Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.5% → 98.0%

Time: 4.3s

Alternatives: 15

Speedup: 8.9×

Specification

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.5% 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: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{if}\;u1 \leq 0.01549999974668026:\\ \;\;\;\;\sqrt{\left(\left(\left(0.25 \cdot u1 + 0.3333333333333333\right) \cdot u1 + 0.5\right) \cdot u1\right) \cdot u1 + u1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* (+ PI PI) u2))))
   (if (<= u1 0.01549999974668026)
     (*
      (sqrt
       (+ (* (* (+ (* (+ (* 0.25 u1) 0.3333333333333333) u1) 0.5) u1) u1) u1))
      t_0)
     (* (sqrt (- (log (- 1.0 u1)))) t_0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	float tmp;
	if (u1 <= 0.01549999974668026f) {
		tmp = sqrtf((((((((0.25f * u1) + 0.3333333333333333f) * u1) + 0.5f) * u1) * u1) + u1)) * t_0;
	} else {
		tmp = sqrtf(-logf((1.0f - u1))) * t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2))
	tmp = Float32(0.0)
	if (u1 <= Float32(0.01549999974668026))
		tmp = Float32(sqrt(Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.25) * u1) + Float32(0.3333333333333333)) * u1) + Float32(0.5)) * u1) * u1) + u1)) * t_0);
	else
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = sin(((single(pi) + single(pi)) * u2));
	tmp = single(0.0);
	if (u1 <= single(0.01549999974668026))
		tmp = sqrt((((((((single(0.25) * u1) + single(0.3333333333333333)) * u1) + single(0.5)) * u1) * u1) + u1)) * t_0;
	else
		tmp = sqrt(-log((single(1.0) - u1))) * t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0154999997

   1. Initial program 48.9%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f32 N/A

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

      13. lower-*.f32 98.4

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

   5. Applied rewrites 98.4%

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

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-+.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-+.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. distribute-lft-in N/A

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

      15. lower-+.f32 N/A

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

      16. lower-*.f32 N/A

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

   7. Applied rewrites 98.6%

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

   9. Applied rewrites 98.6%

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

       1. lift-PI.f32 N/A

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

       2. lift-*.f32 N/A

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

       3. *-commutative N/A

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

       4. count-2-rev N/A

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

       5. lower-+.f32 N/A

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

       6. lift-PI.f32 N/A

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

       7. lift-PI.f32 98.6

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

   11. Applied rewrites 98.6%

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

   if 0.0154999997 < u1

   1. Initial program 97.4%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 97.4

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

   4. Applied rewrites 97.4%

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

Alternative 2: 95.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ t_1 := \left(\pi \cdot 2\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq -0.03500000014901161:\\ \;\;\;\;\sqrt{-t\_0} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(0.3333333333333333 \cdot u1 + 0.5\right) \cdot u1\right) \cdot u1 + u1} \cdot \sin t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))) (t_1 (* (* PI 2.0) u2)))
   (if (<= t_0 -0.03500000014901161)
     (* (sqrt (- t_0)) t_1)
     (*
      (sqrt (+ (* (* (+ (* 0.3333333333333333 u1) 0.5) u1) u1) u1))
      (sin t_1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float t_1 = (((float) M_PI) * 2.0f) * u2;
	float tmp;
	if (t_0 <= -0.03500000014901161f) {
		tmp = sqrtf(-t_0) * t_1;
	} else {
		tmp = sqrtf((((((0.3333333333333333f * u1) + 0.5f) * u1) * u1) + u1)) * sinf(t_1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = log(Float32(Float32(1.0) - u1))
	t_1 = Float32(Float32(Float32(pi) * Float32(2.0)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.03500000014901161))
		tmp = Float32(sqrt(Float32(-t_0)) * t_1);
	else
		tmp = Float32(sqrt(Float32(Float32(Float32(Float32(Float32(Float32(0.3333333333333333) * u1) + Float32(0.5)) * u1) * u1) + u1)) * sin(t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = log((single(1.0) - u1));
	t_1 = (single(pi) * single(2.0)) * u2;
	tmp = single(0.0);
	if (t_0 <= single(-0.03500000014901161))
		tmp = sqrt(-t_0) * t_1;
	else
		tmp = sqrt((((((single(0.3333333333333333) * u1) + single(0.5)) * u1) * u1) + u1)) * sin(t_1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0350000001

