Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.6% → 97.9%

Time: 9.3s

Alternatives: 10

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: 57.6% 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: 97.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.03799999877810478:\\ \;\;\;\;\sqrt{-\left(\frac{1 + 0.5 \cdot u1}{{u1}^{3}} + \left(\frac{0.3333333333333333}{u1} + 0.25\right)\right) \cdot \left(-{u1}^{4}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\log \left(\frac{1}{1 - u1}\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.03799999877810478)
   (*
    (sqrt
     (-
      (*
       (+
        (/ (+ 1.0 (* 0.5 u1)) (pow u1 3.0))
        (+ (/ 0.3333333333333333 u1) 0.25))
       (- (pow u1 4.0)))))
    (sin (* (* 2.0 PI) u2)))
   (*
    (sqrt (log (/ 1.0 (- 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 <= 0.03799999877810478f) {
		tmp = sqrtf(-((((1.0f + (0.5f * u1)) / powf(u1, 3.0f)) + ((0.3333333333333333f / u1) + 0.25f)) * -powf(u1, 4.0f))) * sinf(((2.0f * ((float) M_PI)) * u2));
	} else {
		tmp = sqrtf(logf((1.0f / (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(0.03799999877810478))
		tmp = Float32(sqrt(Float32(-Float32(Float32(Float32(Float32(Float32(1.0) + Float32(Float32(0.5) * u1)) / (u1 ^ Float32(3.0))) + Float32(Float32(Float32(0.3333333333333333) / u1) + Float32(0.25))) * Float32(-(u1 ^ Float32(4.0)))))) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)));
	else
		tmp = Float32(sqrt(log(Float32(Float32(1.0) / Float32(Float32(1.0) - u1)))) * Float32(Float32(2.0) * Float32(sin(Float32(Float32(pi) * u2)) * cos(Float32(Float32(pi) * u2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if (u1 <= single(0.03799999877810478))
		tmp = sqrt(-((((single(1.0) + (single(0.5) * u1)) / (u1 ^ single(3.0))) + ((single(0.3333333333333333) / u1) + single(0.25))) * -(u1 ^ single(4.0)))) * sin(((single(2.0) * single(pi)) * u2));
	else
		tmp = sqrt(log((single(1.0) / (single(1.0) - u1)))) * (single(2.0) * (sin((single(pi) * u2)) * cos((single(pi) * u2))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0379999988

   1. Initial program 48.8%

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

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

   8. Applied rewrites 98.2%

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

   9. Taylor expanded in u1 around 0

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

       1. lower-/.f32 N/A

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

       2. lower-+.f32 N/A

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

       3. lower-*.f32 N/A

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

       4. lower-pow.f32 98.4

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

   11. Applied rewrites 98.4%

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

   if 0.0379999988 < 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-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 97.4

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

   4. Applied rewrites 97.4%

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

      1. lift-sin.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-PI.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. sin-2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-sin.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lower-cos.f32 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f32 N/A

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

      17. lift-PI.f32 97.4

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

   6. Applied rewrites 97.4%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u1 \leq 0.03799999877810478:\\ \;\;\;\;\sqrt{-\left(\frac{1 + 0.5 \cdot u1}{{u1}^{3}} + \left(\frac{0.3333333333333333}{u1} + 0.25\right)\right) \cdot \left(-{u1}^{4}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\log \left(\frac{1}{1 - u1}\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{u1}}\\ t_1 := \sin \left(\left(\pi \cdot 2\right) \cdot u2\right)\\ \mathbf{if}\;u1 \leq 0.02800000086426735:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.25 \cdot t\_0, t\_1, \mathsf{fma}\left(0.5 \cdot \sqrt{u1}, \left(0.25 - \frac{0.0625}{u1}\right) \cdot t\_1, \left(0.16666666666666666 \cdot t\_0\right) \cdot t\_1\right) \cdot u1\right), u1 \cdot u1, t\_1 \cdot \sqrt{u1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\log \left(\frac{1}{1 - u1}\right)} \cdot \left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (/ 1.0 (sqrt u1))) (t_1 (sin (* (* PI 2.0) u2))))
   (if (<= u1 0.02800000086426735)
     (fma
      (fma
       (* 0.25 t_0)
       t_1
       (*
        (fma
         (* 0.5 (sqrt u1))
         (* (- 0.25 (/ 0.0625 u1)) t_1)
         (* (* 0.16666666666666666 t_0) t_1))
        u1))
      (* u1 u1)
      (* t_1 (sqrt u1)))
     (*
      (sqrt (log (/ 1.0 (- 1.0 u1))))
      (* 2.0 (* (sin (* PI u2)) (cos (* PI u2))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = 1.0f / sqrtf(u1);
	float t_1 = sinf(((((float) M_PI) * 2.0f) * u2));
	float tmp;
	if (u1 <= 0.02800000086426735f) {
		tmp = fmaf(fmaf((0.25f * t_0), t_1, (fmaf((0.5f * sqrtf(u1)), ((0.25f - (0.0625f / u1)) * t_1), ((0.16666666666666666f * t_0) * t_1)) * u1)), (u1 * u1), (t_1 * sqrtf(u1)));
	} else {
		tmp = sqrtf(logf((1.0f / (1.0f - u1)))) * (2.0f * (sinf((((float) M_PI) * u2)) * cosf((((float) M_PI) * u2))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(1.0) / sqrt(u1))
	t_1 = sin(Float32(Float32(Float32(pi) * Float32(2.0)) * u2))
	tmp = Float32(0.0)
	if (u1 <= Float32(0.02800000086426735))
		tmp = fma(fma(Float32(Float32(0.25) * t_0), t_1, Float32(fma(Float32(Float32(0.5) * sqrt(u1)), Float32(Float32(Float32(0.25) - Float32(Float32(0.0625) / u1)) * t_1), Float32(Float32(Float32(0.16666666666666666) * t_0) * t_1)) * u1)), Float32(u1 * u1), Float32(t_1 * sqrt(u1)));
	else
		tmp = Float32(sqrt(log(Float32(Float32(1.0) / 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 < 0.0280000009

   1. Initial program 47.9%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 45.4

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

   4. Applied rewrites 45.4%

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

   5. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) + {u1}^{2} \cdot \left(\frac{1}{4} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + u1 \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right)\right)} \]

   7. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.25 \cdot \frac{1}{\sqrt{u1}}, \sin \left(\left(\pi \cdot 2\right) \cdot u2\right), \mathsf{fma}\left(0.5 \cdot \sqrt{u1}, \left(0.25 - \frac{0.0625}{u1}\right) \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right), \left(0.16666666666666666 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(\pi \cdot 2\right) \cdot u2\right) \cdot \sqrt{u1}\right)} \]

   if 0.0280000009 < u1

   1. Initial program 97.2%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 97.0

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

   4. Applied rewrites 97.0%

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

      1. lift-sin.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-PI.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. sin-2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-sin.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lower-cos.f32 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f32 N/A

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

      17. lift-PI.f32 97.0

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

   6. Applied rewrites 97.0%

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

Alternative 3: 97.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{u1}}\\ t_1 := \sin \left(\left(\pi \cdot 2\right) \cdot u2\right)\\ \mathbf{if}\;\log \left(1 - u1\right) \leq -0.028999999165534973:\\ \;\;\;\;\sqrt{\log \left(\frac{1}{1 - u1}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.25 \cdot t\_0, t\_1, \mathsf{fma}\left(0.5 \cdot \sqrt{u1}, \left(0.25 - \frac{0.0625}{u1}\right) \cdot t\_1, \left(0.16666666666666666 \cdot t\_0\right) \cdot t\_1\right) \cdot u1\right), u1 \cdot u1, t\_1 \cdot \sqrt{u1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (/ 1.0 (sqrt u1))) (t_1 (sin (* (* PI 2.0) u2))))
   (if (<= (log (- 1.0 u1)) -0.028999999165534973)
     (* (sqrt (log (/ 1.0 (- 1.0 u1)))) (sin (* (* 2.0 PI) u2)))
     (fma
      (fma
       (* 0.25 t_0)
       t_1
       (*
        (fma
         (* 0.5 (sqrt u1))
         (* (- 0.25 (/ 0.0625 u1)) t_1)
         (* (* 0.16666666666666666 t_0) t_1))
        u1))
      (* u1 u1)
      (* t_1 (sqrt u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = 1.0f / sqrtf(u1);
	float t_1 = sinf(((((float) M_PI) * 2.0f) * u2));
	float tmp;
	if (logf((1.0f - u1)) <= -0.028999999165534973f) {
		tmp = sqrtf(logf((1.0f / (1.0f - u1)))) * sinf(((2.0f * ((float) M_PI)) * u2));
	} else {
		tmp = fmaf(fmaf((0.25f * t_0), t_1, (fmaf((0.5f * sqrtf(u1)), ((0.25f - (0.0625f / u1)) * t_1), ((0.16666666666666666f * t_0) * t_1)) * u1)), (u1 * u1), (t_1 * sqrtf(u1)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(1.0) / sqrt(u1))
	t_1 = sin(Float32(Float32(Float32(pi) * Float32(2.0)) * u2))
	tmp = Float32(0.0)
	if (log(Float32(Float32(1.0) - u1)) <= Float32(-0.028999999165534973))
		tmp = Float32(sqrt(log(Float32(Float32(1.0) / Float32(Float32(1.0) - u1)))) * sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)));
	else
		tmp = fma(fma(Float32(Float32(0.25) * t_0), t_1, Float32(fma(Float32(Float32(0.5) * sqrt(u1)), Float32(Float32(Float32(0.25) - Float32(Float32(0.0625) / u1)) * t_1), Float32(Float32(Float32(0.16666666666666666) * t_0) * t_1)) * u1)), Float32(u1 * u1), Float32(t_1 * sqrt(u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0289999992

