Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.1% → 98.3%

Time: 5.2s

Alternatives: 12

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: 57.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.1%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
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. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. count-2-rev N/A

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

   5. distribute-lft-in N/A

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

   6. distribute-rgt-out N/A

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

   7. lift-PI.f32 N/A

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

   8. add-cube-cbrt N/A

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

   9. lift-PI.f32 N/A

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

   10. pow1/3 N/A

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(\left(\color{blue}{{\pi}^{\frac{1}{3}}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(u2 + u2\right)\right) \]

   11. lift-PI.f32 N/A

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(\left({\pi}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(u2 + u2\right)\right) \]

   12. pow1/3 N/A

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(\left({\pi}^{\frac{1}{3}} \cdot \color{blue}{{\pi}^{\frac{1}{3}}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(u2 + u2\right)\right) \]

   13. pow-prod-up N/A

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(u2 + u2\right)\right) \]

   14. metadata-eval N/A

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

   15. lift-pow.f32 N/A

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

   16. lift-PI.f32 N/A

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

   17. lift-cbrt.f32 N/A

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

   18. lift-+.f32 N/A

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

5. Applied rewrites 98.3%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{\pi}, \sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot u2\right), u2 \cdot \pi\right)\right)} \]
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 57.1%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
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. count-2-rev N/A

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

   6. lift-+.f32 N/A

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

   7. lift-*.f32 98.3

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

5. Applied rewrites 98.3%

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

Alternative 3: 97.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ t_1 := \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq -0.0035000001080334187:\\ \;\;\;\;\sqrt{-t\_0} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))) (t_1 (sin (* (+ PI PI) u2))))
   (if (<= t_0 -0.0035000001080334187)
     (* (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 <= -0.0035000001080334187f) {
		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(-0.0035000001080334187))
		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(-0.0035000001080334187))
		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)) < -0.00350000011

   1. Initial program 57.1%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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 57.1

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

   3. Applied rewrites 57.1%

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

   if -0.00350000011 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 57.1%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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. count-2-rev N/A

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

      6. lift-+.f32 N/A

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

      7. lift-*.f32 98.3

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

   5. Applied rewrites 98.3%

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

   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 88.2

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

   8. Applied rewrites 88.2%

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

Alternative 4: 94.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(1 - u1\right) \leq -0.0017999999690800905:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(u2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (log (- 1.0 u1)) -0.0017999999690800905)
   (* (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 (logf((1.0f - u1)) <= -0.0017999999690800905f) {
		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 (log(Float32(Float32(1.0) - u1)) <= Float32(-0.0017999999690800905))
		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 (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.00179999997

   1. Initial program 57.1%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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 -0.00179999997 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 57.1%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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. count-2-rev N/A

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

      6. lift-+.f32 N/A

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

      7. lift-*.f32 98.3

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

   5. Applied rewrites 98.3%

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

   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 88.2

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

   8. Applied rewrites 88.2%

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

Alternative 5: 93.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.0006062599713914096:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(u2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5 \cdot u1, u1, u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.0006062599713914096)
   (* (sqrt (- (log1p (- u1)))) (* 2.0 (* u2 PI)))
   (* (sqrt (fma (* 0.5 u1) u1 u1)) (sin (* (+ PI PI) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.0006062599713914096f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (u2 * ((float) M_PI)));
	} else {
		tmp = sqrtf(fmaf((0.5f * u1), u1, 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(0.0006062599713914096))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(u2 * Float32(pi))));
	else
		tmp = Float32(sqrt(fma(Float32(Float32(0.5) * u1), u1, u1)) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 6.06259971e-4

   1. Initial program 57.1%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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 6.06259971e-4 < u2

   1. Initial program 57.1%

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

   2. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   3. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \sqrt{u1 \cdot \color{blue}{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. lower-+.f32 N/A

         \[\leadsto \sqrt{u1 \cdot \left(1 + \color{blue}{u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \color{blue}{\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. lower-+.f32 N/A

         \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + \color{blue}{u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)}\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. lower-+.f32 N/A

         \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \color{blue}{\frac{1}{4} \cdot u1}\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      7. lower-*.f32 93.6

         \[\leadsto \sqrt{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 + 0.25 \cdot \color{blue}{u1}\right)\right)\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. Applied rewrites 93.6%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(0.5 + u1 \cdot \left(0.3333333333333333 + 0.25 \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 93.6%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right) \cdot u1, u1, u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)} \]