   1. Initial program 97.5%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lift-PI.f32 77.1

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

   5. Applied rewrites 77.1%

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

   if -0.0350000001 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 50.2%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f32 N/A

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

      13. lower-*.f32 98.4

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

   5. Applied rewrites 98.4%

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

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-+.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-+.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. distribute-lft-in N/A

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

      15. lower-+.f32 N/A

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

      16. lower-*.f32 N/A

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

   7. Applied rewrites 98.5%

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

   9. Applied rewrites 98.5%

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

   10. Taylor expanded in u1 around 0

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

   12. Applied rewrites 98.1%

       \[\leadsto \sqrt{\left(\left(0.3333333333333333 \cdot u1 + 0.5\right) \cdot u1\right) \cdot u1 + u1} \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 95.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ \mathbf{if}\;t\_0 \leq -0.03500000014901161:\\ \;\;\;\;\sqrt{-t\_0} \cdot \left(\left(\pi \cdot 2\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(0.3333333333333333 \cdot u1 + 0.5\right) \cdot u1 + 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1))))
   (if (<= t_0 -0.03500000014901161)
     (* (sqrt (- t_0)) (* (* PI 2.0) u2))
     (*
      (sqrt (* (+ (* (+ (* 0.3333333333333333 u1) 0.5) u1) 1.0) u1))
      (sin (* (* 2.0 PI) u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float tmp;
	if (t_0 <= -0.03500000014901161f) {
		tmp = sqrtf(-t_0) * ((((float) M_PI) * 2.0f) * u2);
	} else {
		tmp = sqrtf((((((0.3333333333333333f * u1) + 0.5f) * u1) + 1.0f) * u1)) * sinf(((2.0f * ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = log(Float32(Float32(1.0) - u1))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.03500000014901161))
		tmp = Float32(sqrt(Float32(-t_0)) * Float32(Float32(Float32(pi) * Float32(2.0)) * u2));
	else
		tmp = Float32(sqrt(Float32(Float32(Float32(Float32(Float32(Float32(0.3333333333333333) * u1) + Float32(0.5)) * u1) + Float32(1.0)) * u1)) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = log((single(1.0) - u1));
	tmp = single(0.0);
	if (t_0 <= single(-0.03500000014901161))
		tmp = sqrt(-t_0) * ((single(pi) * single(2.0)) * u2);
	else
		tmp = sqrt((((((single(0.3333333333333333) * u1) + single(0.5)) * u1) + single(1.0)) * u1)) * sin(((single(2.0) * single(pi)) * u2));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0350000001

   1. Initial program 97.5%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lift-PI.f32 77.1

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

   5. Applied rewrites 77.1%

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

   if -0.0350000001 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 50.2%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f32 N/A

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

      9. lower-*.f32 98.0

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

   5. Applied rewrites 98.0%

      \[\leadsto \sqrt{\color{blue}{\left(\left(0.3333333333333333 \cdot u1 + 0.5\right) \cdot u1 + 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 94.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.25 \cdot u1 + 0.3333333333333333\right) \cdot u1\\ \sqrt{\left(\frac{t\_0 \cdot \left(0.3333333333333333 \cdot u1\right) - 0.25}{t\_0 - 0.5} \cdot u1 + 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (+ (* 0.25 u1) 0.3333333333333333) u1)))
   (*
    (sqrt
     (*
      (+ (* (/ (- (* t_0 (* 0.3333333333333333 u1)) 0.25) (- t_0 0.5)) u1) 1.0)
      u1))
    (sin (* (* 2.0 PI) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = ((0.25f * u1) + 0.3333333333333333f) * u1;
	return sqrtf(((((((t_0 * (0.3333333333333333f * u1)) - 0.25f) / (t_0 - 0.5f)) * u1) + 1.0f) * u1)) * sinf(((2.0f * ((float) M_PI)) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(Float32(0.25) * u1) + Float32(0.3333333333333333)) * u1)
	return Float32(sqrt(Float32(Float32(Float32(Float32(Float32(Float32(t_0 * Float32(Float32(0.3333333333333333) * u1)) - Float32(0.25)) / Float32(t_0 - Float32(0.5))) * u1) + Float32(1.0)) * u1)) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	t_0 = ((single(0.25) * u1) + single(0.3333333333333333)) * u1;
	tmp = sqrt(((((((t_0 * (single(0.3333333333333333) * u1)) - single(0.25)) / (t_0 - single(0.5))) * u1) + single(1.0)) * u1)) * sin(((single(2.0) * single(pi)) * u2));
end