   1. Initial program 97.2%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 97.0

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

   4. Applied rewrites 97.0%

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

   if -0.0289999992 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 47.9%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 45.4

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

   4. Applied rewrites 45.4%

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

   5. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) + {u1}^{2} \cdot \left(\frac{1}{4} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + u1 \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right)\right)} \]

   7. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.25 \cdot \frac{1}{\sqrt{u1}}, \sin \left(\left(\pi \cdot 2\right) \cdot u2\right), \mathsf{fma}\left(0.5 \cdot \sqrt{u1}, \left(0.25 - \frac{0.0625}{u1}\right) \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right), \left(0.16666666666666666 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(\pi \cdot 2\right) \cdot u2\right) \cdot \sqrt{u1}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 97.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 - u1\right)\\ t_1 := \sqrt{-t\_0}\\ t_2 := \frac{1}{\sqrt{u1}}\\ t_3 := \sin \left(\left(\pi \cdot 2\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq -0.029999999329447746:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({\pi}^{5} \cdot t\_1, 0.26666666666666666, \left(\left(-0.025396825396825397 \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{7}\right) \cdot t\_1\right), u2 \cdot u2, \left({\pi}^{3} \cdot -1.3333333333333333\right) \cdot t\_1\right), u2 \cdot u2, \left(\pi \cdot 2\right) \cdot t\_1\right) \cdot u2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.25 \cdot t\_2, t\_3, \mathsf{fma}\left(0.5 \cdot \sqrt{u1}, \left(0.25 - \frac{0.0625}{u1}\right) \cdot t\_3, \left(0.16666666666666666 \cdot t\_2\right) \cdot t\_3\right) \cdot u1\right), u1 \cdot u1, t\_3 \cdot \sqrt{u1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (log (- 1.0 u1)))
        (t_1 (sqrt (- t_0)))
        (t_2 (/ 1.0 (sqrt u1)))
        (t_3 (sin (* (* PI 2.0) u2))))
   (if (<= t_0 -0.029999999329447746)
     (*
      (fma
       (fma
        (fma
         (* (pow PI 5.0) t_1)
         0.26666666666666666
         (* (* (* -0.025396825396825397 (* u2 u2)) (pow PI 7.0)) t_1))
        (* u2 u2)
        (* (* (pow PI 3.0) -1.3333333333333333) t_1))
       (* u2 u2)
       (* (* PI 2.0) t_1))
      u2)
     (fma
      (fma
       (* 0.25 t_2)
       t_3
       (*
        (fma
         (* 0.5 (sqrt u1))
         (* (- 0.25 (/ 0.0625 u1)) t_3)
         (* (* 0.16666666666666666 t_2) t_3))
        u1))
      (* u1 u1)
      (* t_3 (sqrt u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = logf((1.0f - u1));
	float t_1 = sqrtf(-t_0);
	float t_2 = 1.0f / sqrtf(u1);
	float t_3 = sinf(((((float) M_PI) * 2.0f) * u2));
	float tmp;
	if (t_0 <= -0.029999999329447746f) {
		tmp = fmaf(fmaf(fmaf((powf(((float) M_PI), 5.0f) * t_1), 0.26666666666666666f, (((-0.025396825396825397f * (u2 * u2)) * powf(((float) M_PI), 7.0f)) * t_1)), (u2 * u2), ((powf(((float) M_PI), 3.0f) * -1.3333333333333333f) * t_1)), (u2 * u2), ((((float) M_PI) * 2.0f) * t_1)) * u2;
	} else {
		tmp = fmaf(fmaf((0.25f * t_2), t_3, (fmaf((0.5f * sqrtf(u1)), ((0.25f - (0.0625f / u1)) * t_3), ((0.16666666666666666f * t_2) * t_3)) * u1)), (u1 * u1), (t_3 * sqrtf(u1)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = log(Float32(Float32(1.0) - u1))
	t_1 = sqrt(Float32(-t_0))
	t_2 = Float32(Float32(1.0) / sqrt(u1))
	t_3 = sin(Float32(Float32(Float32(pi) * Float32(2.0)) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.029999999329447746))
		tmp = Float32(fma(fma(fma(Float32((Float32(pi) ^ Float32(5.0)) * t_1), Float32(0.26666666666666666), Float32(Float32(Float32(Float32(-0.025396825396825397) * Float32(u2 * u2)) * (Float32(pi) ^ Float32(7.0))) * t_1)), Float32(u2 * u2), Float32(Float32((Float32(pi) ^ Float32(3.0)) * Float32(-1.3333333333333333)) * t_1)), Float32(u2 * u2), Float32(Float32(Float32(pi) * Float32(2.0)) * t_1)) * u2);
	else
		tmp = fma(fma(Float32(Float32(0.25) * t_2), t_3, Float32(fma(Float32(Float32(0.5) * sqrt(u1)), Float32(Float32(Float32(0.25) - Float32(Float32(0.0625) / u1)) * t_3), Float32(Float32(Float32(0.16666666666666666) * t_2) * t_3)) * u1)), Float32(u1 * u1), Float32(t_3 * sqrt(u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (log.f32 (-.f32 #s(literal 1 binary32) u1)) < -0.0299999993

   1. Initial program 97.3%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 97.1

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

   4. Applied rewrites 97.1%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{u2 \cdot \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right) + {u2}^{2} \cdot \left(\frac{-4}{3} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right) + {u2}^{2} \cdot \left(\frac{-8}{315} \cdot \left(\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right) + \frac{4}{15} \cdot \left({\mathsf{PI}\left(\right)}^{5} \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right)\right)\right)\right)} \]

   7. Applied rewrites 94.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({\pi}^{5} \cdot \sqrt{-\log \left(1 - u1\right)}, 0.26666666666666666, \left(\left(-0.025396825396825397 \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{7}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right), u2 \cdot u2, \left({\pi}^{3} \cdot -1.3333333333333333\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right), u2 \cdot u2, \left(\pi \cdot 2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right) \cdot u2} \]

   if -0.0299999993 < (log.f32 (-.f32 #s(literal 1 binary32) u1))

   1. Initial program 48.1%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 45.6

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

   4. Applied rewrites 45.6%

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

   5. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) + {u1}^{2} \cdot \left(\frac{1}{4} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + u1 \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right)\right)} \]

