Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.8% → 98.3%

Time: 14.6s

Alternatives: 20

Speedup: 8.9×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.1%

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

   1. sub-neg N/A

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

   2. accelerator-lowering-log1p.f32 N/A

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

   3. neg-lowering-neg.f32 98.4

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

4. Applied egg-rr 98.4%

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

Alternative 2: 97.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.09000000357627869:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, \left(u2 \cdot u2\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \left(\sqrt{-\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)} \cdot \sqrt{u1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.09000000357627869)
     (*
      (sqrt (- (log1p (- u1))))
      (*
       u2
       (fma -1.3333333333333333 (* (* u2 u2) (* PI (* PI PI))) (* 2.0 PI))))
     (*
      (sin t_0)
      (*
       (sqrt
        (- (fma u1 (fma u1 (fma u1 -0.25 -0.3333333333333333) -0.5) -1.0)))
       (sqrt u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.09000000357627869f) {
		tmp = sqrtf(-log1pf(-u1)) * (u2 * fmaf(-1.3333333333333333f, ((u2 * u2) * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))), (2.0f * ((float) M_PI))));
	} else {
		tmp = sinf(t_0) * (sqrtf(-fmaf(u1, fmaf(u1, fmaf(u1, -0.25f, -0.3333333333333333f), -0.5f), -1.0f)) * sqrtf(u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(2.0) * Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.09000000357627869))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(u2 * fma(Float32(-1.3333333333333333), Float32(Float32(u2 * u2) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(2.0) * Float32(pi)))));
	else
		tmp = Float32(sin(t_0) * Float32(sqrt(Float32(-fma(u1, fma(u1, fma(u1, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5)), Float32(-1.0)))) * sqrt(u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.0900000036

   1. Initial program 56.3%

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

      1. sub-neg N/A

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

      2. accelerator-lowering-log1p.f32 N/A

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

      3. neg-lowering-neg.f32 98.7

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right) + \frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \left(2 \cdot \mathsf{PI}\left(\right) + \frac{-4}{3} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{3} \cdot {u2}^{2}\right)}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \left(2 \cdot \mathsf{PI}\left(\right) + \color{blue}{\left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {u2}^{2}}\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(u2 \cdot \left(2 \cdot \mathsf{PI}\left(\right) + \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {u2}^{2}\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \left(2 \cdot \mathsf{PI}\left(\right) + \color{blue}{\frac{-4}{3} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot {u2}^{2}\right)}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \left(2 \cdot \mathsf{PI}\left(\right) + \frac{-4}{3} \cdot \color{blue}{\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \color{blue}{\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      8. accelerator-lowering-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left(\frac{-4}{3}, {u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)}\right) \]

   7. Simplified 98.6%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, \left(u2 \cdot u2\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right)} \]

   if 0.0900000036 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 55.2%

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

   3. Taylor expanded in u1 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f32 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f32 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f32 91.7

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

   5. Simplified 91.7%

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

      1. *-commutative N/A

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

      2. distribute-lft-neg-in N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. *-lowering-*.f32 N/A