Derivation

1. Initial program 58.9%

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

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f32 N/A

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

   13. lower-*.f32 92.4

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

5. Applied rewrites 92.4%

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

   1. lift-+.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-+.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. flip-+ N/A

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

   6. lower-/.f32 N/A

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

   7. metadata-eval N/A

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

   8. lower--.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-+.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-+.f32 N/A

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

   15. lift-*.f32 N/A

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

7. Applied rewrites 92.4%

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

8. Taylor expanded in u1 around 0

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

10. Applied rewrites 93.5%

    \[\leadsto \sqrt{\left(\frac{\left(\left(0.25 \cdot u1 + 0.3333333333333333\right) \cdot u1\right) \cdot \left(0.3333333333333333 \cdot u1\right) - 0.25}{\left(0.25 \cdot u1 + 0.3333333333333333\right) \cdot u1 - 0.5} \cdot u1 + 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
11. Add Preprocessing

Alternative 5: 93.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(\left(\left(0.25 \cdot u1 + 0.3333333333333333\right) \cdot u1 + 0.5\right) \cdot u1\right) \cdot u1 + u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (+ (* (* (+ (* (+ (* 0.25 u1) 0.3333333333333333) u1) 0.5) u1) u1) u1))
  (sin (* (+ PI PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((((((((0.25f * u1) + 0.3333333333333333f) * u1) + 0.5f) * u1) * u1) + u1)) * sinf(((((float) M_PI) + ((float) M_PI)) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.25) * u1) + Float32(0.3333333333333333)) * u1) + Float32(0.5)) * u1) * u1) + u1)) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((((((((single(0.25) * u1) + single(0.3333333333333333)) * u1) + single(0.5)) * u1) * u1) + u1)) * sin(((single(pi) + single(pi)) * u2));
end

Derivation

1. Initial program 58.9%

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

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f32 N/A

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

   13. lower-*.f32 92.4

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

5. Applied rewrites 92.4%

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

   1. lift-*.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-+.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-+.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. +-commutative N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. +-commutative N/A

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

   13. *-commutative N/A

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

   14. distribute-lft-in N/A

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

   15. lower-+.f32 N/A

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

   16. lower-*.f32 N/A

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

7. Applied rewrites 92.5%

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

9. Applied rewrites 92.5%

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

    1. lift-PI.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. *-commutative N/A

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

    4. count-2-rev N/A

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

    5. lower-+.f32 N/A

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

    6. lift-PI.f32 N/A

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

    7. lift-PI.f32 92.5

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

11. Applied rewrites 92.5%

    \[\leadsto \sqrt{\left(\left(\left(0.25 \cdot u1 + 0.3333333333333333\right) \cdot u1 + 0.5\right) \cdot u1\right) \cdot u1 + u1} \cdot \sin \left(\color{blue}{\left(\pi + \pi\right)} \cdot u2\right) \]
12. Add Preprocessing

Alternative 6: 93.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(\left(\left(0.25 \cdot u1 + 0.3333333333333333\right) \cdot u1 + 0.5\right) \cdot u1 + 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt
   (* (+ (* (+ (* (+ (* 0.25 u1) 0.3333333333333333) u1) 0.5) u1) 1.0) u1))
  (sin (* (+ PI PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((((((((0.25f * u1) + 0.3333333333333333f) * u1) + 0.5f) * u1) + 1.0f) * u1)) * sinf(((((float) M_PI) + ((float) M_PI)) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.25) * u1) + Float32(0.3333333333333333)) * u1) + Float32(0.5)) * u1) + Float32(1.0)) * u1)) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((((((((single(0.25) * u1) + single(0.3333333333333333)) * u1) + single(0.5)) * u1) + single(1.0)) * u1)) * sin(((single(pi) + single(pi)) * u2));
end