   7. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.25 \cdot \frac{1}{\sqrt{u1}}, \sin \left(\left(\pi \cdot 2\right) \cdot u2\right), \mathsf{fma}\left(0.5 \cdot \sqrt{u1}, \left(0.25 - \frac{0.0625}{u1}\right) \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right), \left(0.16666666666666666 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(\pi \cdot 2\right) \cdot u2\right) \cdot \sqrt{u1}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 97.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ t_1 := \frac{1}{\sqrt{u1}}\\ t_2 := \sin \left(\left(\pi \cdot 2\right) \cdot u2\right)\\ \mathbf{if}\;u1 \leq 0.029999999329447746:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.25 \cdot t\_1, t\_2, \mathsf{fma}\left(0.5 \cdot \sqrt{u1}, \frac{-0.0625}{u1} \cdot t\_2, \left(0.16666666666666666 \cdot t\_1\right) \cdot t\_2\right) \cdot u1\right), u1 \cdot u1, t\_2 \cdot \sqrt{u1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({\pi}^{5} \cdot t\_0, 0.26666666666666666, \left(\left(-0.025396825396825397 \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{7}\right) \cdot t\_0\right), u2 \cdot u2, \left({\pi}^{3} \cdot -1.3333333333333333\right) \cdot t\_0\right), u2 \cdot u2, \left(\pi \cdot 2\right) \cdot t\_0\right) \cdot u2\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1)))))
        (t_1 (/ 1.0 (sqrt u1)))
        (t_2 (sin (* (* PI 2.0) u2))))
   (if (<= u1 0.029999999329447746)
     (fma
      (fma
       (* 0.25 t_1)
       t_2
       (*
        (fma
         (* 0.5 (sqrt u1))
         (* (/ -0.0625 u1) t_2)
         (* (* 0.16666666666666666 t_1) t_2))
        u1))
      (* u1 u1)
      (* t_2 (sqrt u1)))
     (*
      (fma
       (fma
        (fma
         (* (pow PI 5.0) t_0)
         0.26666666666666666
         (* (* (* -0.025396825396825397 (* u2 u2)) (pow PI 7.0)) t_0))
        (* u2 u2)
        (* (* (pow PI 3.0) -1.3333333333333333) t_0))
       (* u2 u2)
       (* (* PI 2.0) t_0))
      u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf(-logf((1.0f - u1)));
	float t_1 = 1.0f / sqrtf(u1);
	float t_2 = sinf(((((float) M_PI) * 2.0f) * u2));
	float tmp;
	if (u1 <= 0.029999999329447746f) {
		tmp = fmaf(fmaf((0.25f * t_1), t_2, (fmaf((0.5f * sqrtf(u1)), ((-0.0625f / u1) * t_2), ((0.16666666666666666f * t_1) * t_2)) * u1)), (u1 * u1), (t_2 * sqrtf(u1)));
	} else {
		tmp = fmaf(fmaf(fmaf((powf(((float) M_PI), 5.0f) * t_0), 0.26666666666666666f, (((-0.025396825396825397f * (u2 * u2)) * powf(((float) M_PI), 7.0f)) * t_0)), (u2 * u2), ((powf(((float) M_PI), 3.0f) * -1.3333333333333333f) * t_0)), (u2 * u2), ((((float) M_PI) * 2.0f) * t_0)) * u2;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(-log(Float32(Float32(1.0) - u1))))
	t_1 = Float32(Float32(1.0) / sqrt(u1))
	t_2 = sin(Float32(Float32(Float32(pi) * Float32(2.0)) * u2))
	tmp = Float32(0.0)
	if (u1 <= Float32(0.029999999329447746))
		tmp = fma(fma(Float32(Float32(0.25) * t_1), t_2, Float32(fma(Float32(Float32(0.5) * sqrt(u1)), Float32(Float32(Float32(-0.0625) / u1) * t_2), Float32(Float32(Float32(0.16666666666666666) * t_1) * t_2)) * u1)), Float32(u1 * u1), Float32(t_2 * sqrt(u1)));
	else
		tmp = Float32(fma(fma(fma(Float32((Float32(pi) ^ Float32(5.0)) * t_0), Float32(0.26666666666666666), Float32(Float32(Float32(Float32(-0.025396825396825397) * Float32(u2 * u2)) * (Float32(pi) ^ Float32(7.0))) * t_0)), Float32(u2 * u2), Float32(Float32((Float32(pi) ^ Float32(3.0)) * Float32(-1.3333333333333333)) * t_0)), Float32(u2 * u2), Float32(Float32(Float32(pi) * Float32(2.0)) * t_0)) * u2);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0299999993

   1. Initial program 48.1%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 45.6

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

   4. Applied rewrites 45.6%

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

   5. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) + {u1}^{2} \cdot \left(\frac{1}{4} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + u1 \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right)\right)} \]

   7. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.25 \cdot \frac{1}{\sqrt{u1}}, \sin \left(\left(\pi \cdot 2\right) \cdot u2\right), \mathsf{fma}\left(0.5 \cdot \sqrt{u1}, \left(0.25 - \frac{0.0625}{u1}\right) \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right), \left(0.16666666666666666 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(\pi \cdot 2\right) \cdot u2\right) \cdot \sqrt{u1}\right)} \]

   8. Taylor expanded in u1 around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{\sqrt{u1}}, \sin \left(\left(\pi \cdot 2\right) \cdot u2\right), \mathsf{fma}\left(\frac{1}{2} \cdot \sqrt{u1}, \frac{\frac{-1}{16}}{u1} \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right), \left(\frac{1}{6} \cdot \frac{1}{\sqrt{u1}}\right) \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(\pi \cdot 2\right) \cdot u2\right) \cdot \sqrt{u1}\right) \]
   9. Step-by-step derivation

      1. lower-/.f32 98.1

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.25 \cdot \frac{1}{\sqrt{u1}}, \sin \left(\left(\pi \cdot 2\right) \cdot u2\right), \mathsf{fma}\left(0.5 \cdot \sqrt{u1}, \frac{-0.0625}{u1} \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right), \left(0.16666666666666666 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(\pi \cdot 2\right) \cdot u2\right) \cdot \sqrt{u1}\right) \]

   10. Applied rewrites 98.1%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.25 \cdot \frac{1}{\sqrt{u1}}, \sin \left(\left(\pi \cdot 2\right) \cdot u2\right), \mathsf{fma}\left(0.5 \cdot \sqrt{u1}, \frac{-0.0625}{u1} \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right), \left(0.16666666666666666 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(\pi \cdot 2\right) \cdot u2\right) \cdot \sqrt{u1}\right) \]

   if 0.0299999993 < u1

   1. Initial program 97.3%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 97.1

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

   4. Applied rewrites 97.1%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{u2 \cdot \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right) + {u2}^{2} \cdot \left(\frac{-4}{3} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right) + {u2}^{2} \cdot \left(\frac{-8}{315} \cdot \left(\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right) + \frac{4}{15} \cdot \left({\mathsf{PI}\left(\right)}^{5} \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right)\right)\right)\right)} \]

   7. Applied rewrites 94.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({\pi}^{5} \cdot \sqrt{-\log \left(1 - u1\right)}, 0.26666666666666666, \left(\left(-0.025396825396825397 \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{7}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right), u2 \cdot u2, \left({\pi}^{3} \cdot -1.3333333333333333\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right), u2 \cdot u2, \left(\pi \cdot 2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right) \cdot u2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 96.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{u1}}\\ t_1 := \sin \left(\left(\pi \cdot 2\right) \cdot u2\right)\\ t_2 := t\_1 \cdot \sqrt{u1}\\ t_3 := \mathsf{fma}\left(0.25 \cdot t\_0, t\_1, \mathsf{fma}\left(0.5 \cdot \sqrt{u1}, \left(0.25 - \frac{0.0625}{u1}\right) \cdot t\_1, \left(0.16666666666666666 \cdot t\_0\right) \cdot t\_1\right) \cdot u1\right) \cdot \left(u1 \cdot u1\right)\\ t_4 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;u1 \leq 0.029999999329447746:\\ \;\;\;\;\frac{{t\_3}^{3} + {t\_2}^{3}}{\mathsf{fma}\left(t\_3, t\_3, t\_2 \cdot t\_2 - t\_3 \cdot t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({\pi}^{5} \cdot t\_4, 0.26666666666666666, \left(\left(-0.025396825396825397 \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{7}\right) \cdot t\_4\right), u2 \cdot u2, \left({\pi}^{3} \cdot -1.3333333333333333\right) \cdot t\_4\right), u2 \cdot u2, \left(\pi \cdot 2\right) \cdot t\_4\right) \cdot u2\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (/ 1.0 (sqrt u1)))
        (t_1 (sin (* (* PI 2.0) u2)))
        (t_2 (* t_1 (sqrt u1)))
        (t_3
         (*
          (fma
           (* 0.25 t_0)
           t_1
           (*
            (fma
             (* 0.5 (sqrt u1))
             (* (- 0.25 (/ 0.0625 u1)) t_1)
             (* (* 0.16666666666666666 t_0) t_1))
            u1))
          (* u1 u1)))
        (t_4 (sqrt (- (log (- 1.0 u1))))))
   (if (<= u1 0.029999999329447746)
     (/
      (+ (pow t_3 3.0) (pow t_2 3.0))
      (fma t_3 t_3 (- (* t_2 t_2) (* t_3 t_2))))
     (*
      (fma
       (fma
        (fma
         (* (pow PI 5.0) t_4)
         0.26666666666666666
         (* (* (* -0.025396825396825397 (* u2 u2)) (pow PI 7.0)) t_4))
        (* u2 u2)
        (* (* (pow PI 3.0) -1.3333333333333333) t_4))
       (* u2 u2)
       (* (* PI 2.0) t_4))
      u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = 1.0f / sqrtf(u1);
	float t_1 = sinf(((((float) M_PI) * 2.0f) * u2));
	float t_2 = t_1 * sqrtf(u1);
	float t_3 = fmaf((0.25f * t_0), t_1, (fmaf((0.5f * sqrtf(u1)), ((0.25f - (0.0625f / u1)) * t_1), ((0.16666666666666666f * t_0) * t_1)) * u1)) * (u1 * u1);
	float t_4 = sqrtf(-logf((1.0f - u1)));
	float tmp;
	if (u1 <= 0.029999999329447746f) {
		tmp = (powf(t_3, 3.0f) + powf(t_2, 3.0f)) / fmaf(t_3, t_3, ((t_2 * t_2) - (t_3 * t_2)));
	} else {
		tmp = fmaf(fmaf(fmaf((powf(((float) M_PI), 5.0f) * t_4), 0.26666666666666666f, (((-0.025396825396825397f * (u2 * u2)) * powf(((float) M_PI), 7.0f)) * t_4)), (u2 * u2), ((powf(((float) M_PI), 3.0f) * -1.3333333333333333f) * t_4)), (u2 * u2), ((((float) M_PI) * 2.0f) * t_4)) * u2;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(1.0) / sqrt(u1))
	t_1 = sin(Float32(Float32(Float32(pi) * Float32(2.0)) * u2))
	t_2 = Float32(t_1 * sqrt(u1))
	t_3 = Float32(fma(Float32(Float32(0.25) * t_0), t_1, Float32(fma(Float32(Float32(0.5) * sqrt(u1)), Float32(Float32(Float32(0.25) - Float32(Float32(0.0625) / u1)) * t_1), Float32(Float32(Float32(0.16666666666666666) * t_0) * t_1)) * u1)) * Float32(u1 * u1))
	t_4 = sqrt(Float32(-log(Float32(Float32(1.0) - u1))))
	tmp = Float32(0.0)
	if (u1 <= Float32(0.029999999329447746))
		tmp = Float32(Float32((t_3 ^ Float32(3.0)) + (t_2 ^ Float32(3.0))) / fma(t_3, t_3, Float32(Float32(t_2 * t_2) - Float32(t_3 * t_2))));
	else
		tmp = Float32(fma(fma(fma(Float32((Float32(pi) ^ Float32(5.0)) * t_4), Float32(0.26666666666666666), Float32(Float32(Float32(Float32(-0.025396825396825397) * Float32(u2 * u2)) * (Float32(pi) ^ Float32(7.0))) * t_4)), Float32(u2 * u2), Float32(Float32((Float32(pi) ^ Float32(3.0)) * Float32(-1.3333333333333333)) * t_4)), Float32(u2 * u2), Float32(Float32(Float32(pi) * Float32(2.0)) * t_4)) * u2);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0299999993