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

      6. pow1/2 N/A

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

      7. sqrt-lowering-sqrt.f32 N/A

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

      8. neg-lowering-neg.f32 N/A

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

      9. accelerator-lowering-fma.f32 N/A

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

      10. accelerator-lowering-fma.f32 N/A

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

      11. accelerator-lowering-fma.f32 N/A

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

      12. sqrt-lowering-sqrt.f32 91.8

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

   7. Applied egg-rr 91.8%

      \[\leadsto \color{blue}{\left(\sqrt{-\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)} \cdot \sqrt{u1}\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.09000000357627869:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, \left(u2 \cdot u2\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \left(\sqrt{-\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)} \cdot \sqrt{u1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.09000000357627869:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, \left(u2 \cdot u2\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.09000000357627869)
     (*
      (sqrt (- (log1p (- u1))))
      (*
       u2
       (fma -1.3333333333333333 (* (* u2 u2) (* PI (* PI PI))) (* 2.0 PI))))
     (*
      (sin t_0)
      (sqrt
       (fma (* u1 u1) (fma u1 (fma u1 0.25 0.3333333333333333) 0.5) u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.09000000357627869f) {
		tmp = sqrtf(-log1pf(-u1)) * (u2 * fmaf(-1.3333333333333333f, ((u2 * u2) * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))), (2.0f * ((float) M_PI))));
	} else {
		tmp = sinf(t_0) * sqrtf(fmaf((u1 * u1), fmaf(u1, fmaf(u1, 0.25f, 0.3333333333333333f), 0.5f), u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(2.0) * Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.09000000357627869))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(u2 * fma(Float32(-1.3333333333333333), Float32(Float32(u2 * u2) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(2.0) * Float32(pi)))));
	else
		tmp = Float32(sin(t_0) * sqrt(fma(Float32(u1 * u1), fma(u1, fma(u1, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.0900000036

   1. Initial program 56.3%

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

      1. sub-neg N/A

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

      2. accelerator-lowering-log1p.f32 N/A

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

      3. neg-lowering-neg.f32 98.7

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right) + \frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \left(2 \cdot \mathsf{PI}\left(\right) + \frac{-4}{3} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{3} \cdot {u2}^{2}\right)}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \left(2 \cdot \mathsf{PI}\left(\right) + \color{blue}{\left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {u2}^{2}}\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(u2 \cdot \left(2 \cdot \mathsf{PI}\left(\right) + \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {u2}^{2}\right)\right)} \]

      5. associate-*r* N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \left(2 \cdot \mathsf{PI}\left(\right) + \color{blue}{\frac{-4}{3} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot {u2}^{2}\right)}\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \left(2 \cdot \mathsf{PI}\left(\right) + \frac{-4}{3} \cdot \color{blue}{\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \color{blue}{\left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      8. accelerator-lowering-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left(\frac{-4}{3}, {u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)}\right) \]

   7. Simplified 98.6%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, \left(u2 \cdot u2\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right)} \]

   if 0.0900000036 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 55.2%

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

   3. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-rgt-identity N/A

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

      6. accelerator-lowering-fma.f32 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f32 N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f32 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f32 91.7

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

   5. Simplified 91.7%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.09000000357627869:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, \left(u2 \cdot u2\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.003000000026077032:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.003000000026077032)
     (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))
     (*
      (sin t_0)
      (sqrt
       (fma (* u1 u1) (fma u1 (fma u1 0.25 0.3333333333333333) 0.5) u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.003000000026077032f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	} else {
		tmp = sinf(t_0) * sqrtf(fmaf((u1 * u1), fmaf(u1, fmaf(u1, 0.25f, 0.3333333333333333f), 0.5f), u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(2.0) * Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.003000000026077032))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(sin(t_0) * sqrt(fma(Float32(u1 * u1), fma(u1, fma(u1, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00300000003

   1. Initial program 57.8%

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

      1. sub-neg N/A

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

      2. accelerator-lowering-log1p.f32 N/A

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

      3. neg-lowering-neg.f32 98.8

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

   4. Applied egg-rr 98.8%

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

   5. Taylor expanded in u2 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. *-lowering-*.f32 N/A

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

      3. PI-lowering-PI.f32 98.3

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

   7. Simplified 98.3%

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

   if 0.00300000003 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 53.1%

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

   3. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-rgt-identity N/A

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

      6. accelerator-lowering-fma.f32 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f32 N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f32 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f32 93.9

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

   5. Simplified 93.9%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.003000000026077032:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.003000000026077032:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.003000000026077032)
     (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))
     (* (sin t_0) (sqrt (fma (* u1 u1) (fma u1 0.3333333333333333 0.5) u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.003000000026077032f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	} else {
		tmp = sinf(t_0) * sqrtf(fmaf((u1 * u1), fmaf(u1, 0.3333333333333333f, 0.5f), u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(2.0) * Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.003000000026077032))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(sin(t_0) * sqrt(fma(Float32(u1 * u1), fma(u1, Float32(0.3333333333333333), Float32(0.5)), u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00300000003