Derivation

1. Initial program 58.9%

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

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f32 N/A

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

   13. lower-*.f32 92.4

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

5. Applied rewrites 92.4%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. count-2-rev N/A

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

   4. lift-+.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-PI.f32 92.4

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

7. Applied rewrites 92.4%

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

Alternative 7: 93.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot 2\right) \cdot u2\\ \mathbf{if}\;u1 \leq 0.026000000536441803:\\ \;\;\;\;\sqrt{\left(0.5 \cdot u1\right) \cdot u1 + u1} \cdot \sin t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* PI 2.0) u2)))
   (if (<= u1 0.026000000536441803)
     (* (sqrt (+ (* (* 0.5 u1) u1) u1)) (sin t_0))
     (* (sqrt (- (log (- 1.0 u1)))) t_0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (((float) M_PI) * 2.0f) * u2;
	float tmp;
	if (u1 <= 0.026000000536441803f) {
		tmp = sqrtf((((0.5f * u1) * u1) + u1)) * sinf(t_0);
	} else {
		tmp = sqrtf(-logf((1.0f - u1))) * t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(pi) * Float32(2.0)) * u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.026000000536441803))
		tmp = Float32(sqrt(Float32(Float32(Float32(Float32(0.5) * u1) * u1) + u1)) * sin(t_0));
	else
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = (single(pi) * single(2.0)) * u2;
	tmp = single(0.0);
	if (u1 <= single(0.026000000536441803))
		tmp = sqrt((((single(0.5) * u1) * u1) + u1)) * sin(t_0);
	else
		tmp = sqrt(-log((single(1.0) - u1))) * t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0260000005

   1. Initial program 49.6%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f32 N/A

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

      13. lower-*.f32 98.4

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

   5. Applied rewrites 98.4%

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

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-+.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-+.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. distribute-lft-in N/A

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

      15. lower-+.f32 N/A

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

      16. lower-*.f32 N/A

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

   7. Applied rewrites 98.5%

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

   9. Applied rewrites 98.5%

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

   10. Taylor expanded in u1 around 0

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

   12. Applied rewrites 95.6%

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

   if 0.0260000005 < u1

   1. Initial program 97.4%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lift-PI.f32 76.8

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

   5. Applied rewrites 76.8%

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

Alternative 8: 93.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.026000000536441803:\\ \;\;\;\;\sqrt{\left(0.5 \cdot u1 + 1\right) \cdot u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\pi \cdot 2\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.026000000536441803)
   (* (sqrt (* (+ (* 0.5 u1) 1.0) u1)) (sin (* (+ PI PI) u2)))
   (* (sqrt (- (log (- 1.0 u1)))) (* (* PI 2.0) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.026000000536441803f) {
		tmp = sqrtf((((0.5f * u1) + 1.0f) * u1)) * sinf(((((float) M_PI) + ((float) M_PI)) * u2));
	} else {
		tmp = sqrtf(-logf((1.0f - u1))) * ((((float) M_PI) * 2.0f) * u2);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.026000000536441803))
		tmp = Float32(sqrt(Float32(Float32(Float32(Float32(0.5) * u1) + Float32(1.0)) * u1)) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	else
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * Float32(Float32(Float32(pi) * Float32(2.0)) * u2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (u1 <= single(0.026000000536441803))
		tmp = sqrt((((single(0.5) * u1) + single(1.0)) * u1)) * sin(((single(pi) + single(pi)) * u2));
	else
		tmp = sqrt(-log((single(1.0) - u1))) * ((single(pi) * single(2.0)) * u2);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0260000005

   1. Initial program 49.6%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f32 N/A

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

      5. lower-*.f32 95.5

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

   5. Applied rewrites 95.5%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 95.5

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

   7. Applied rewrites 95.5%

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

   if 0.0260000005 < u1

   1. Initial program 97.4%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lift-PI.f32 76.8

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

   5. Applied rewrites 76.8%

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

Alternative 9: 87.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.001500000013038516:\\ \;\;\;\;\sqrt{\left(\left(\left(0.25 \cdot u1 + 0.3333333333333333\right) \cdot u1 + 0.5\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\pi \cdot 2\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\left(\pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.001500000013038516)
   (*
    (sqrt
     (* (+ (* (+ (* (+ (* 0.25 u1) 0.3333333333333333) u1) 0.5) u1) 1.0) u1))
    (* (* PI 2.0) u2))
   (* (sqrt u1) (sin (* (+ PI PI) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.001500000013038516f) {
		tmp = sqrtf((((((((0.25f * u1) + 0.3333333333333333f) * u1) + 0.5f) * u1) + 1.0f) * u1)) * ((((float) M_PI) * 2.0f) * u2);
	} 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(0.001500000013038516))
		tmp = Float32(sqrt(Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.25) * u1) + Float32(0.3333333333333333)) * u1) + Float32(0.5)) * u1) + Float32(1.0)) * u1)) * Float32(Float32(Float32(pi) * Float32(2.0)) * u2));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (u2 <= single(0.001500000013038516))
		tmp = sqrt((((((((single(0.25) * u1) + single(0.3333333333333333)) * u1) + single(0.5)) * u1) + single(1.0)) * u1)) * ((single(pi) * single(2.0)) * u2);
	else
		tmp = sqrt(u1) * sin(((single(pi) + single(pi)) * u2));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.00150000001