   1. Initial program 48.1%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 45.6

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

   4. Applied rewrites 45.6%

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

   5. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) + {u1}^{2} \cdot \left(\frac{1}{4} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + u1 \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right)\right)} \]

   7. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.25 \cdot \frac{1}{\sqrt{u1}}, \sin \left(\left(\pi \cdot 2\right) \cdot u2\right), \mathsf{fma}\left(0.5 \cdot \sqrt{u1}, \left(0.25 - \frac{0.0625}{u1}\right) \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right), \left(0.16666666666666666 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(\pi \cdot 2\right) \cdot u2\right) \cdot \sqrt{u1}\right)} \]

   9. Applied rewrites 97.9%

      \[\leadsto \frac{{\left(\mathsf{fma}\left(0.25 \cdot \frac{1}{\sqrt{u1}}, \sin \left(\left(\pi \cdot 2\right) \cdot u2\right), \mathsf{fma}\left(0.5 \cdot \sqrt{u1}, \left(0.25 - \frac{0.0625}{u1}\right) \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right), \left(0.16666666666666666 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right)\right) \cdot u1\right) \cdot \left(u1 \cdot u1\right)\right)}^{3} + {\left(\sin \left(\left(\pi \cdot 2\right) \cdot u2\right) \cdot \sqrt{u1}\right)}^{3}}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.25 \cdot \frac{1}{\sqrt{u1}}, \sin \left(\left(\pi \cdot 2\right) \cdot u2\right), \mathsf{fma}\left(0.5 \cdot \sqrt{u1}, \left(0.25 - \frac{0.0625}{u1}\right) \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right), \left(0.16666666666666666 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right)\right) \cdot u1\right) \cdot \left(u1 \cdot u1\right), \mathsf{fma}\left(0.25 \cdot \frac{1}{\sqrt{u1}}, \sin \left(\left(\pi \cdot 2\right) \cdot u2\right), \mathsf{fma}\left(0.5 \cdot \sqrt{u1}, \left(0.25 - \frac{0.0625}{u1}\right) \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right), \left(0.16666666666666666 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right)\right) \cdot u1\right) \cdot \left(u1 \cdot u1\right), \left(\sin \left(\left(\pi \cdot 2\right) \cdot u2\right) \cdot \sqrt{u1}\right) \cdot \left(\sin \left(\left(\pi \cdot 2\right) \cdot u2\right) \cdot \sqrt{u1}\right) - \left(\mathsf{fma}\left(0.25 \cdot \frac{1}{\sqrt{u1}}, \sin \left(\left(\pi \cdot 2\right) \cdot u2\right), \mathsf{fma}\left(0.5 \cdot \sqrt{u1}, \left(0.25 - \frac{0.0625}{u1}\right) \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right), \left(0.16666666666666666 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right)\right) \cdot u1\right) \cdot \left(u1 \cdot u1\right)\right) \cdot \left(\sin \left(\left(\pi \cdot 2\right) \cdot u2\right) \cdot \sqrt{u1}\right)\right)}} \]

   if 0.0299999993 < u1

   1. Initial program 97.3%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 97.1

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

   4. Applied rewrites 97.1%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{u2 \cdot \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right) + {u2}^{2} \cdot \left(\frac{-4}{3} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right) + {u2}^{2} \cdot \left(\frac{-8}{315} \cdot \left(\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right) + \frac{4}{15} \cdot \left({\mathsf{PI}\left(\right)}^{5} \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right)\right)\right)\right)} \]

   7. Applied rewrites 94.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({\pi}^{5} \cdot \sqrt{-\log \left(1 - u1\right)}, 0.26666666666666666, \left(\left(-0.025396825396825397 \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{7}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right), u2 \cdot u2, \left({\pi}^{3} \cdot -1.3333333333333333\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right), u2 \cdot u2, \left(\pi \cdot 2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right) \cdot u2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 96.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\\ t_1 := \sqrt{u1} \cdot t\_0\\ t_2 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;u1 \leq 0.029999999329447746:\\ \;\;\;\;{u1}^{4} \cdot \mathsf{fma}\left(-1, \frac{\frac{\mathsf{fma}\left(-1, t\_1, u1 \cdot \mathsf{fma}\left(-1, u1 \cdot \mathsf{fma}\left(-0.03125, t\_1, 0.16666666666666666 \cdot t\_1\right), -0.25 \cdot t\_1\right)\right)}{{u1}^{3}}}{u1}, -0.125 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(t\_0 \cdot -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({\pi}^{5} \cdot t\_2, 0.26666666666666666, \left(\left(-0.025396825396825397 \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{7}\right) \cdot t\_2\right), u2 \cdot u2, \left({\pi}^{3} \cdot -1.3333333333333333\right) \cdot t\_2\right), u2 \cdot u2, \left(\pi \cdot 2\right) \cdot t\_2\right) \cdot u2\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* 2.0 (* u2 PI))))
        (t_1 (* (sqrt u1) t_0))
        (t_2 (sqrt (- (log (- 1.0 u1))))))
   (if (<= u1 0.029999999329447746)
     (*
      (pow u1 4.0)
      (fma
       -1.0
       (/
        (/
         (fma
          -1.0
          t_1
          (*
           u1
           (fma
            -1.0
            (* u1 (fma -0.03125 t_1 (* 0.16666666666666666 t_1)))
            (* -0.25 t_1))))
         (pow u1 3.0))
        u1)
       (* -0.125 (* (/ 1.0 (sqrt u1)) (* t_0 -1.0)))))
     (*
      (fma
       (fma
        (fma
         (* (pow PI 5.0) t_2)
         0.26666666666666666
         (* (* (* -0.025396825396825397 (* u2 u2)) (pow PI 7.0)) t_2))
        (* u2 u2)
        (* (* (pow PI 3.0) -1.3333333333333333) t_2))
       (* u2 u2)
       (* (* PI 2.0) t_2))
      u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sinf((2.0f * (u2 * ((float) M_PI))));
	float t_1 = sqrtf(u1) * t_0;
	float t_2 = sqrtf(-logf((1.0f - u1)));
	float tmp;
	if (u1 <= 0.029999999329447746f) {
		tmp = powf(u1, 4.0f) * fmaf(-1.0f, ((fmaf(-1.0f, t_1, (u1 * fmaf(-1.0f, (u1 * fmaf(-0.03125f, t_1, (0.16666666666666666f * t_1))), (-0.25f * t_1)))) / powf(u1, 3.0f)) / u1), (-0.125f * ((1.0f / sqrtf(u1)) * (t_0 * -1.0f))));
	} else {
		tmp = fmaf(fmaf(fmaf((powf(((float) M_PI), 5.0f) * t_2), 0.26666666666666666f, (((-0.025396825396825397f * (u2 * u2)) * powf(((float) M_PI), 7.0f)) * t_2)), (u2 * u2), ((powf(((float) M_PI), 3.0f) * -1.3333333333333333f) * t_2)), (u2 * u2), ((((float) M_PI) * 2.0f) * t_2)) * u2;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sin(Float32(Float32(2.0) * Float32(u2 * Float32(pi))))
	t_1 = Float32(sqrt(u1) * t_0)
	t_2 = sqrt(Float32(-log(Float32(Float32(1.0) - u1))))
	tmp = Float32(0.0)
	if (u1 <= Float32(0.029999999329447746))
		tmp = Float32((u1 ^ Float32(4.0)) * fma(Float32(-1.0), Float32(Float32(fma(Float32(-1.0), t_1, Float32(u1 * fma(Float32(-1.0), Float32(u1 * fma(Float32(-0.03125), t_1, Float32(Float32(0.16666666666666666) * t_1))), Float32(Float32(-0.25) * t_1)))) / (u1 ^ Float32(3.0))) / u1), Float32(Float32(-0.125) * Float32(Float32(Float32(1.0) / sqrt(u1)) * Float32(t_0 * Float32(-1.0))))));
	else
		tmp = Float32(fma(fma(fma(Float32((Float32(pi) ^ Float32(5.0)) * t_2), Float32(0.26666666666666666), Float32(Float32(Float32(Float32(-0.025396825396825397) * Float32(u2 * u2)) * (Float32(pi) ^ Float32(7.0))) * t_2)), Float32(u2 * u2), Float32(Float32((Float32(pi) ^ Float32(3.0)) * Float32(-1.3333333333333333)) * t_2)), Float32(u2 * u2), Float32(Float32(Float32(pi) * Float32(2.0)) * t_2)) * u2);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0299999993