   1. Initial program 57.8%

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

      1. sub-neg N/A

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

      2. accelerator-lowering-log1p.f32 N/A

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

      3. neg-lowering-neg.f32 98.8

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

   4. Applied egg-rr 98.8%

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

   5. Taylor expanded in u2 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. *-lowering-*.f32 N/A

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

      3. PI-lowering-PI.f32 98.3

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

   7. Simplified 98.3%

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

   if 0.00300000003 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 53.1%

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

   3. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-rgt-identity N/A

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

      6. accelerator-lowering-fma.f32 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f32 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. accelerator-lowering-fma.f32 92.2

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

   5. Simplified 92.2%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), u1\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.003000000026077032:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.003000000026077032:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{-u1 \cdot \mathsf{fma}\left(u1, -0.5, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.003000000026077032)
     (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))
     (* (sin t_0) (sqrt (- (* u1 (fma u1 -0.5 -1.0))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.003000000026077032f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	} else {
		tmp = sinf(t_0) * sqrtf(-(u1 * fmaf(u1, -0.5f, -1.0f)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(2.0) * Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.003000000026077032))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(sin(t_0) * sqrt(Float32(-Float32(u1 * fma(u1, Float32(-0.5), Float32(-1.0))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00300000003

   1. Initial program 57.8%

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

      1. sub-neg N/A

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

      2. accelerator-lowering-log1p.f32 N/A

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

      3. neg-lowering-neg.f32 98.8

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

   4. Applied egg-rr 98.8%

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

   5. Taylor expanded in u2 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. *-lowering-*.f32 N/A

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

      3. PI-lowering-PI.f32 98.3

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

   7. Simplified 98.3%

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

   if 0.00300000003 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 53.1%

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

   3. Taylor expanded in u1 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f32 89.2

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

   5. Simplified 89.2%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.003000000026077032:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{-u1 \cdot \mathsf{fma}\left(u1, -0.5, -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.003000000026077032:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{\mathsf{fma}\left(u1, u1 \cdot 0.5, u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.003000000026077032)
     (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))
     (* (sin t_0) (sqrt (fma u1 (* u1 0.5) u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.003000000026077032f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	} else {
		tmp = sinf(t_0) * sqrtf(fmaf(u1, (u1 * 0.5f), u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(2.0) * Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.003000000026077032))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(sin(t_0) * sqrt(fma(u1, Float32(u1 * Float32(0.5)), u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00300000003

   1. Initial program 57.8%

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

      1. sub-neg N/A

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

      2. accelerator-lowering-log1p.f32 N/A

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

      3. neg-lowering-neg.f32 98.8

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

   4. Applied egg-rr 98.8%

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

   5. Taylor expanded in u2 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. *-lowering-*.f32 N/A

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

      3. PI-lowering-PI.f32 98.3

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

   7. Simplified 98.3%

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

   if 0.00300000003 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 53.1%

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

   3. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. accelerator-lowering-fma.f32 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f32 89.2

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

   5. Simplified 89.2%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.003000000026077032:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1 \cdot 0.5, u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.00800000037997961:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.00800000037997961)
     (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))
     (* (sin t_0) (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.00800000037997961f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	} else {
		tmp = sinf(t_0) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(2.0) * Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.00800000037997961))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(sin(t_0) * sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00800000038