   1. Initial program 58.0%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f32 N/A

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

      13. lower-*.f32 93.2

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

   5. Applied rewrites 93.2%

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

   6. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      6. lift-PI.f32 91.8

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

   8. Applied rewrites 91.8%

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

   if 0.00150000001 < u2

   1. Initial program 61.2%

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

   3. Taylor expanded in u1 around 0

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

   5. Applied rewrites 73.9%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 73.9

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

   7. Applied rewrites 73.9%

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

Alternative 10: 81.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.001500000013038516:\\ \;\;\;\;\sqrt{\left(\left(\left(0.25 \cdot u1 + 0.3333333333333333\right) \cdot u1 + 0.5\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\pi \cdot 2\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(\left(\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right) - -2 \cdot \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.001500000013038516)
   (*
    (sqrt
     (* (+ (* (+ (* (+ (* 0.25 u1) 0.3333333333333333) u1) 0.5) u1) 1.0) u1))
    (* (* PI 2.0) u2))
   (*
    (sqrt u1)
    (*
     (- (* (* -1.3333333333333333 (* u2 u2)) (* (* PI PI) PI)) (* -2.0 PI))
     u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.001500000013038516f) {
		tmp = sqrtf((((((((0.25f * u1) + 0.3333333333333333f) * u1) + 0.5f) * u1) + 1.0f) * u1)) * ((((float) M_PI) * 2.0f) * u2);
	} else {
		tmp = sqrtf(u1) * ((((-1.3333333333333333f * (u2 * u2)) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))) - (-2.0f * ((float) M_PI))) * u2);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.001500000013038516))
		tmp = Float32(sqrt(Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.25) * u1) + Float32(0.3333333333333333)) * u1) + Float32(0.5)) * u1) + Float32(1.0)) * u1)) * Float32(Float32(Float32(pi) * Float32(2.0)) * u2));
	else
		tmp = Float32(sqrt(u1) * Float32(Float32(Float32(Float32(Float32(-1.3333333333333333) * Float32(u2 * u2)) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))) - Float32(Float32(-2.0) * Float32(pi))) * u2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (u2 <= single(0.001500000013038516))
		tmp = sqrt((((((((single(0.25) * u1) + single(0.3333333333333333)) * u1) + single(0.5)) * u1) + single(1.0)) * u1)) * ((single(pi) * single(2.0)) * u2);
	else
		tmp = sqrt(u1) * ((((single(-1.3333333333333333) * (u2 * u2)) * ((single(pi) * single(pi)) * single(pi))) - (single(-2.0) * single(pi))) * u2);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.00150000001

   1. Initial program 58.0%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f32 N/A

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

      13. lower-*.f32 93.2

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

   5. Applied rewrites 93.2%

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

   6. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \sqrt{\left(\left(\left(\frac{1}{4} \cdot u1 + \frac{1}{3}\right) \cdot u1 + \frac{1}{2}\right) \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      6. lift-PI.f32 91.8

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

   8. Applied rewrites 91.8%

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

   if 0.00150000001 < u2

   1. Initial program 61.2%

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

   3. Taylor expanded in u1 around 0

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

   5. Applied rewrites 73.9%

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

   6. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{u1} \cdot \left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{u1} \cdot \left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) - \left(\mathsf{neg}\left(2\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. lower--.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) - \left(\mathsf{neg}\left(2\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. associate-*r* N/A

         \[\leadsto \sqrt{u1} \cdot \left(\left(\left(\frac{-4}{3} \cdot {u2}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3} - \left(\mathsf{neg}\left(2\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(\left(\left(\frac{-4}{3} \cdot {u2}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3} - \left(\mathsf{neg}\left(2\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. lower-*.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(\left(\left(\frac{-4}{3} \cdot {u2}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3} - \left(\mathsf{neg}\left(2\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. unpow2 N/A

