Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.8% → 98.0%

Time: 5.5s

Alternatives: 7

Speedup: N/A×

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: 57.8% 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, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.035999998450279236:\\ \;\;\;\;\sqrt{-1 \cdot \left(\left(\frac{-1 \cdot \left(\frac{1}{u1} + 0.5\right)}{u1 \cdot u1} - \mathsf{fma}\left(\frac{1}{u1}, 0.3333333333333333, 0.25\right)\right) \cdot {u1}^{4}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 \cdot \log \left(1 - u1\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.035999998450279236)
   (*
    (sqrt
     (*
      -1.0
      (*
       (-
        (/ (* -1.0 (+ (/ 1.0 u1) 0.5)) (* u1 u1))
        (fma (/ 1.0 u1) 0.3333333333333333 0.25))
       (pow u1 4.0))))
    (sin (* (* 2.0 PI) u2)))
   (*
    (sqrt (* -1.0 (log (- 1.0 u1))))
    (* 2.0 (* (sin (* PI u2)) (cos (* PI u2)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u1 <= 0.035999998450279236f) {
		tmp = sqrtf((-1.0f * ((((-1.0f * ((1.0f / u1) + 0.5f)) / (u1 * u1)) - fmaf((1.0f / u1), 0.3333333333333333f, 0.25f)) * powf(u1, 4.0f)))) * sinf(((2.0f * ((float) M_PI)) * u2));
	} else {
		tmp = sqrtf((-1.0f * logf((1.0f - u1)))) * (2.0f * (sinf((((float) M_PI) * u2)) * cosf((((float) M_PI) * u2))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.035999998450279236))
		tmp = Float32(sqrt(Float32(Float32(-1.0) * Float32(Float32(Float32(Float32(Float32(-1.0) * Float32(Float32(Float32(1.0) / u1) + Float32(0.5))) / Float32(u1 * u1)) - fma(Float32(Float32(1.0) / u1), Float32(0.3333333333333333), Float32(0.25))) * (u1 ^ Float32(4.0))))) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)));
	else
		tmp = Float32(sqrt(Float32(Float32(-1.0) * log(Float32(Float32(1.0) - u1)))) * Float32(Float32(2.0) * Float32(sin(Float32(Float32(pi) * u2)) * cos(Float32(Float32(pi) * u2)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0359999985

   1. Initial program 48.2%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lower--.f32 N/A

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

      10. 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. Taylor expanded in u1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f32 N/A

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

      9. lift-/.f32 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f32 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f32 N/A

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

      15. lift-/.f32 N/A

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

      16. lower-pow.f32 98.4

          \[\leadsto \sqrt{-\left(\frac{-1 \cdot \left(\frac{1}{u1} + 0.5\right)}{u1 \cdot u1} - \mathsf{fma}\left(\frac{1}{u1}, 0.3333333333333333, 0.25\right)\right) \cdot {u1}^{4}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   8. Applied rewrites 98.4%

      \[\leadsto \sqrt{-\left(\frac{-1 \cdot \left(\frac{1}{u1} + 0.5\right)}{u1 \cdot u1} - \mathsf{fma}\left(\frac{1}{u1}, 0.3333333333333333, 0.25\right)\right) \cdot \color{blue}{{u1}^{4}}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   if 0.0359999985 < u1

   1. Initial program 96.7%

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

      1. lift-sin.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-PI.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. sin-2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-sin.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lower-cos.f32 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f32 N/A