   1. Initial program 58.1%

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

      1. sub-neg N/A

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

      2. accelerator-lowering-log1p.f32 N/A

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

      3. neg-lowering-neg.f32 98.8

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

   4. Applied egg-rr 98.8%

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

   5. Taylor expanded in u2 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. *-lowering-*.f32 N/A

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

      3. PI-lowering-PI.f32 97.2

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

   7. Simplified 97.2%

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

   if 0.00800000038 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 52.0%

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

   3. Taylor expanded in u1 around 0

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

   5. Simplified 79.6%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.00800000037997961:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.15000000596046448:\\ \;\;\;\;\left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, \left(u2 \cdot u2\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right) \cdot \sqrt{\left(-u1\right) \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.15000000596046448)
     (*
      (*
       u2
       (fma -1.3333333333333333 (* (* u2 u2) (* PI (* PI PI))) (* 2.0 PI)))
      (sqrt
       (*
        (- u1)
        (fma u1 (fma u1 (fma u1 -0.25 -0.3333333333333333) -0.5) -1.0))))
     (* (sin t_0) (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.15000000596046448f) {
		tmp = (u2 * fmaf(-1.3333333333333333f, ((u2 * u2) * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))), (2.0f * ((float) M_PI)))) * sqrtf((-u1 * fmaf(u1, fmaf(u1, fmaf(u1, -0.25f, -0.3333333333333333f), -0.5f), -1.0f)));
	} else {
		tmp = sinf(t_0) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(2.0) * Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.15000000596046448))
		tmp = Float32(Float32(u2 * fma(Float32(-1.3333333333333333), Float32(Float32(u2 * u2) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(-u1) * fma(u1, fma(u1, fma(u1, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5)), Float32(-1.0)))));
	else
		tmp = Float32(sin(t_0) * sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.150000006

   1. Initial program 56.1%

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

   3. Taylor expanded in u1 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f32 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f32 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f32 93.8

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

   5. Simplified 93.8%

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

   6. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f32 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f32 N/A

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

   8. Simplified 93.5%

      \[\leadsto \sqrt{-u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)} \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, \left(u2 \cdot u2\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right)} \]

   if 0.150000006 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 55.7%

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

   3. Taylor expanded in u1 around 0

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

   5. Simplified 75.5%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.15000000596046448:\\ \;\;\;\;\left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, \left(u2 \cdot u2\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right) \cdot \sqrt{\left(-u1\right) \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.0020000000949949026:\\ \;\;\;\;\left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{\left(-u1\right) \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(u2 \cdot \left(\pi \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 2\right)\right)\right) \cdot \mathsf{fma}\left(0.25, \sqrt{u1 \cdot \left(u1 \cdot u1\right)}, \sqrt{u1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* (* 2.0 PI) u2) 0.0020000000949949026)
   (*
    (* 2.0 (* PI u2))
    (sqrt
     (*
      (- u1)
      (fma u1 (fma u1 (fma u1 -0.25 -0.3333333333333333) -0.5) -1.0))))
   (*
    (* u2 (* PI (fma (* -1.3333333333333333 (* u2 u2)) (* PI PI) 2.0)))
    (fma 0.25 (sqrt (* u1 (* u1 u1))) (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.0020000000949949026f) {
		tmp = (2.0f * (((float) M_PI) * u2)) * sqrtf((-u1 * fmaf(u1, fmaf(u1, fmaf(u1, -0.25f, -0.3333333333333333f), -0.5f), -1.0f)));
	} else {
		tmp = (u2 * (((float) M_PI) * fmaf((-1.3333333333333333f * (u2 * u2)), (((float) M_PI) * ((float) M_PI)), 2.0f))) * fmaf(0.25f, sqrtf((u1 * (u1 * u1))), sqrtf(u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.0020000000949949026))
		tmp = Float32(Float32(Float32(2.0) * Float32(Float32(pi) * u2)) * sqrt(Float32(Float32(-u1) * fma(u1, fma(u1, fma(u1, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5)), Float32(-1.0)))));
	else
		tmp = Float32(Float32(u2 * Float32(Float32(pi) * fma(Float32(Float32(-1.3333333333333333) * Float32(u2 * u2)), Float32(Float32(pi) * Float32(pi)), Float32(2.0)))) * fma(Float32(0.25), sqrt(Float32(u1 * Float32(u1 * u1))), sqrt(u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00200000009

   1. Initial program 57.5%

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

   3. Taylor expanded in u1 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f32 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f32 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f32 93.3