         \[\leadsto \sqrt{u1} \cdot \left(\left(\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot {\mathsf{PI}\left(\right)}^{3} - \left(\mathsf{neg}\left(2\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. lower-*.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(\left(\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot {\mathsf{PI}\left(\right)}^{3} - \left(\mathsf{neg}\left(2\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      10. lower-pow.f32 N/A

          \[\leadsto \sqrt{u1} \cdot \left(\left(\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot {\mathsf{PI}\left(\right)}^{3} - \left(\mathsf{neg}\left(2\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      11. lift-PI.f32 N/A

          \[\leadsto \sqrt{u1} \cdot \left(\left(\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{3} - \left(\mathsf{neg}\left(2\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \sqrt{u1} \cdot \left(\left(\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{3} - \left(\mathsf{neg}\left(2\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      13. metadata-eval N/A

          \[\leadsto \sqrt{u1} \cdot \left(\left(\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{3} - -2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      14. lift-PI.f32 55.2

          \[\leadsto \sqrt{u1} \cdot \left(\left(\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{3} - -2 \cdot \pi\right) \cdot u2\right) \]

   8. Applied rewrites 55.2%

      \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(\left(\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{3} - -2 \cdot \pi\right) \cdot u2\right)} \]
   9. Step-by-step derivation

      1. lift-PI.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(\left(\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot {\mathsf{PI}\left(\right)}^{3} - -2 \cdot \pi\right) \cdot u2\right) \]

      2. lift-pow.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(\left(\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot {\mathsf{PI}\left(\right)}^{3} - -2 \cdot \pi\right) \cdot u2\right) \]

      3. unpow3 N/A

         \[\leadsto \sqrt{u1} \cdot \left(\left(\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) - -2 \cdot \pi\right) \cdot u2\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(\left(\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) - -2 \cdot \pi\right) \cdot u2\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(\left(\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) - -2 \cdot \pi\right) \cdot u2\right) \]

      6. lift-PI.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(\left(\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\pi \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) - -2 \cdot \pi\right) \cdot u2\right) \]

      7. lift-PI.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(\left(\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{PI}\left(\right)\right) - -2 \cdot \pi\right) \cdot u2\right) \]

      8. lift-PI.f32 55.2

         \[\leadsto \sqrt{u1} \cdot \left(\left(\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right) - -2 \cdot \pi\right) \cdot u2\right) \]

   10. Applied rewrites 55.2%

       \[\leadsto \sqrt{u1} \cdot \left(\left(\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right) - -2 \cdot \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 78.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.001500000013038516:\\ \;\;\;\;\sqrt{\left(\left(\frac{1}{u1} + 0.5\right) \cdot u1\right) \cdot u1} \cdot \left(\left(\pi \cdot 2\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(\left(\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right) - -2 \cdot \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.001500000013038516)
   (* (sqrt (* (* (+ (/ 1.0 u1) 0.5) u1) u1)) (* (* PI 2.0) u2))
   (*
    (sqrt u1)
    (*
     (- (* (* -1.3333333333333333 (* u2 u2)) (* (* PI PI) PI)) (* -2.0 PI))
     u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.001500000013038516f) {
		tmp = sqrtf(((((1.0f / u1) + 0.5f) * u1) * u1)) * ((((float) M_PI) * 2.0f) * u2);
	} else {
		tmp = sqrtf(u1) * ((((-1.3333333333333333f * (u2 * u2)) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))) - (-2.0f * ((float) M_PI))) * u2);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.001500000013038516))
		tmp = Float32(sqrt(Float32(Float32(Float32(Float32(Float32(1.0) / u1) + Float32(0.5)) * u1) * u1)) * Float32(Float32(Float32(pi) * Float32(2.0)) * u2));
	else
		tmp = Float32(sqrt(u1) * Float32(Float32(Float32(Float32(Float32(-1.3333333333333333) * Float32(u2 * u2)) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))) - Float32(Float32(-2.0) * Float32(pi))) * u2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (u2 <= single(0.001500000013038516))
		tmp = sqrt(((((single(1.0) / u1) + single(0.5)) * u1) * u1)) * ((single(pi) * single(2.0)) * u2);
	else
		tmp = sqrt(u1) * ((((single(-1.3333333333333333) * (u2 * u2)) * ((single(pi) * single(pi)) * single(pi))) - (single(-2.0) * single(pi))) * u2);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.00150000001

