
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * u1)))) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 1.40000004e-4

   1. Initial program 57.4%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.3

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

   3. Applied rewrites 98.3%

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

   4. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 81.6

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

   6. Applied rewrites 81.6%

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

   if 1.40000004e-4 < u2

   1. Initial program 57.4%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.3

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

   3. Applied rewrites 98.3%

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

      1. lift-*.f32 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f32 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 87.9

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

   8. Applied rewrites 87.9%

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

Alternative 5: 90.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.0020000000949949026f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (u2 * ((float) M_PI)));
	} else {
		tmp = sqrtf(u1) * sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.0020000000949949026))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(u2 * Float32(pi))));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.00200000009

   1. Initial program 57.4%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.3

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

   3. Applied rewrites 98.3%

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

   4. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 81.6

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

   6. Applied rewrites 81.6%

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

   if 0.00200000009 < u2

   1. Initial program 57.4%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.3

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

   3. Applied rewrites 98.3%

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

      1. lift-*.f32 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f32 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in u1 around 0

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

   8. Applied rewrites 76.7%

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

Alternative 6: 81.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.4%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-flip N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 98.3

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

3. Applied rewrites 98.3%

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

4. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 81.6

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

6. Applied rewrites 81.6%

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

Alternative 7: 77.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.00016999999934341758:\\ \;\;\;\;2 \cdot \left(u2 \cdot \sqrt{\left(\pi \cdot \pi\right) \cdot u1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(u2 + u2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right) \cdot \pi\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.00016999999934341758)
   (* 2.0 (* u2 (sqrt (* (* PI PI) u1))))
   (* (* (+ u2 u2) (sqrt (- (log (- 1.0 u1))))) PI)))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.00016999999934341758f) {
		tmp = 2.0f * (u2 * sqrtf(((((float) M_PI) * ((float) M_PI)) * u1)));
	} else {
		tmp = ((u2 + u2) * sqrtf(-logf((1.0f - u1)))) * ((float) M_PI);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.00016999999934341758))
		tmp = Float32(Float32(2.0) * Float32(u2 * sqrt(Float32(Float32(Float32(pi) * Float32(pi)) * u1))));
	else
		tmp = Float32(Float32(Float32(u2 + u2) * sqrt(Float32(-log(Float32(Float32(1.0) - u1))))) * Float32(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (u1 <= single(0.00016999999934341758))
		tmp = single(2.0) * (u2 * sqrt(((single(pi) * single(pi)) * u1)));
	else
		tmp = ((u2 + u2) * sqrt(-log((single(1.0) - u1)))) * single(pi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u1 < 1.69999999e-4

   1. Initial program 57.4%

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

   2. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-neg.f32 N/A

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

      7. lower-log.f32 N/A

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

      8. lower--.f32 50.5

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

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-PI.f32 N/A

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

      3. lower-sqrt.f32 66.6

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

   7. Applied rewrites 66.6%

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

      1. lift-*.f32 N/A

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

      2. lift-PI.f32 N/A

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

      3. add-sqr-sqrt N/A

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

      4. lift-PI.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. sqrt-prod N/A

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

      7. lift-PI.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. lift-sqrt.f32 N/A