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

   5. Simplified 93.3%

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

   6. Taylor expanded in u2 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. *-lowering-*.f32 N/A

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

      3. PI-lowering-PI.f32 93.1

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

   8. Simplified 93.1%

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

   if 0.00200000009 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 53.8%

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

   4. Applied egg-rr 50.2%

      \[\leadsto \sqrt{-\color{blue}{\left(\log \left(\left(1 + \left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(\left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \left(u1 \cdot \left(u1 \cdot u1\right)\right)\right)\right) \cdot \frac{1}{1 + \mathsf{fma}\left(u1, u1, u1\right)}\right) - \mathsf{log1p}\left(\left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \mathsf{fma}\left(u1, u1 \cdot u1, 1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Taylor expanded in u1 around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. *-lowering-*.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. sin-lowering-sin.f32 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f32 N/A

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

      12. *-lowering-*.f32 N/A

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

      13. PI-lowering-PI.f32 N/A

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

      14. accelerator-lowering-fma.f32 N/A

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

   7. Simplified 88.9%

      \[\leadsto \color{blue}{\sin \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \mathsf{fma}\left(0.25, \sqrt{u1 \cdot \left(u1 \cdot u1\right)}, \sqrt{u1}\right)} \]

   8. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. *-lowering-*.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. unpow3 N/A

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

      12. unpow2 N/A

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

      13. associate-*r* N/A

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

   10. Simplified 68.5%

       \[\leadsto \color{blue}{\left(u2 \cdot \left(\pi \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 2\right)\right)\right)} \cdot \mathsf{fma}\left(0.25, \sqrt{u1 \cdot \left(u1 \cdot u1\right)}, \sqrt{u1}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.0020000000949949026:\\ \;\;\;\;\left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{\left(-u1\right) \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(u2 \cdot \left(\pi \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 2\right)\right)\right) \cdot \mathsf{fma}\left(0.25, \sqrt{u1 \cdot \left(u1 \cdot u1\right)}, \sqrt{u1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 83.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.0020000000949949026:\\ \;\;\;\;\left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{\left(-u1\right) \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-u1 \cdot \mathsf{fma}\left(u1, -0.5, -1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* (* 2.0 PI) u2) 0.0020000000949949026)
   (*
    (* 2.0 (* PI u2))
    (sqrt
     (*
      (- u1)
      (fma u1 (fma u1 (fma u1 -0.25 -0.3333333333333333) -0.5) -1.0))))
   (*
    (sqrt (- (* u1 (fma u1 -0.5 -1.0))))
    (*
     u2
     (fma (* -1.3333333333333333 (* u2 u2)) (* PI (* PI PI)) (* 2.0 PI))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.0020000000949949026f) {
		tmp = (2.0f * (((float) M_PI) * u2)) * sqrtf((-u1 * fmaf(u1, fmaf(u1, fmaf(u1, -0.25f, -0.3333333333333333f), -0.5f), -1.0f)));
	} else {
		tmp = sqrtf(-(u1 * fmaf(u1, -0.5f, -1.0f))) * (u2 * fmaf((-1.3333333333333333f * (u2 * u2)), (((float) M_PI) * (((float) M_PI) * ((float) M_PI))), (2.0f * ((float) M_PI))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.0020000000949949026))
		tmp = Float32(Float32(Float32(2.0) * Float32(Float32(pi) * u2)) * sqrt(Float32(Float32(-u1) * fma(u1, fma(u1, fma(u1, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5)), Float32(-1.0)))));
	else
		tmp = Float32(sqrt(Float32(-Float32(u1 * fma(u1, Float32(-0.5), Float32(-1.0))))) * Float32(u2 * fma(Float32(Float32(-1.3333333333333333) * Float32(u2 * u2)), Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))), Float32(Float32(2.0) * Float32(pi)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00200000009

   1. Initial program 57.5%

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

   3. Taylor expanded in u1 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f32 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f32 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f32 93.3