   1. Initial program 48.1%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 45.6

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

   4. Applied rewrites 45.6%

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

   5. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) + {u1}^{2} \cdot \left(\frac{1}{4} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + u1 \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right)\right)} \]

   7. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.25 \cdot \frac{1}{\sqrt{u1}}, \sin \left(\left(\pi \cdot 2\right) \cdot u2\right), \mathsf{fma}\left(0.5 \cdot \sqrt{u1}, \left(0.25 - \frac{0.0625}{u1}\right) \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right), \left(0.16666666666666666 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(\pi \cdot 2\right) \cdot u2\right) \cdot \sqrt{u1}\right)} \]

   8. Taylor expanded in u1 around -inf

      \[\leadsto {u1}^{4} \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \left(\frac{1}{32} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) + \frac{1}{6} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + -1 \cdot \frac{-1 \cdot \left(\sqrt{\frac{1}{{u1}^{3}}} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) + \frac{1}{4} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{u1}}{u1} + \frac{-1}{8} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)\right)} \]

   10. Applied rewrites 97.7%

       \[\leadsto {u1}^{4} \cdot \color{blue}{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \mathsf{fma}\left(0.03125, \frac{1}{\sqrt{u1}} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot -1\right), 0.16666666666666666 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)\right), -1 \cdot \frac{\mathsf{fma}\left(-1, \sqrt{{u1}^{-3}} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot -1\right), 0.25 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)\right)}{u1}\right)}{u1}, -0.125 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot -1\right)\right)\right)} \]

   11. Taylor expanded in u1 around 0

       \[\leadsto {u1}^{4} \cdot \mathsf{fma}\left(-1, \frac{\frac{-1 \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + u1 \cdot \left(-1 \cdot \left(u1 \cdot \left(\frac{-1}{32} \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{1}{6} \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) + \frac{-1}{4} \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}{{u1}^{3}}}{u1}, \frac{-1}{8} \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot -1\right)\right)\right) \]
   12. Step-by-step derivation

       1. lower-/.f32 N/A

          \[\leadsto {u1}^{4} \cdot \mathsf{fma}\left(-1, \frac{\frac{-1 \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + u1 \cdot \left(-1 \cdot \left(u1 \cdot \left(\frac{-1}{32} \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{1}{6} \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) + \frac{-1}{4} \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}{{u1}^{3}}}{u1}, \frac{-1}{8} \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot -1\right)\right)\right) \]

   13. Applied rewrites 97.8%

       \[\leadsto {u1}^{4} \cdot \mathsf{fma}\left(-1, \frac{\frac{\mathsf{fma}\left(-1, \sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right), u1 \cdot \mathsf{fma}\left(-1, u1 \cdot \mathsf{fma}\left(-0.03125, \sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right), 0.16666666666666666 \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)\right), -0.25 \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)\right)\right)}{{u1}^{3}}}{u1}, -0.125 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot -1\right)\right)\right) \]

   if 0.0299999993 < u1

   1. Initial program 97.3%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 97.1

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

   4. Applied rewrites 97.1%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{u2 \cdot \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right) + {u2}^{2} \cdot \left(\frac{-4}{3} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right) + {u2}^{2} \cdot \left(\frac{-8}{315} \cdot \left(\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right) + \frac{4}{15} \cdot \left({\mathsf{PI}\left(\right)}^{5} \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right)\right)\right)\right)} \]

   7. Applied rewrites 94.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({\pi}^{5} \cdot \sqrt{-\log \left(1 - u1\right)}, 0.26666666666666666, \left(\left(-0.025396825396825397 \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{7}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right), u2 \cdot u2, \left({\pi}^{3} \cdot -1.3333333333333333\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right), u2 \cdot u2, \left(\pi \cdot 2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right) \cdot u2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 96.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\\ t_1 := \sqrt{u1} \cdot t\_0\\ t_2 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;u1 \leq 0.029999999329447746:\\ \;\;\;\;{u1}^{4} \cdot \mathsf{fma}\left(-1, \frac{\frac{\mathsf{fma}\left(-1, t\_1, u1 \cdot \mathsf{fma}\left(-1, u1 \cdot \mathsf{fma}\left(-0.03125, t\_1, 0.16666666666666666 \cdot t\_1\right), -0.25 \cdot t\_1\right)\right)}{{u1}^{3}}}{u1}, -0.125 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(t\_0 \cdot -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({\pi}^{5} \cdot t\_2, 0.26666666666666666, \left(\left(-0.025396825396825397 \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{7}\right) \cdot t\_2\right), u2 \cdot u2, \left({\pi}^{3} \cdot -1.3333333333333333\right) \cdot t\_2\right), u2 \cdot u2, \left(\pi \cdot 2\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right) \cdot u2\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* 2.0 (* u2 PI))))
        (t_1 (* (sqrt u1) t_0))
        (t_2 (sqrt (- (log (- 1.0 u1))))))
   (if (<= u1 0.029999999329447746)
     (*
      (pow u1 4.0)
      (fma
       -1.0
       (/
        (/
         (fma
          -1.0
          t_1
          (*
           u1
           (fma
            -1.0
            (* u1 (fma -0.03125 t_1 (* 0.16666666666666666 t_1)))
            (* -0.25 t_1))))
         (pow u1 3.0))
        u1)
       (* -0.125 (* (/ 1.0 (sqrt u1)) (* t_0 -1.0)))))
     (*
      (fma
       (fma
        (fma
         (* (pow PI 5.0) t_2)
         0.26666666666666666
         (* (* (* -0.025396825396825397 (* u2 u2)) (pow PI 7.0)) t_2))
        (* u2 u2)
        (* (* (pow PI 3.0) -1.3333333333333333) t_2))
       (* u2 u2)
       (* (* PI 2.0) (sqrt (log (/ 1.0 (- 1.0 u1))))))
      u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sinf((2.0f * (u2 * ((float) M_PI))));
	float t_1 = sqrtf(u1) * t_0;
	float t_2 = sqrtf(-logf((1.0f - u1)));
	float tmp;
	if (u1 <= 0.029999999329447746f) {
		tmp = powf(u1, 4.0f) * fmaf(-1.0f, ((fmaf(-1.0f, t_1, (u1 * fmaf(-1.0f, (u1 * fmaf(-0.03125f, t_1, (0.16666666666666666f * t_1))), (-0.25f * t_1)))) / powf(u1, 3.0f)) / u1), (-0.125f * ((1.0f / sqrtf(u1)) * (t_0 * -1.0f))));
	} else {
		tmp = fmaf(fmaf(fmaf((powf(((float) M_PI), 5.0f) * t_2), 0.26666666666666666f, (((-0.025396825396825397f * (u2 * u2)) * powf(((float) M_PI), 7.0f)) * t_2)), (u2 * u2), ((powf(((float) M_PI), 3.0f) * -1.3333333333333333f) * t_2)), (u2 * u2), ((((float) M_PI) * 2.0f) * sqrtf(logf((1.0f / (1.0f - u1)))))) * u2;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sin(Float32(Float32(2.0) * Float32(u2 * Float32(pi))))
	t_1 = Float32(sqrt(u1) * t_0)
	t_2 = sqrt(Float32(-log(Float32(Float32(1.0) - u1))))
	tmp = Float32(0.0)
	if (u1 <= Float32(0.029999999329447746))
		tmp = Float32((u1 ^ Float32(4.0)) * fma(Float32(-1.0), Float32(Float32(fma(Float32(-1.0), t_1, Float32(u1 * fma(Float32(-1.0), Float32(u1 * fma(Float32(-0.03125), t_1, Float32(Float32(0.16666666666666666) * t_1))), Float32(Float32(-0.25) * t_1)))) / (u1 ^ Float32(3.0))) / u1), Float32(Float32(-0.125) * Float32(Float32(Float32(1.0) / sqrt(u1)) * Float32(t_0 * Float32(-1.0))))));
	else
		tmp = Float32(fma(fma(fma(Float32((Float32(pi) ^ Float32(5.0)) * t_2), Float32(0.26666666666666666), Float32(Float32(Float32(Float32(-0.025396825396825397) * Float32(u2 * u2)) * (Float32(pi) ^ Float32(7.0))) * t_2)), Float32(u2 * u2), Float32(Float32((Float32(pi) ^ Float32(3.0)) * Float32(-1.3333333333333333)) * t_2)), Float32(u2 * u2), Float32(Float32(Float32(pi) * Float32(2.0)) * sqrt(log(Float32(Float32(1.0) / Float32(Float32(1.0) - u1)))))) * u2);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0299999993