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

      10. sqrt-unprod N/A

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

      11. lower-sqrt.f32 N/A

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

      12. lower-*.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lift-PI.f32 N/A

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

      15. lower-*.f32 66.6

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

   9. Applied rewrites 66.6%

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

   if 1.69999999e-4 < u1

   1. Initial program 57.4%

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

   2. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-neg.f32 N/A

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

      7. lower-log.f32 N/A

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

      8. lower--.f32 50.5

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

   4. Applied rewrites 50.5%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f32 50.5

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

   6. Applied rewrites 50.5%

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

Alternative 8: 77.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.00016999999934341758:\\ \;\;\;\;2 \cdot \left(u2 \cdot \sqrt{\left(\pi \cdot \pi\right) \cdot u1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{-\log \left(1 - u1\right)} \cdot \left(\pi + \pi\right)\right) \cdot u2\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.00016999999934341758)
   (* 2.0 (* u2 (sqrt (* (* PI PI) u1))))
   (* (* (sqrt (- (log (- 1.0 u1)))) (+ PI PI)) u2)))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.00016999999934341758f) {
		tmp = 2.0f * (u2 * sqrtf(((((float) M_PI) * ((float) M_PI)) * u1)));
	} else {
		tmp = (sqrtf(-logf((1.0f - u1))) * (((float) M_PI) + ((float) M_PI))) * u2;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.00016999999934341758))
		tmp = Float32(Float32(2.0) * Float32(u2 * sqrt(Float32(Float32(Float32(pi) * Float32(pi)) * u1))));
	else
		tmp = Float32(Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * Float32(Float32(pi) + Float32(pi))) * u2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (u1 <= single(0.00016999999934341758))
		tmp = single(2.0) * (u2 * sqrt(((single(pi) * single(pi)) * u1)));
	else
		tmp = (sqrt(-log((single(1.0) - u1))) * (single(pi) + single(pi))) * u2;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u1 < 1.69999999e-4

   1. Initial program 57.4%

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

   2. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-neg.f32 N/A

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

      7. lower-log.f32 N/A

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

      8. lower--.f32 50.5

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

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-PI.f32 N/A

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

      3. lower-sqrt.f32 66.6

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

   7. Applied rewrites 66.6%

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

      1. lift-*.f32 N/A

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

      2. lift-PI.f32 N/A

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

      3. add-sqr-sqrt N/A

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

      4. lift-PI.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. sqrt-prod N/A

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

      7. lift-PI.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. lift-sqrt.f32 N/A

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

      10. sqrt-unprod N/A

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

      11. lower-sqrt.f32 N/A

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

      12. lower-*.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lift-PI.f32 N/A

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

      15. lower-*.f32 66.6

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

   9. Applied rewrites 66.6%

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

   if 1.69999999e-4 < u1

   1. Initial program 57.4%

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

   2. Taylor expanded in u2 around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-neg.f32 N/A

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

      7. lower-log.f32 N/A

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

      8. lower--.f32 50.5

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

   4. Applied rewrites 50.5%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f32 N/A

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

      6. count-2-rev N/A

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

      7. lift-*.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. distribute-rgt-out N/A

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

      10. count-2-rev N/A

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

      11. lift-*.f32 N/A

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

      12. lower-*.f32 50.5

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

      13. lift-*.f32 N/A

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

      14. count-2-rev N/A

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

      15. lower-+.f32 50.5

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

   6. Applied rewrites 50.5%

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

Alternative 9: 66.6% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

2. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-neg.f32 N/A

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

   7. lower-log.f32 N/A

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

   8. lower--.f32 50.5

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

4. Applied rewrites 50.5%

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

5. Taylor expanded in u1 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 N/A

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

   3. lower-sqrt.f32 66.6

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

7. Applied rewrites 66.6%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. add-sqr-sqrt N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. sqrt-prod N/A

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

   7. lift-PI.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-sqrt.f32 N/A

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

   10. sqrt-unprod N/A

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

   11. lower-sqrt.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lift-PI.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lower-*.f32 66.6

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

9. Applied rewrites 66.6%

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

Alternative 10: 66.6% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

2. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-neg.f32 N/A

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

   7. lower-log.f32 N/A

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

   8. lower--.f32 50.5

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

4. Applied rewrites 50.5%

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

5. Taylor expanded in u1 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 N/A

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

   3. lower-sqrt.f32 66.6

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

7. Applied rewrites 66.6%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. count-2 N/A

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

   5. lift-+.f32 N/A

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

   6. lower-*.f32 66.6

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 66.6

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

9. Applied rewrites 66.6%

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

Reproduce

herbie shell --seed 2025162 
(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))))