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

   5. Simplified 93.3%

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

   6. Taylor expanded in u2 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. *-lowering-*.f32 N/A

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

      3. PI-lowering-PI.f32 93.1

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

   8. Simplified 93.1%

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

   if 0.00200000009 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 53.8%

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

   3. Taylor expanded in u1 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f32 88.7

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

   5. Simplified 88.7%

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

   6. Taylor expanded in u2 around 0

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

      1. distribute-rgt-in N/A

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

      2. *-rgt-identity N/A

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

      3. metadata-eval N/A

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

      4. log-E N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \frac{-1}{2}, -1\right)\right)} \cdot \left(\left(\frac{-4}{3} \cdot \left(\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {\color{blue}{\log \mathsf{E}\left(\right)}}^{3}\right)\right) \cdot u2 + \left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. associate-*r* N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \frac{-1}{2}, -1\right)\right)} \cdot \left(\left(\frac{-4}{3} \cdot \color{blue}{\left({u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}\right)\right)}\right) \cdot u2 + \left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. distribute-rgt-in N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \frac{-1}{2}, -1\right)\right)} \cdot \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}\right)\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \frac{-1}{2}, -1\right)\right)} \cdot \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}\right)\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \frac{-1}{2}, -1\right)\right)} \cdot \left(u2 \cdot \left(\frac{-4}{3} \cdot \color{blue}{\left(\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {\log \mathsf{E}\left(\right)}^{3}\right)} + 2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      9. log-E N/A

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

      10. metadata-eval N/A

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

      11. *-rgt-identity N/A

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

   8. Simplified 68.5%

      \[\leadsto \sqrt{-u1 \cdot \mathsf{fma}\left(u1, -0.5, -1\right)} \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.0020000000949949026:\\ \;\;\;\;\left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{\left(-u1\right) \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-u1 \cdot \mathsf{fma}\left(u1, -0.5, -1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 85.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, \left(u2 \cdot u2\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right) \cdot \sqrt{\left(-u1\right) \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (* u2 (fma -1.3333333333333333 (* (* u2 u2) (* PI (* PI PI))) (* 2.0 PI)))
  (sqrt
   (* (- u1) (fma u1 (fma u1 (fma u1 -0.25 -0.3333333333333333) -0.5) -1.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return (u2 * fmaf(-1.3333333333333333f, ((u2 * u2) * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))), (2.0f * ((float) M_PI)))) * sqrtf((-u1 * fmaf(u1, fmaf(u1, fmaf(u1, -0.25f, -0.3333333333333333f), -0.5f), -1.0f)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(u2 * fma(Float32(-1.3333333333333333), Float32(Float32(u2 * u2) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(-u1) * fma(u1, fma(u1, fma(u1, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5)), Float32(-1.0)))))
end

Derivation

1. Initial program 56.1%

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

3. Taylor expanded in u1 around 0

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

   1. *-lowering-*.f32 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. accelerator-lowering-fma.f32 N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. accelerator-lowering-fma.f32 N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. accelerator-lowering-fma.f32 93.3

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

5. Simplified 93.3%

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

6. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-lowering-*.f32 N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. +-commutative N/A

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

   8. accelerator-lowering-fma.f32 N/A

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

8. Simplified 84.7%

   \[\leadsto \sqrt{-u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)} \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, \left(u2 \cdot u2\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right)} \]

9. Final simplification 84.7%

   \[\leadsto \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, \left(u2 \cdot u2\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right) \cdot \sqrt{\left(-u1\right) \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)} \]
10. Add Preprocessing