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

      17. lift-PI.f32 96.7

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

   4. Applied rewrites 96.7%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u1 \leq 0.035999998450279236:\\ \;\;\;\;\sqrt{-1 \cdot \left(\left(\frac{-1 \cdot \left(\frac{1}{u1} + 0.5\right)}{u1 \cdot u1} - \mathsf{fma}\left(\frac{1}{u1}, 0.3333333333333333, 0.25\right)\right) \cdot {u1}^{4}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 \cdot \log \left(1 - u1\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;u1 \leq 0.035999998450279236:\\ \;\;\;\;\sqrt{-1 \cdot \left(\left(\frac{-1 \cdot \left(\frac{1}{u1} + 0.5\right)}{u1 \cdot u1} - \mathsf{fma}\left(\frac{1}{u1}, 0.3333333333333333, 0.25\right)\right) \cdot {u1}^{4}\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 \cdot \log \left(1 - u1\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* (* 2.0 PI) u2))))
   (if (<= u1 0.035999998450279236)
     (*
      (sqrt
       (*
        -1.0
        (*
         (-
          (/ (* -1.0 (+ (/ 1.0 u1) 0.5)) (* u1 u1))
          (fma (/ 1.0 u1) 0.3333333333333333 0.25))
         (pow u1 4.0))))
      t_0)
     (* (sqrt (* -1.0 (log (- 1.0 u1)))) t_0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sinf(((2.0f * ((float) M_PI)) * u2));
	float tmp;
	if (u1 <= 0.035999998450279236f) {
		tmp = sqrtf((-1.0f * ((((-1.0f * ((1.0f / u1) + 0.5f)) / (u1 * u1)) - fmaf((1.0f / u1), 0.3333333333333333f, 0.25f)) * powf(u1, 4.0f)))) * t_0;
	} else {
		tmp = sqrtf((-1.0f * logf((1.0f - u1)))) * t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))
	tmp = Float32(0.0)
	if (u1 <= Float32(0.035999998450279236))
		tmp = Float32(sqrt(Float32(Float32(-1.0) * Float32(Float32(Float32(Float32(Float32(-1.0) * Float32(Float32(Float32(1.0) / u1) + Float32(0.5))) / Float32(u1 * u1)) - fma(Float32(Float32(1.0) / u1), Float32(0.3333333333333333), Float32(0.25))) * (u1 ^ Float32(4.0))))) * t_0);
	else
		tmp = Float32(sqrt(Float32(Float32(-1.0) * log(Float32(Float32(1.0) - u1)))) * t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0359999985

   1. Initial program 48.2%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lower--.f32 N/A

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

      10. 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. Taylor expanded in u1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f32 N/A

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

      9. lift-/.f32 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f32 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f32 N/A

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

      15. lift-/.f32 N/A

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

      16. lower-pow.f32 98.4

          \[\leadsto \sqrt{-\left(\frac{-1 \cdot \left(\frac{1}{u1} + 0.5\right)}{u1 \cdot u1} - \mathsf{fma}\left(\frac{1}{u1}, 0.3333333333333333, 0.25\right)\right) \cdot {u1}^{4}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   8. Applied rewrites 98.4%

      \[\leadsto \sqrt{-\left(\frac{-1 \cdot \left(\frac{1}{u1} + 0.5\right)}{u1 \cdot u1} - \mathsf{fma}\left(\frac{1}{u1}, 0.3333333333333333, 0.25\right)\right) \cdot \color{blue}{{u1}^{4}}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   if 0.0359999985 < u1

   1. Initial program 96.7%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u1 \leq 0.035999998450279236:\\ \;\;\;\;\sqrt{-1 \cdot \left(\left(\frac{-1 \cdot \left(\frac{1}{u1} + 0.5\right)}{u1 \cdot u1} - \mathsf{fma}\left(\frac{1}{u1}, 0.3333333333333333, 0.25\right)\right) \cdot {u1}^{4}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 \cdot \log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;u1 \leq 0.03799999877810478:\\ \;\;\;\;\sqrt{-1 \cdot \left(\left(\frac{-1 \cdot \left(\frac{1}{u1} + 0.5\right)}{u1 \cdot u1} - \mathsf{fma}\left(\frac{1}{u1}, 0.3333333333333333, 0.25\right)\right) \cdot {u1}^{4}\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 \cdot \log \left(\left(\frac{1}{u1} - 1\right) \cdot u1\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* (* 2.0 PI) u2))))
   (if (<= u1 0.03799999877810478)
     (*
      (sqrt
       (*
        -1.0
        (*
         (-
          (/ (* -1.0 (+ (/ 1.0 u1) 0.5)) (* u1 u1))
          (fma (/ 1.0 u1) 0.3333333333333333 0.25))
         (pow u1 4.0))))
      t_0)
     (* (sqrt (* -1.0 (log (* (- (/ 1.0 u1) 1.0) u1)))) t_0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sinf(((2.0f * ((float) M_PI)) * u2));
	float tmp;
	if (u1 <= 0.03799999877810478f) {
		tmp = sqrtf((-1.0f * ((((-1.0f * ((1.0f / u1) + 0.5f)) / (u1 * u1)) - fmaf((1.0f / u1), 0.3333333333333333f, 0.25f)) * powf(u1, 4.0f)))) * t_0;
	} else {
		tmp = sqrtf((-1.0f * logf((((1.0f / u1) - 1.0f) * u1)))) * t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))
	tmp = Float32(0.0)
	if (u1 <= Float32(0.03799999877810478))
		tmp = Float32(sqrt(Float32(Float32(-1.0) * Float32(Float32(Float32(Float32(Float32(-1.0) * Float32(Float32(Float32(1.0) / u1) + Float32(0.5))) / Float32(u1 * u1)) - fma(Float32(Float32(1.0) / u1), Float32(0.3333333333333333), Float32(0.25))) * (u1 ^ Float32(4.0))))) * t_0);
	else
		tmp = Float32(sqrt(Float32(Float32(-1.0) * log(Float32(Float32(Float32(Float32(1.0) / u1) - Float32(1.0)) * u1)))) * t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0379999988