   1. Initial program 58.0%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f32 N/A

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

      5. lower-*.f32 87.6

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

   5. Applied rewrites 87.6%

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

   6. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

      6. lift-PI.f32 86.5

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

   8. Applied rewrites 86.5%

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

   9. Taylor expanded in u1 around inf

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-*.f32 N/A

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

       4. lift-/.f32 N/A

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

       5. lift-+.f32 86.6

          \[\leadsto \sqrt{\left(\left(\frac{1}{u1} + 0.5\right) \cdot u1\right) \cdot u1} \cdot \left(\left(\pi \cdot 2\right) \cdot u2\right) \]

   11. Applied rewrites 86.6%

       \[\leadsto \sqrt{\left(\left(\frac{1}{u1} + 0.5\right) \cdot u1\right) \cdot u1} \cdot \left(\left(\pi \cdot 2\right) \cdot u2\right) \]

   if 0.00150000001 < u2

   1. Initial program 61.2%

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

   3. Taylor expanded in u1 around 0

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

   5. Applied rewrites 73.9%

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

   6. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{u1} \cdot \left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{u2}\right) \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \sqrt{u1} \cdot \left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) - \left(\mathsf{neg}\left(2\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. lower--.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) - \left(\mathsf{neg}\left(2\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. associate-*r* N/A

         \[\leadsto \sqrt{u1} \cdot \left(\left(\left(\frac{-4}{3} \cdot {u2}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3} - \left(\mathsf{neg}\left(2\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(\left(\left(\frac{-4}{3} \cdot {u2}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3} - \left(\mathsf{neg}\left(2\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. lower-*.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(\left(\left(\frac{-4}{3} \cdot {u2}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3} - \left(\mathsf{neg}\left(2\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. unpow2 N/A

         \[\leadsto \sqrt{u1} \cdot \left(\left(\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot {\mathsf{PI}\left(\right)}^{3} - \left(\mathsf{neg}\left(2\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. lower-*.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(\left(\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot {\mathsf{PI}\left(\right)}^{3} - \left(\mathsf{neg}\left(2\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      10. lower-pow.f32 N/A

          \[\leadsto \sqrt{u1} \cdot \left(\left(\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot {\mathsf{PI}\left(\right)}^{3} - \left(\mathsf{neg}\left(2\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      11. lift-PI.f32 N/A

          \[\leadsto \sqrt{u1} \cdot \left(\left(\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{3} - \left(\mathsf{neg}\left(2\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      12. lower-*.f32 N/A

          \[\leadsto \sqrt{u1} \cdot \left(\left(\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{3} - \left(\mathsf{neg}\left(2\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      13. metadata-eval N/A

          \[\leadsto \sqrt{u1} \cdot \left(\left(\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{3} - -2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      14. lift-PI.f32 55.2

          \[\leadsto \sqrt{u1} \cdot \left(\left(\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{3} - -2 \cdot \pi\right) \cdot u2\right) \]

   8. Applied rewrites 55.2%

      \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(\left(\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{3} - -2 \cdot \pi\right) \cdot u2\right)} \]
   9. Step-by-step derivation

      1. lift-PI.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(\left(\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot {\mathsf{PI}\left(\right)}^{3} - -2 \cdot \pi\right) \cdot u2\right) \]

      2. lift-pow.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(\left(\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot {\mathsf{PI}\left(\right)}^{3} - -2 \cdot \pi\right) \cdot u2\right) \]

      3. unpow3 N/A

         \[\leadsto \sqrt{u1} \cdot \left(\left(\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) - -2 \cdot \pi\right) \cdot u2\right) \]

      4. lower-*.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(\left(\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) - -2 \cdot \pi\right) \cdot u2\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(\left(\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) - -2 \cdot \pi\right) \cdot u2\right) \]

      6. lift-PI.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(\left(\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\pi \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) - -2 \cdot \pi\right) \cdot u2\right) \]

      7. lift-PI.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(\left(\left(\frac{-4}{3} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{PI}\left(\right)\right) - -2 \cdot \pi\right) \cdot u2\right) \]

      8. lift-PI.f32 55.2

         \[\leadsto \sqrt{u1} \cdot \left(\left(\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right) - -2 \cdot \pi\right) \cdot u2\right) \]