   1. Initial program 48.1%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 45.6

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

   4. Applied rewrites 45.6%

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

   5. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) + {u1}^{2} \cdot \left(\frac{1}{4} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + u1 \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right)\right)} \]

   7. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.25 \cdot \frac{1}{\sqrt{u1}}, \sin \left(\left(\pi \cdot 2\right) \cdot u2\right), \mathsf{fma}\left(0.5 \cdot \sqrt{u1}, \left(0.25 - \frac{0.0625}{u1}\right) \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right), \left(0.16666666666666666 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(\pi \cdot 2\right) \cdot u2\right) \cdot \sqrt{u1}\right)} \]

   8. Taylor expanded in u1 around -inf

      \[\leadsto {u1}^{4} \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \left(\frac{1}{32} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) + \frac{1}{6} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + -1 \cdot \frac{-1 \cdot \left(\sqrt{\frac{1}{{u1}^{3}}} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) + \frac{1}{4} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{u1}}{u1} + \frac{-1}{8} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)\right)} \]

   10. Applied rewrites 97.7%

       \[\leadsto {u1}^{4} \cdot \color{blue}{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \mathsf{fma}\left(0.03125, \frac{1}{\sqrt{u1}} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot -1\right), 0.16666666666666666 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)\right), -1 \cdot \frac{\mathsf{fma}\left(-1, \sqrt{{u1}^{-3}} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot -1\right), 0.25 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)\right)}{u1}\right)}{u1}, -0.125 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot -1\right)\right)\right)} \]

   11. Taylor expanded in u1 around 0

       \[\leadsto {u1}^{4} \cdot \mathsf{fma}\left(-1, \frac{\frac{-1 \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + u1 \cdot \left(-1 \cdot \left(u1 \cdot \left(\frac{-1}{32} \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{1}{6} \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) + \frac{-1}{4} \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}{{u1}^{3}}}{u1}, \frac{-1}{8} \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot -1\right)\right)\right) \]
   12. Step-by-step derivation

       1. lower-/.f32 N/A

          \[\leadsto {u1}^{4} \cdot \mathsf{fma}\left(-1, \frac{\frac{-1 \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + u1 \cdot \left(-1 \cdot \left(u1 \cdot \left(\frac{-1}{32} \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{1}{6} \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) + \frac{-1}{4} \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}{{u1}^{3}}}{u1}, \frac{-1}{8} \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot -1\right)\right)\right) \]

   13. Applied rewrites 97.8%

       \[\leadsto {u1}^{4} \cdot \mathsf{fma}\left(-1, \frac{\frac{\mathsf{fma}\left(-1, \sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right), u1 \cdot \mathsf{fma}\left(-1, u1 \cdot \mathsf{fma}\left(-0.03125, \sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right), 0.16666666666666666 \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)\right), -0.25 \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)\right)\right)}{{u1}^{3}}}{u1}, -0.125 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot -1\right)\right)\right) \]

   if 0.0299999993 < u1

   1. Initial program 97.3%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 97.1

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

   4. Applied rewrites 97.1%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{u2 \cdot \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right) + {u2}^{2} \cdot \left(\frac{-4}{3} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right) + {u2}^{2} \cdot \left(\frac{-8}{315} \cdot \left(\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right) + \frac{4}{15} \cdot \left({\mathsf{PI}\left(\right)}^{5} \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right)\right)\right)\right)} \]

   7. Applied rewrites 94.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({\pi}^{5} \cdot \sqrt{-\log \left(1 - u1\right)}, 0.26666666666666666, \left(\left(-0.025396825396825397 \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{7}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right), u2 \cdot u2, \left({\pi}^{3} \cdot -1.3333333333333333\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right), u2 \cdot u2, \left(\pi \cdot 2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right) \cdot u2} \]
   8. Step-by-step derivation

      1. lift-neg.f32 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({\pi}^{5} \cdot \sqrt{-\log \left(1 - u1\right)}, \frac{4}{15}, \left(\left(\frac{-8}{315} \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{7}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right), u2 \cdot u2, \left({\pi}^{3} \cdot \frac{-4}{3}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right), u2 \cdot u2, \left(\pi \cdot 2\right) \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right) \cdot u2 \]

      2. lift--.f32 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({\pi}^{5} \cdot \sqrt{-\log \left(1 - u1\right)}, \frac{4}{15}, \left(\left(\frac{-8}{315} \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{7}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right), u2 \cdot u2, \left({\pi}^{3} \cdot \frac{-4}{3}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right), u2 \cdot u2, \left(\pi \cdot 2\right) \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right) \cdot u2 \]

      3. lift-log.f32 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({\pi}^{5} \cdot \sqrt{-\log \left(1 - u1\right)}, \frac{4}{15}, \left(\left(\frac{-8}{315} \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{7}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right), u2 \cdot u2, \left({\pi}^{3} \cdot \frac{-4}{3}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right), u2 \cdot u2, \left(\pi \cdot 2\right) \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right) \cdot u2 \]

      4. neg-log N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({\pi}^{5} \cdot \sqrt{-\log \left(1 - u1\right)}, \frac{4}{15}, \left(\left(\frac{-8}{315} \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{7}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right), u2 \cdot u2, \left({\pi}^{3} \cdot \frac{-4}{3}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right), u2 \cdot u2, \left(\pi \cdot 2\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right) \cdot u2 \]

      5. lower-log.f32 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({\pi}^{5} \cdot \sqrt{-\log \left(1 - u1\right)}, \frac{4}{15}, \left(\left(\frac{-8}{315} \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{7}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right), u2 \cdot u2, \left({\pi}^{3} \cdot \frac{-4}{3}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right), u2 \cdot u2, \left(\pi \cdot 2\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right) \cdot u2 \]

      6. lower-/.f32 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({\pi}^{5} \cdot \sqrt{-\log \left(1 - u1\right)}, \frac{4}{15}, \left(\left(\frac{-8}{315} \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{7}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right), u2 \cdot u2, \left({\pi}^{3} \cdot \frac{-4}{3}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right), u2 \cdot u2, \left(\pi \cdot 2\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right) \cdot u2 \]

      7. lift--.f32 94.0

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({\pi}^{5} \cdot \sqrt{-\log \left(1 - u1\right)}, 0.26666666666666666, \left(\left(-0.025396825396825397 \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{7}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right), u2 \cdot u2, \left({\pi}^{3} \cdot -1.3333333333333333\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right), u2 \cdot u2, \left(\pi \cdot 2\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right) \cdot u2 \]