   1. Initial program 48.4%

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

   3. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower--.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lower--.f32 N/A

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

      10. lower-*.f32 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in u1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f32 N/A

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

      9. lift-/.f32 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f32 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f32 N/A

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

      15. lift-/.f32 N/A

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

      16. lower-pow.f32 98.4

          \[\leadsto \sqrt{-\left(\frac{-1 \cdot \left(\frac{1}{u1} + 0.5\right)}{u1 \cdot u1} - \mathsf{fma}\left(\frac{1}{u1}, 0.3333333333333333, 0.25\right)\right) \cdot {u1}^{4}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   8. Applied rewrites 98.4%

      \[\leadsto \sqrt{-\left(\frac{-1 \cdot \left(\frac{1}{u1} + 0.5\right)}{u1 \cdot u1} - \mathsf{fma}\left(\frac{1}{u1}, 0.3333333333333333, 0.25\right)\right) \cdot \color{blue}{{u1}^{4}}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   if 0.0379999988 < u1

   1. Initial program 96.7%

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

   3. Taylor expanded in u1 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 96.2

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

   5. Applied rewrites 96.2%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u1 \leq 0.03799999877810478:\\ \;\;\;\;\sqrt{-1 \cdot \left(\left(\frac{-1 \cdot \left(\frac{1}{u1} + 0.5\right)}{u1 \cdot u1} - \mathsf{fma}\left(\frac{1}{u1}, 0.3333333333333333, 0.25\right)\right) \cdot {u1}^{4}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 \cdot \log \left(\left(\frac{1}{u1} - 1\right) \cdot u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.2% accurate, N/A× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (- (* (- (* -0.25 u1) 0.3333333333333333) u1) 0.5)))
   (*
    (sqrt
     (*
      -1.0
      (*
       (/
        (- (* (* t_0 u1) (* (- (* -0.3333333333333333 u1) 0.5) u1)) 1.0)
        (fma t_0 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) - 0.5f;
	return sqrtf((-1.0f * (((((t_0 * u1) * (((-0.3333333333333333f * u1) - 0.5f) * u1)) - 1.0f) / fmaf(t_0, u1, 1.0f)) * u1))) * sinf(((2.0f * ((float) M_PI)) * u2));
}

Julia

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

Derivation

1. Initial program 54.8%

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

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower--.f32 N/A

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

   10. lower-*.f32 94.1

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

5. Applied rewrites 94.1%

   \[\leadsto \sqrt{-\color{blue}{\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
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. 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. flip-- N/A

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

   8. lower-/.f32 N/A

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

7. Applied rewrites 94.0%

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

8. Taylor expanded in u1 around 0

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

10. Applied rewrites 94.8%

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

11. Final simplification 94.8%

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

Alternative 5: 93.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \sqrt{-1 \cdot \left(\left(\frac{-1 \cdot \left(\frac{1}{u1} + 0.5\right)}{u1 \cdot u1} - \mathsf{fma}\left(\frac{1}{u1}, 0.3333333333333333, 0.25\right)\right) \cdot {u1}^{4}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt
   (*
    -1.0
    (*
     (-
      (/ (* -1.0 (+ (/ 1.0 u1) 0.5)) (* u1 u1))
      (fma (/ 1.0 u1) 0.3333333333333333 0.25))
     (pow u1 4.0))))
  (sin (* (* 2.0 PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((-1.0f * ((((-1.0f * ((1.0f / u1) + 0.5f)) / (u1 * u1)) - fmaf((1.0f / u1), 0.3333333333333333f, 0.25f)) * powf(u1, 4.0f)))) * sinf(((2.0f * ((float) M_PI)) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(Float32(-1.0) * Float32(Float32(Float32(Float32(Float32(-1.0) * Float32(Float32(Float32(1.0) / u1) + Float32(0.5))) / Float32(u1 * u1)) - fma(Float32(Float32(1.0) / u1), Float32(0.3333333333333333), Float32(0.25))) * (u1 ^ Float32(4.0))))) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end