   10. Applied rewrites 55.2%

       \[\leadsto \sqrt{u1} \cdot \left(\left(\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right) - -2 \cdot \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 74.5% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(\left(\frac{1}{u1} + 0.5\right) \cdot u1\right) \cdot u1} \cdot \left(\left(\pi \cdot 2\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (* (* (+ (/ 1.0 u1) 0.5) u1) u1)) (* (* PI 2.0) u2)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(((((1.0f / u1) + 0.5f) * u1) * u1)) * ((((float) M_PI) * 2.0f) * u2);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(Float32(Float32(Float32(Float32(1.0) / u1) + Float32(0.5)) * u1) * u1)) * Float32(Float32(Float32(pi) * Float32(2.0)) * u2))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(((((single(1.0) / u1) + single(0.5)) * u1) * u1)) * ((single(pi) * single(2.0)) * u2);
end

Derivation

1. Initial program 58.9%

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

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f32 N/A

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

   5. lower-*.f32 86.6

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

5. Applied rewrites 86.6%

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

6. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

      \[\leadsto \sqrt{\left(\frac{1}{2} \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \sqrt{\left(\frac{1}{2} \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

   6. lift-PI.f32 73.7

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

8. Applied rewrites 73.7%

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

9. Taylor expanded in u1 around inf

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-*.f32 N/A

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

    4. lift-/.f32 N/A

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

    5. lift-+.f32 73.8

       \[\leadsto \sqrt{\left(\left(\frac{1}{u1} + 0.5\right) \cdot u1\right) \cdot u1} \cdot \left(\left(\pi \cdot 2\right) \cdot u2\right) \]

11. Applied rewrites 73.8%

    \[\leadsto \sqrt{\left(\left(\frac{1}{u1} + 0.5\right) \cdot u1\right) \cdot u1} \cdot \left(\left(\pi \cdot 2\right) \cdot u2\right) \]
12. Add Preprocessing

Alternative 13: 74.5% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.9%

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

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f32 N/A

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

   5. lower-*.f32 86.6

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

5. Applied rewrites 86.6%

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

6. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

      \[\leadsto \sqrt{\left(\frac{1}{2} \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \sqrt{\left(\frac{1}{2} \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

   6. lift-PI.f32 73.7

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

8. Applied rewrites 73.7%

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

   1. lift-*.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. +-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. lower-+.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lift-*.f32 73.8

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

10. Applied rewrites 73.8%

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

11. Final simplification 73.8%

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

Alternative 14: 74.5% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(0.5 \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (* (+ (* 0.5 u1) 1.0) u1)) (* (+ PI PI) u2)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((((0.5f * u1) + 1.0f) * u1)) * ((((float) M_PI) + ((float) M_PI)) * u2);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(Float32(Float32(Float32(0.5) * u1) + Float32(1.0)) * u1)) * Float32(Float32(Float32(pi) + Float32(pi)) * u2))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((((single(0.5) * u1) + single(1.0)) * u1)) * ((single(pi) + single(pi)) * u2);
end

Derivation

1. Initial program 58.9%

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

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f32 N/A

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

   5. lower-*.f32 86.6

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

5. Applied rewrites 86.6%

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

6. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

      \[\leadsto \sqrt{\left(\frac{1}{2} \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \sqrt{\left(\frac{1}{2} \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

   6. lift-PI.f32 73.7

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

8. Applied rewrites 73.7%

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

   1. lift-PI.f32 N/A

      \[\leadsto \sqrt{\left(\frac{1}{2} \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

   2. lift-*.f32 N/A

      \[\leadsto \sqrt{\left(\frac{1}{2} \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \]

   3. *-commutative N/A

      \[\leadsto \sqrt{\left(\frac{1}{2} \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. count-2-rev N/A

      \[\leadsto \sqrt{\left(\frac{1}{2} \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. lower-+.f32 N/A

      \[\leadsto \sqrt{\left(\frac{1}{2} \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   6. lift-PI.f32 N/A

      \[\leadsto \sqrt{\left(\frac{1}{2} \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\pi + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. lift-PI.f32 73.7

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

10. Applied rewrites 73.7%

    \[\leadsto \sqrt{\left(0.5 \cdot u1 + 1\right) \cdot u1} \cdot \left(\left(\pi + \pi\right) \cdot u2\right) \]
11. Add Preprocessing

Alternative 15: 66.4% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.9%

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

3. Taylor expanded in u1 around 0

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

5. Applied rewrites 75.6%

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

6. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 66.1

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

8. Applied rewrites 66.1%

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

Reproduce

herbie shell --seed 2025058 
(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))))