Alternative 13: 81.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.00800000037997961:\\ \;\;\;\;\left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{\left(-u1\right) \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\pi, 2, u2 \cdot \left(u2 \cdot \left(\pi \cdot \left(-1.3333333333333333 \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right) \cdot \left(u2 \cdot \sqrt{u1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* (* 2.0 PI) u2) 0.00800000037997961)
   (*
    (* 2.0 (* PI u2))
    (sqrt
     (*
      (- u1)
      (fma u1 (fma u1 (fma u1 -0.25 -0.3333333333333333) -0.5) -1.0))))
   (*
    (fma PI 2.0 (* u2 (* u2 (* PI (* -1.3333333333333333 (* PI PI))))))
    (* u2 (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.00800000037997961f) {
		tmp = (2.0f * (((float) M_PI) * u2)) * sqrtf((-u1 * fmaf(u1, fmaf(u1, fmaf(u1, -0.25f, -0.3333333333333333f), -0.5f), -1.0f)));
	} else {
		tmp = fmaf(((float) M_PI), 2.0f, (u2 * (u2 * (((float) M_PI) * (-1.3333333333333333f * (((float) M_PI) * ((float) M_PI))))))) * (u2 * sqrtf(u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.00800000037997961))
		tmp = Float32(Float32(Float32(2.0) * Float32(Float32(pi) * u2)) * sqrt(Float32(Float32(-u1) * fma(u1, fma(u1, fma(u1, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5)), Float32(-1.0)))));
	else
		tmp = Float32(fma(Float32(pi), Float32(2.0), Float32(u2 * Float32(u2 * Float32(Float32(pi) * Float32(Float32(-1.3333333333333333) * Float32(Float32(pi) * Float32(pi))))))) * Float32(u2 * sqrt(u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00800000038

   1. Initial program 58.1%

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

   3. Taylor expanded in u1 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f32 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f32 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f32 93.1

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

   5. Simplified 93.1%

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

   6. Taylor expanded in u2 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. *-lowering-*.f32 N/A

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

      3. PI-lowering-PI.f32 91.9

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

   8. Simplified 91.9%

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

   if 0.00800000038 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 52.0%

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

   3. Taylor expanded in u1 around 0

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

   5. Simplified 79.6%

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

   6. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(u2 \cdot \left(2 \cdot \mathsf{PI}\left(\right) + {u2}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{4}{15} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f32 N/A

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

      2. accelerator-lowering-fma.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left(2, \mathsf{PI}\left(\right), {u2}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{4}{15} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)}\right) \]

      3. PI-lowering-PI.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \color{blue}{\mathsf{PI}\left(\right)}, {u2}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{4}{15} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \color{blue}{{u2}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{4}{15} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \color{blue}{\left(u2 \cdot u2\right)} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{4}{15} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \color{blue}{\left(u2 \cdot u2\right)} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{4}{15} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \left(u2 \cdot u2\right) \cdot \color{blue}{\left(\frac{4}{15} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right) + \frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \left(u2 \cdot u2\right) \cdot \left(\color{blue}{\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right) \cdot \frac{4}{15}} + \frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \left(u2 \cdot u2\right) \cdot \left(\color{blue}{{u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{5} \cdot \frac{4}{15}\right)} + \frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \left(u2 \cdot u2\right) \cdot \left({u2}^{2} \cdot \color{blue}{\left(\frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)} + \frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \left(u2 \cdot u2\right) \cdot \left(\color{blue}{\left(u2 \cdot u2\right)} \cdot \left(\frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right) + \frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \left(u2 \cdot u2\right) \cdot \left(\color{blue}{u2 \cdot \left(u2 \cdot \left(\frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)} + \frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) \]

      13. accelerator-lowering-fma.f32 N/A

          \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \left(u2 \cdot u2\right) \cdot \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(\frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right), \frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right)}\right)\right) \]

   8. Simplified 64.5%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f32 N/A

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

   10. Applied egg-rr 64.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\pi, 2, u2 \cdot \left(u2 \cdot \mathsf{fma}\left(\pi, \pi \cdot \left(\pi \cdot -1.3333333333333333\right), \left(u2 \cdot \left(u2 \cdot 0.26666666666666666\right)\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \sqrt{\pi}\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{\pi}\right)\right)\right)\right)\right) \cdot \left(u2 \cdot \sqrt{u1}\right)} \]

   11. Taylor expanded in u2 around 0

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

       1. *-commutative N/A

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-lowering-*.f32 N/A

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

       6. PI-lowering-PI.f32 N/A

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

       7. *-lowering-*.f32 N/A

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

       8. unpow2 N/A

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

       9. *-lowering-*.f32 N/A

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

       10. PI-lowering-PI.f32 N/A

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

       11. PI-lowering-PI.f32 60.9

           \[\leadsto \mathsf{fma}\left(\pi, 2, u2 \cdot \left(u2 \cdot \left(\pi \cdot \left(\left(\pi \cdot \color{blue}{\pi}\right) \cdot -1.3333333333333333\right)\right)\right)\right) \cdot \left(u2 \cdot \sqrt{u1}\right) \]