Derivation

1. Initial program 54.8%

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

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower--.f32 N/A

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

   10. lower-*.f32 94.1

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

5. Applied rewrites 94.1%

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

6. Taylor expanded in u1 around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 N/A

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

   12. +-commutative N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f32 N/A

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

   15. lift-/.f32 N/A

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

   16. lower-pow.f32 94.2

       \[\leadsto \sqrt{-\left(\frac{-1 \cdot \left(\frac{1}{u1} + 0.5\right)}{u1 \cdot u1} - \mathsf{fma}\left(\frac{1}{u1}, 0.3333333333333333, 0.25\right)\right) \cdot {u1}^{4}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

8. Applied rewrites 94.2%

   \[\leadsto \sqrt{-\left(\frac{-1 \cdot \left(\frac{1}{u1} + 0.5\right)}{u1 \cdot u1} - \mathsf{fma}\left(\frac{1}{u1}, 0.3333333333333333, 0.25\right)\right) \cdot \color{blue}{{u1}^{4}}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

9. Final simplification 94.2%

   \[\leadsto \sqrt{-1 \cdot \left(\left(\frac{-1 \cdot \left(\frac{1}{u1} + 0.5\right)}{u1 \cdot u1} - \mathsf{fma}\left(\frac{1}{u1}, 0.3333333333333333, 0.25\right)\right) \cdot {u1}^{4}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
10. Add Preprocessing

Alternative 6: 93.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \sqrt{-1 \cdot \left(\left(\frac{-0.5 \cdot u1 - 1}{{u1}^{3}} - \mathsf{fma}\left(\frac{1}{u1}, 0.3333333333333333, 0.25\right)\right) \cdot {u1}^{4}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt
   (*
    -1.0
    (*
     (-
      (/ (- (* -0.5 u1) 1.0) (pow u1 3.0))
      (fma (/ 1.0 u1) 0.3333333333333333 0.25))
     (pow u1 4.0))))
  (sin (* (* 2.0 PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((-1.0f * (((((-0.5f * u1) - 1.0f) / powf(u1, 3.0f)) - fmaf((1.0f / u1), 0.3333333333333333f, 0.25f)) * powf(u1, 4.0f)))) * sinf(((2.0f * ((float) M_PI)) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(Float32(-1.0) * Float32(Float32(Float32(Float32(Float32(Float32(-0.5) * u1) - Float32(1.0)) / (u1 ^ Float32(3.0))) - fma(Float32(Float32(1.0) / u1), Float32(0.3333333333333333), Float32(0.25))) * (u1 ^ Float32(4.0))))) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end

Derivation

1. Initial program 54.8%

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

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower--.f32 N/A

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

   10. lower-*.f32 94.1

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

5. Applied rewrites 94.1%

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

6. Taylor expanded in u1 around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 N/A

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

   12. +-commutative N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f32 N/A

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

   15. lift-/.f32 N/A

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

   16. lower-pow.f32 94.2

       \[\leadsto \sqrt{-\left(\frac{-1 \cdot \left(\frac{1}{u1} + 0.5\right)}{u1 \cdot u1} - \mathsf{fma}\left(\frac{1}{u1}, 0.3333333333333333, 0.25\right)\right) \cdot {u1}^{4}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

8. Applied rewrites 94.2%

   \[\leadsto \sqrt{-\left(\frac{-1 \cdot \left(\frac{1}{u1} + 0.5\right)}{u1 \cdot u1} - \mathsf{fma}\left(\frac{1}{u1}, 0.3333333333333333, 0.25\right)\right) \cdot \color{blue}{{u1}^{4}}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

9. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{-\left(\frac{\frac{-1}{2} \cdot u1 - 1}{{u1}^{3}} - \mathsf{fma}\left(\frac{1}{u1}, \frac{1}{3}, \frac{1}{4}\right)\right) \cdot {u1}^{4}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
10. Step-by-step derivation