   7. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{2} \cdot u1, u1, u1\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 88.2%

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

Alternative 6: 90.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.003000000026077032:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(u2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.003000000026077032)
   (* (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 <= 0.003000000026077032f) {
		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(0.003000000026077032))
		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 < 0.00300000003

   1. Initial program 57.1%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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 0.00300000003 < u2

   1. Initial program 57.1%

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

   2. Taylor expanded in u1 around 0

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

   4. Applied rewrites 76.9%

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

      1. lift-*.f32 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f32 76.9

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

   6. Applied rewrites 76.9%

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

Alternative 7: 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 57.1%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
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 8: 77.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.00011999999696854502:\\ \;\;\;\;\left(u2 + u2\right) \cdot \left(\left(u1 \cdot \sqrt{\frac{1}{u1}}\right) \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u2 \cdot \left(\pi + \pi\right)\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.00011999999696854502)
   (* (+ u2 u2) (* (* u1 (sqrt (/ 1.0 u1))) PI))
   (* (* u2 (+ PI PI)) (sqrt (- (log (- 1.0 u1)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.00011999999696854502f) {
		tmp = (u2 + u2) * ((u1 * sqrtf((1.0f / u1))) * ((float) M_PI));
	} else {
		tmp = (u2 * (((float) M_PI) + ((float) M_PI))) * sqrtf(-logf((1.0f - u1)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.00011999999696854502))
		tmp = Float32(Float32(u2 + u2) * Float32(Float32(u1 * sqrt(Float32(Float32(1.0) / u1))) * Float32(pi)));
	else
		tmp = Float32(Float32(u2 * Float32(Float32(pi) + Float32(pi))) * sqrt(Float32(-log(Float32(Float32(1.0) - u1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (u1 <= single(0.00011999999696854502))
		tmp = (u2 + u2) * ((u1 * sqrt((single(1.0) / u1))) * single(pi));
	else
		tmp = (u2 * (single(pi) + single(pi))) * sqrt(-log((single(1.0) - u1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u1 < 1.19999997e-4

   1. Initial program 57.1%

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

   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 50.4

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

   4. Applied rewrites 50.4%

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

   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. lower-*.f32 N/A

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

      5. count-2-rev N/A

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

      6. lower-+.f32 66.6

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

      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%

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

   10. Taylor expanded in u1 around inf

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

       1. lower-*.f32 N/A

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

       2. lower-sqrt.f32 N/A

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

       3. lower-/.f32 66.6

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

   12. Applied rewrites 66.6%

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

   if 1.19999997e-4 < u1

   1. Initial program 57.1%

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

   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 50.4

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

   4. Applied rewrites 50.4%

      \[\leadsto \color{blue}{2 \cdot \left(u2 \cdot \left(\pi \cdot \sqrt{-\log \left(1 - u1\right)}\right)\right)} \]
   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. lift-*.f32 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f32 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f32 N/A

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

      11. lower-*.f32 50.4

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 50.4

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

      15. lift-*.f32 N/A

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

      16. count-2-rev N/A

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

      17. lift-+.f32 50.4

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

   6. Applied rewrites 50.4%

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

Alternative 9: 77.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.00011999999696854502:\\ \;\;\;\;\left(u2 + u2\right) \cdot \left(\left(u1 \cdot \sqrt{\frac{1}{u1}}\right) \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{-\log \left(1 - u1\right)} \cdot \left(\pi + \pi\right)\right) \cdot u2\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.00011999999696854502)
   (* (+ u2 u2) (* (* u1 (sqrt (/ 1.0 u1))) PI))
   (* (* (sqrt (- (log (- 1.0 u1)))) (+ PI PI)) u2)))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.00011999999696854502f) {
		tmp = (u2 + u2) * ((u1 * sqrtf((1.0f / u1))) * ((float) M_PI));
	} 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(0.00011999999696854502))
		tmp = Float32(Float32(u2 + u2) * Float32(Float32(u1 * sqrt(Float32(Float32(1.0) / u1))) * Float32(pi)));
	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(0.00011999999696854502))
		tmp = (u2 + u2) * ((u1 * sqrt((single(1.0) / u1))) * single(pi));
	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 < 1.19999997e-4