   9. Applied rewrites 94.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({\pi}^{5} \cdot \sqrt{-\log \left(1 - u1\right)}, 0.26666666666666666, \left(\left(-0.025396825396825397 \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{7}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right), u2 \cdot u2, \left({\pi}^{3} \cdot -1.3333333333333333\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right), u2 \cdot u2, \left(\pi \cdot 2\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right) \cdot u2 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 96.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\\ t_1 := \sqrt{-\log \left(1 - u1\right)}\\ t_2 := \left(\pi \cdot 2\right) \cdot t\_1\\ t_3 := \mathsf{fma}\left(\mathsf{fma}\left({\pi}^{5} \cdot t\_1, 0.26666666666666666, \left(\left(-0.025396825396825397 \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{7}\right) \cdot t\_1\right), u2 \cdot u2, \left({\pi}^{3} \cdot -1.3333333333333333\right) \cdot t\_1\right) \cdot \left(u2 \cdot u2\right)\\ t_4 := \sqrt{u1} \cdot t\_0\\ \mathbf{if}\;u1 \leq 0.029999999329447746:\\ \;\;\;\;{u1}^{4} \cdot \mathsf{fma}\left(-1, \frac{\frac{\mathsf{fma}\left(-1, t\_4, u1 \cdot \mathsf{fma}\left(-1, u1 \cdot \mathsf{fma}\left(-0.03125, t\_4, 0.16666666666666666 \cdot t\_4\right), -0.25 \cdot t\_4\right)\right)}{{u1}^{3}}}{u1}, -0.125 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(t\_0 \cdot -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{t\_3}^{3} + {t\_2}^{3}}{\mathsf{fma}\left(t\_3, t\_3, t\_2 \cdot t\_2 - t\_3 \cdot t\_2\right)} \cdot u2\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* 2.0 (* u2 PI))))
        (t_1 (sqrt (- (log (- 1.0 u1)))))
        (t_2 (* (* PI 2.0) t_1))
        (t_3
         (*
          (fma
           (fma
            (* (pow PI 5.0) t_1)
            0.26666666666666666
            (* (* (* -0.025396825396825397 (* u2 u2)) (pow PI 7.0)) t_1))
           (* u2 u2)
           (* (* (pow PI 3.0) -1.3333333333333333) t_1))
          (* u2 u2)))
        (t_4 (* (sqrt u1) t_0)))
   (if (<= u1 0.029999999329447746)
     (*
      (pow u1 4.0)
      (fma
       -1.0
       (/
        (/
         (fma
          -1.0
          t_4
          (*
           u1
           (fma
            -1.0
            (* u1 (fma -0.03125 t_4 (* 0.16666666666666666 t_4)))
            (* -0.25 t_4))))
         (pow u1 3.0))
        u1)
       (* -0.125 (* (/ 1.0 (sqrt u1)) (* t_0 -1.0)))))
     (*
      (/
       (+ (pow t_3 3.0) (pow t_2 3.0))
       (fma t_3 t_3 (- (* t_2 t_2) (* t_3 t_2))))
      u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sinf((2.0f * (u2 * ((float) M_PI))));
	float t_1 = sqrtf(-logf((1.0f - u1)));
	float t_2 = (((float) M_PI) * 2.0f) * t_1;
	float t_3 = fmaf(fmaf((powf(((float) M_PI), 5.0f) * t_1), 0.26666666666666666f, (((-0.025396825396825397f * (u2 * u2)) * powf(((float) M_PI), 7.0f)) * t_1)), (u2 * u2), ((powf(((float) M_PI), 3.0f) * -1.3333333333333333f) * t_1)) * (u2 * u2);
	float t_4 = sqrtf(u1) * t_0;
	float tmp;
	if (u1 <= 0.029999999329447746f) {
		tmp = powf(u1, 4.0f) * fmaf(-1.0f, ((fmaf(-1.0f, t_4, (u1 * fmaf(-1.0f, (u1 * fmaf(-0.03125f, t_4, (0.16666666666666666f * t_4))), (-0.25f * t_4)))) / powf(u1, 3.0f)) / u1), (-0.125f * ((1.0f / sqrtf(u1)) * (t_0 * -1.0f))));
	} else {
		tmp = ((powf(t_3, 3.0f) + powf(t_2, 3.0f)) / fmaf(t_3, t_3, ((t_2 * t_2) - (t_3 * t_2)))) * u2;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sin(Float32(Float32(2.0) * Float32(u2 * Float32(pi))))
	t_1 = sqrt(Float32(-log(Float32(Float32(1.0) - u1))))
	t_2 = Float32(Float32(Float32(pi) * Float32(2.0)) * t_1)
	t_3 = Float32(fma(fma(Float32((Float32(pi) ^ Float32(5.0)) * t_1), Float32(0.26666666666666666), Float32(Float32(Float32(Float32(-0.025396825396825397) * Float32(u2 * u2)) * (Float32(pi) ^ Float32(7.0))) * t_1)), Float32(u2 * u2), Float32(Float32((Float32(pi) ^ Float32(3.0)) * Float32(-1.3333333333333333)) * t_1)) * Float32(u2 * u2))
	t_4 = Float32(sqrt(u1) * t_0)
	tmp = Float32(0.0)
	if (u1 <= Float32(0.029999999329447746))
		tmp = Float32((u1 ^ Float32(4.0)) * fma(Float32(-1.0), Float32(Float32(fma(Float32(-1.0), t_4, Float32(u1 * fma(Float32(-1.0), Float32(u1 * fma(Float32(-0.03125), t_4, Float32(Float32(0.16666666666666666) * t_4))), Float32(Float32(-0.25) * t_4)))) / (u1 ^ Float32(3.0))) / u1), Float32(Float32(-0.125) * Float32(Float32(Float32(1.0) / sqrt(u1)) * Float32(t_0 * Float32(-1.0))))));
	else
		tmp = Float32(Float32(Float32((t_3 ^ Float32(3.0)) + (t_2 ^ Float32(3.0))) / fma(t_3, t_3, Float32(Float32(t_2 * t_2) - Float32(t_3 * t_2)))) * u2);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0299999993

   1. Initial program 48.1%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 45.6

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

   4. Applied rewrites 45.6%

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

   5. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) + {u1}^{2} \cdot \left(\frac{1}{4} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + u1 \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right)\right)} \]

   7. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.25 \cdot \frac{1}{\sqrt{u1}}, \sin \left(\left(\pi \cdot 2\right) \cdot u2\right), \mathsf{fma}\left(0.5 \cdot \sqrt{u1}, \left(0.25 - \frac{0.0625}{u1}\right) \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right), \left(0.16666666666666666 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(\pi \cdot 2\right) \cdot u2\right) \cdot \sqrt{u1}\right)} \]

   8. Taylor expanded in u1 around -inf

      \[\leadsto {u1}^{4} \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \left(\frac{1}{32} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) + \frac{1}{6} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + -1 \cdot \frac{-1 \cdot \left(\sqrt{\frac{1}{{u1}^{3}}} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) + \frac{1}{4} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{u1}}{u1} + \frac{-1}{8} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)\right)} \]

   10. Applied rewrites 97.7%

       \[\leadsto {u1}^{4} \cdot \color{blue}{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \mathsf{fma}\left(0.03125, \frac{1}{\sqrt{u1}} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot -1\right), 0.16666666666666666 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)\right), -1 \cdot \frac{\mathsf{fma}\left(-1, \sqrt{{u1}^{-3}} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot -1\right), 0.25 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)\right)}{u1}\right)}{u1}, -0.125 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot -1\right)\right)\right)} \]

   11. Taylor expanded in u1 around 0

       \[\leadsto {u1}^{4} \cdot \mathsf{fma}\left(-1, \frac{\frac{-1 \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + u1 \cdot \left(-1 \cdot \left(u1 \cdot \left(\frac{-1}{32} \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{1}{6} \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) + \frac{-1}{4} \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}{{u1}^{3}}}{u1}, \frac{-1}{8} \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot -1\right)\right)\right) \]
   12. Step-by-step derivation

       1. lower-/.f32 N/A

          \[\leadsto {u1}^{4} \cdot \mathsf{fma}\left(-1, \frac{\frac{-1 \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + u1 \cdot \left(-1 \cdot \left(u1 \cdot \left(\frac{-1}{32} \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{1}{6} \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) + \frac{-1}{4} \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}{{u1}^{3}}}{u1}, \frac{-1}{8} \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot -1\right)\right)\right) \]

   13. Applied rewrites 97.8%

       \[\leadsto {u1}^{4} \cdot \mathsf{fma}\left(-1, \frac{\frac{\mathsf{fma}\left(-1, \sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right), u1 \cdot \mathsf{fma}\left(-1, u1 \cdot \mathsf{fma}\left(-0.03125, \sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right), 0.16666666666666666 \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)\right), -0.25 \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)\right)\right)}{{u1}^{3}}}{u1}, -0.125 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot -1\right)\right)\right) \]

   if 0.0299999993 < u1

   1. Initial program 97.3%

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

      1. lift-neg.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-log.f32 N/A

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

      4. neg-log N/A

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

      5. lower-log.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lift--.f32 97.1