    1. lower-/.f32 N/A

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

    2. lower--.f32 N/A

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

    3. lower-*.f32 N/A

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

    4. lift-pow.f32 94.1

       \[\leadsto \sqrt{-\left(\frac{-0.5 \cdot u1 - 1}{{u1}^{3}} - \mathsf{fma}\left(\frac{1}{u1}, 0.3333333333333333, 0.25\right)\right) \cdot {u1}^{4}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

11. Applied rewrites 94.1%

    \[\leadsto \sqrt{-\left(\frac{-0.5 \cdot u1 - 1}{{u1}^{3}} - \mathsf{fma}\left(\frac{1}{u1}, 0.3333333333333333, 0.25\right)\right) \cdot {u1}^{4}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

12. Final simplification 94.1%

    \[\leadsto \sqrt{-1 \cdot \left(\left(\frac{-0.5 \cdot u1 - 1}{{u1}^{3}} - \mathsf{fma}\left(\frac{1}{u1}, 0.3333333333333333, 0.25\right)\right) \cdot {u1}^{4}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
13. Add Preprocessing

Alternative 7: 92.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\\ t_1 := \mathsf{fma}\left(t\_0, u1, 1\right)\\ e^{\log \left(-1 \cdot \left(\left(\frac{\left(t\_0 \cdot u1\right) \cdot \left(\left(-0.3333333333333333 \cdot u1 - 0.5\right) \cdot u1\right)}{t\_1} - \frac{1}{t\_1}\right) \cdot u1\right)\right) \cdot 0.5} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (- (* (- (* -0.25 u1) 0.3333333333333333) u1) 0.5))
        (t_1 (fma t_0 u1 1.0)))
   (*
    (exp
     (*
      (log
       (*
        -1.0
        (*
         (-
          (/ (* (* t_0 u1) (* (- (* -0.3333333333333333 u1) 0.5) u1)) t_1)
          (/ 1.0 t_1))
         u1)))
      0.5))
    (sin (* (* 2.0 PI) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (((-0.25f * u1) - 0.3333333333333333f) * u1) - 0.5f;
	float t_1 = fmaf(t_0, u1, 1.0f);
	return expf((logf((-1.0f * (((((t_0 * u1) * (((-0.3333333333333333f * u1) - 0.5f) * u1)) / t_1) - (1.0f / t_1)) * u1))) * 0.5f)) * sinf(((2.0f * ((float) M_PI)) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(Float32(Float32(-0.25) * u1) - Float32(0.3333333333333333)) * u1) - Float32(0.5))
	t_1 = fma(t_0, u1, Float32(1.0))
	return Float32(exp(Float32(log(Float32(Float32(-1.0) * Float32(Float32(Float32(Float32(Float32(t_0 * u1) * Float32(Float32(Float32(Float32(-0.3333333333333333) * u1) - Float32(0.5)) * u1)) / t_1) - Float32(Float32(1.0) / t_1)) * u1))) * Float32(0.5))) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end

Derivation

1. Initial program 54.8%

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

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower--.f32 N/A

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

   10. lower-*.f32 94.1

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

5. Applied rewrites 94.1%

   \[\leadsto \sqrt{-\color{blue}{\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1 - 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
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. 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. flip-- N/A

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

   8. lower-/.f32 N/A

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

7. Applied rewrites 94.0%

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

8. Taylor expanded in u1 around 0

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

10. Applied rewrites 94.8%

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

12. Applied rewrites 92.6%

    \[\leadsto \color{blue}{e^{\log \left(-\left(\frac{\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1\right) \cdot \left(\left(-0.3333333333333333 \cdot u1 - 0.5\right) \cdot u1\right)}{\mathsf{fma}\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5, u1, 1\right)} - \frac{1}{\mathsf{fma}\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5, u1, 1\right)}\right) \cdot u1\right) \cdot 0.5}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

13. Final simplification 92.6%

    \[\leadsto e^{\log \left(-1 \cdot \left(\left(\frac{\left(\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5\right) \cdot u1\right) \cdot \left(\left(-0.3333333333333333 \cdot u1 - 0.5\right) \cdot u1\right)}{\mathsf{fma}\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5, u1, 1\right)} - \frac{1}{\mathsf{fma}\left(\left(-0.25 \cdot u1 - 0.3333333333333333\right) \cdot u1 - 0.5, u1, 1\right)}\right) \cdot u1\right)\right) \cdot 0.5} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025064 
(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))))