   1. Initial program 57.1%

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

   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 50.4

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

   4. Applied rewrites 50.4%

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

   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. lower-*.f32 N/A

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

      5. count-2-rev N/A

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

      6. lower-+.f32 66.6

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

      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%

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

   10. Taylor expanded in u1 around inf

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

       1. lower-*.f32 N/A

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

       2. lower-sqrt.f32 N/A

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

       3. lower-/.f32 66.6

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

   12. Applied rewrites 66.6%

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

   if 1.19999997e-4 < u1

   1. Initial program 57.1%

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

   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 50.4

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

   4. Applied rewrites 50.4%

      \[\leadsto \color{blue}{2 \cdot \left(u2 \cdot \left(\pi \cdot \sqrt{-\log \left(1 - u1\right)}\right)\right)} \]
   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 50.4

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

      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. lift-+.f32 50.4

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

   6. Applied rewrites 50.4%

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

Alternative 10: 66.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \left(u2 + u2\right) \cdot \left(\left(u1 \cdot \sqrt{\frac{1}{u1}}\right) \cdot \pi\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (+ u2 u2) (* (* u1 (sqrt (/ 1.0 u1))) PI)))

C

float code(float cosTheta_i, float u1, float u2) {
	return (u2 + u2) * ((u1 * sqrtf((1.0f / u1))) * ((float) M_PI));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(u2 + u2) * Float32(Float32(u1 * sqrt(Float32(Float32(1.0) / u1))) * Float32(pi)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (u2 + u2) * ((u1 * sqrt((single(1.0) / u1))) * single(pi));
end

Derivation

1. Initial program 57.1%

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

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 50.4

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

4. Applied rewrites 50.4%

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

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. lower-*.f32 N/A

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

   5. count-2-rev N/A

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

   6. lower-+.f32 66.6

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

   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%

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

10. Taylor expanded in u1 around inf

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

    1. lower-*.f32 N/A

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

    2. lower-sqrt.f32 N/A

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

    3. lower-/.f32 66.6

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

12. Applied rewrites 66.6%

    \[\leadsto \left(u2 + u2\right) \cdot \left(\left(u1 \cdot \sqrt{\frac{1}{u1}}\right) \cdot \pi\right) \]
13. Add Preprocessing

Alternative 11: 66.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \left(u2 + u2\right) \cdot \left(u1 \cdot \left(\pi \cdot \sqrt{\frac{1}{u1}}\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (+ u2 u2) (* u1 (* PI (sqrt (/ 1.0 u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return (u2 + u2) * (u1 * (((float) M_PI) * sqrtf((1.0f / u1))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(u2 + u2) * Float32(u1 * Float32(Float32(pi) * sqrt(Float32(Float32(1.0) / u1)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (u2 + u2) * (u1 * (single(pi) * sqrt((single(1.0) / u1))));
end

Derivation

1. Initial program 57.1%

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

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 50.4

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

4. Applied rewrites 50.4%

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

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. lower-*.f32 N/A

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

   5. count-2-rev N/A

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

   6. lower-+.f32 66.6

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

   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%

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

10. Taylor expanded in u1 around inf

    \[\leadsto \left(u2 + u2\right) \cdot \left(u1 \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\sqrt{\frac{1}{u1}}}\right)\right) \]
11. Step-by-step derivation

    1. lower-*.f32 N/A

       \[\leadsto \left(u2 + u2\right) \cdot \left(u1 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\frac{1}{u1}}\right)\right) \]

    2. lower-*.f32 N/A

       \[\leadsto \left(u2 + u2\right) \cdot \left(u1 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\frac{1}{u1}}\right)\right) \]

    3. lower-PI.f32 N/A

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

    4. lower-sqrt.f32 N/A

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

    5. lower-/.f32 66.6

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

12. Applied rewrites 66.6%

    \[\leadsto \left(u2 + u2\right) \cdot \left(u1 \cdot \left(\pi \cdot \color{blue}{\sqrt{\frac{1}{u1}}}\right)\right) \]
13. Add Preprocessing

Alternative 12: 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 57.1%

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

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 50.4

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

4. Applied rewrites 50.4%

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

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. lower-*.f32 N/A

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

   5. count-2-rev N/A

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

   6. lower-+.f32 66.6

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

   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%

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

Reproduce

herbie shell --seed 2025157 
(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))))