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

   4. Applied rewrites 97.1%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{u2 \cdot \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right) + {u2}^{2} \cdot \left(\frac{-4}{3} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right) + {u2}^{2} \cdot \left(\frac{-8}{315} \cdot \left(\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right) + \frac{4}{15} \cdot \left({\mathsf{PI}\left(\right)}^{5} \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right)\right)\right)\right)} \]

   7. Applied rewrites 94.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({\pi}^{5} \cdot \sqrt{-\log \left(1 - u1\right)}, 0.26666666666666666, \left(\left(-0.025396825396825397 \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{7}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right), u2 \cdot u2, \left({\pi}^{3} \cdot -1.3333333333333333\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right), u2 \cdot u2, \left(\pi \cdot 2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right) \cdot u2} \]

   9. Applied rewrites 93.9%

      \[\leadsto \frac{{\left(\mathsf{fma}\left(\mathsf{fma}\left({\pi}^{5} \cdot \sqrt{-\log \left(1 - u1\right)}, 0.26666666666666666, \left(\left(-0.025396825396825397 \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{7}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right), u2 \cdot u2, \left({\pi}^{3} \cdot -1.3333333333333333\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right) \cdot \left(u2 \cdot u2\right)\right)}^{3} + {\left(\left(\pi \cdot 2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right)}^{3}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({\pi}^{5} \cdot \sqrt{-\log \left(1 - u1\right)}, 0.26666666666666666, \left(\left(-0.025396825396825397 \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{7}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right), u2 \cdot u2, \left({\pi}^{3} \cdot -1.3333333333333333\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right) \cdot \left(u2 \cdot u2\right), \mathsf{fma}\left(\mathsf{fma}\left({\pi}^{5} \cdot \sqrt{-\log \left(1 - u1\right)}, 0.26666666666666666, \left(\left(-0.025396825396825397 \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{7}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right), u2 \cdot u2, \left({\pi}^{3} \cdot -1.3333333333333333\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right) \cdot \left(u2 \cdot u2\right), \left(\left(\pi \cdot 2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right) \cdot \left(\left(\pi \cdot 2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right) - \left(\mathsf{fma}\left(\mathsf{fma}\left({\pi}^{5} \cdot \sqrt{-\log \left(1 - u1\right)}, 0.26666666666666666, \left(\left(-0.025396825396825397 \cdot \left(u2 \cdot u2\right)\right) \cdot {\pi}^{7}\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right), u2 \cdot u2, \left({\pi}^{3} \cdot -1.3333333333333333\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right) \cdot \left(u2 \cdot u2\right)\right) \cdot \left(\left(\pi \cdot 2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\right)\right)} \cdot u2 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 92.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\\ t_1 := \sqrt{u1} \cdot t\_0\\ {u1}^{4} \cdot \mathsf{fma}\left(-1, \frac{\frac{\mathsf{fma}\left(-1, t\_1, u1 \cdot \mathsf{fma}\left(-1, u1 \cdot \mathsf{fma}\left(-0.03125, t\_1, 0.16666666666666666 \cdot t\_1\right), -0.25 \cdot t\_1\right)\right)}{{u1}^{3}}}{u1}, -0.125 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(t\_0 \cdot -1\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* 2.0 (* u2 PI)))) (t_1 (* (sqrt u1) t_0)))
   (*
    (pow u1 4.0)
    (fma
     -1.0
     (/
      (/
       (fma
        -1.0
        t_1
        (*
         u1
         (fma
          -1.0
          (* u1 (fma -0.03125 t_1 (* 0.16666666666666666 t_1)))
          (* -0.25 t_1))))
       (pow u1 3.0))
      u1)
     (* -0.125 (* (/ 1.0 (sqrt u1)) (* t_0 -1.0)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sinf((2.0f * (u2 * ((float) M_PI))));
	float t_1 = sqrtf(u1) * t_0;
	return powf(u1, 4.0f) * fmaf(-1.0f, ((fmaf(-1.0f, t_1, (u1 * fmaf(-1.0f, (u1 * fmaf(-0.03125f, t_1, (0.16666666666666666f * t_1))), (-0.25f * t_1)))) / powf(u1, 3.0f)) / u1), (-0.125f * ((1.0f / sqrtf(u1)) * (t_0 * -1.0f))));
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sin(Float32(Float32(2.0) * Float32(u2 * Float32(pi))))
	t_1 = Float32(sqrt(u1) * t_0)
	return Float32((u1 ^ Float32(4.0)) * fma(Float32(-1.0), Float32(Float32(fma(Float32(-1.0), t_1, Float32(u1 * fma(Float32(-1.0), Float32(u1 * fma(Float32(-0.03125), t_1, Float32(Float32(0.16666666666666666) * t_1))), Float32(Float32(-0.25) * t_1)))) / (u1 ^ Float32(3.0))) / u1), Float32(Float32(-0.125) * Float32(Float32(Float32(1.0) / sqrt(u1)) * Float32(t_0 * Float32(-1.0))))))
end

Derivation

1. Initial program 57.4%

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

   1. lift-neg.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-log.f32 N/A

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

   4. neg-log N/A

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

   5. lower-log.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lift--.f32 55.3

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

4. Applied rewrites 55.3%

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

5. Taylor expanded in u1 around 0

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) + {u1}^{2} \cdot \left(\frac{1}{4} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + u1 \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{1}{2} \cdot \left(\sqrt{u1} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\frac{1}{4} - \frac{1}{16} \cdot \frac{1}{u1}\right)\right)\right)\right)\right)} \]

7. Applied rewrites 92.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.25 \cdot \frac{1}{\sqrt{u1}}, \sin \left(\left(\pi \cdot 2\right) \cdot u2\right), \mathsf{fma}\left(0.5 \cdot \sqrt{u1}, \left(0.25 - \frac{0.0625}{u1}\right) \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right), \left(0.16666666666666666 \cdot \frac{1}{\sqrt{u1}}\right) \cdot \sin \left(\left(\pi \cdot 2\right) \cdot u2\right)\right) \cdot u1\right), u1 \cdot u1, \sin \left(\left(\pi \cdot 2\right) \cdot u2\right) \cdot \sqrt{u1}\right)} \]

8. Taylor expanded in u1 around -inf

   \[\leadsto {u1}^{4} \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \left(\frac{1}{32} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) + \frac{1}{6} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + -1 \cdot \frac{-1 \cdot \left(\sqrt{\frac{1}{{u1}^{3}}} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right) + \frac{1}{4} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{u1}}{u1} + \frac{-1}{8} \cdot \left(\sqrt{\frac{1}{u1}} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)\right)} \]

10. Applied rewrites 91.6%

    \[\leadsto {u1}^{4} \cdot \color{blue}{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \mathsf{fma}\left(0.03125, \frac{1}{\sqrt{u1}} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot -1\right), 0.16666666666666666 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)\right), -1 \cdot \frac{\mathsf{fma}\left(-1, \sqrt{{u1}^{-3}} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot -1\right), 0.25 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)\right)}{u1}\right)}{u1}, -0.125 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot -1\right)\right)\right)} \]

11. Taylor expanded in u1 around 0

    \[\leadsto {u1}^{4} \cdot \mathsf{fma}\left(-1, \frac{\frac{-1 \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + u1 \cdot \left(-1 \cdot \left(u1 \cdot \left(\frac{-1}{32} \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{1}{6} \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) + \frac{-1}{4} \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}{{u1}^{3}}}{u1}, \frac{-1}{8} \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot -1\right)\right)\right) \]
12. Step-by-step derivation

    1. lower-/.f32 N/A

       \[\leadsto {u1}^{4} \cdot \mathsf{fma}\left(-1, \frac{\frac{-1 \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + u1 \cdot \left(-1 \cdot \left(u1 \cdot \left(\frac{-1}{32} \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{1}{6} \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) + \frac{-1}{4} \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}{{u1}^{3}}}{u1}, \frac{-1}{8} \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot -1\right)\right)\right) \]

13. Applied rewrites 91.7%

    \[\leadsto {u1}^{4} \cdot \mathsf{fma}\left(-1, \frac{\frac{\mathsf{fma}\left(-1, \sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right), u1 \cdot \mathsf{fma}\left(-1, u1 \cdot \mathsf{fma}\left(-0.03125, \sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right), 0.16666666666666666 \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)\right), -0.25 \cdot \left(\sqrt{u1} \cdot \sin \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)\right)\right)}{{u1}^{3}}}{u1}, -0.125 \cdot \left(\frac{1}{\sqrt{u1}} \cdot \left(\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot -1\right)\right)\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(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))))