   13. Simplified 60.9%

       \[\leadsto \mathsf{fma}\left(\pi, 2, u2 \cdot \left(u2 \cdot \color{blue}{\left(\pi \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.3333333333333333\right)\right)}\right)\right) \cdot \left(u2 \cdot \sqrt{u1}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.00800000037997961:\\ \;\;\;\;\left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{\left(-u1\right) \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\pi, 2, u2 \cdot \left(u2 \cdot \left(\pi \cdot \left(-1.3333333333333333 \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right) \cdot \left(u2 \cdot \sqrt{u1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 81.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.00800000037997961:\\ \;\;\;\;\left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{\left(-u1\right) \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 2\right)\right) \cdot \left(u2 \cdot \sqrt{u1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* (* 2.0 PI) u2) 0.00800000037997961)
   (*
    (* 2.0 (* PI u2))
    (sqrt
     (*
      (- u1)
      (fma u1 (fma u1 (fma u1 -0.25 -0.3333333333333333) -0.5) -1.0))))
   (*
    (* PI (fma (* -1.3333333333333333 (* u2 u2)) (* PI PI) 2.0))
    (* u2 (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.00800000037997961f) {
		tmp = (2.0f * (((float) M_PI) * u2)) * sqrtf((-u1 * fmaf(u1, fmaf(u1, fmaf(u1, -0.25f, -0.3333333333333333f), -0.5f), -1.0f)));
	} else {
		tmp = (((float) M_PI) * fmaf((-1.3333333333333333f * (u2 * u2)), (((float) M_PI) * ((float) M_PI)), 2.0f)) * (u2 * sqrtf(u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.00800000037997961))
		tmp = Float32(Float32(Float32(2.0) * Float32(Float32(pi) * u2)) * sqrt(Float32(Float32(-u1) * fma(u1, fma(u1, fma(u1, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5)), Float32(-1.0)))));
	else
		tmp = Float32(Float32(Float32(pi) * fma(Float32(Float32(-1.3333333333333333) * Float32(u2 * u2)), Float32(Float32(pi) * Float32(pi)), Float32(2.0))) * Float32(u2 * sqrt(u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00800000038

   1. Initial program 58.1%

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

   3. Taylor expanded in u1 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f32 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. accelerator-lowering-fma.f32 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f32 93.1

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

   5. Simplified 93.1%

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

   6. Taylor expanded in u2 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. *-lowering-*.f32 N/A

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

      3. PI-lowering-PI.f32 91.9

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

   8. Simplified 91.9%

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

   if 0.00800000038 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 52.0%

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

   3. Taylor expanded in u1 around 0

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

   5. Simplified 79.6%

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

   6. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(u2 \cdot \left(2 \cdot \mathsf{PI}\left(\right) + {u2}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{4}{15} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f32 N/A

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

      2. accelerator-lowering-fma.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left(2, \mathsf{PI}\left(\right), {u2}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{4}{15} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)}\right) \]

      3. PI-lowering-PI.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \color{blue}{\mathsf{PI}\left(\right)}, {u2}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{4}{15} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \color{blue}{{u2}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{4}{15} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \color{blue}{\left(u2 \cdot u2\right)} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{4}{15} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \color{blue}{\left(u2 \cdot u2\right)} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{4}{15} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \left(u2 \cdot u2\right) \cdot \color{blue}{\left(\frac{4}{15} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right) + \frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \left(u2 \cdot u2\right) \cdot \left(\color{blue}{\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right) \cdot \frac{4}{15}} + \frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \left(u2 \cdot u2\right) \cdot \left(\color{blue}{{u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{5} \cdot \frac{4}{15}\right)} + \frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \left(u2 \cdot u2\right) \cdot \left({u2}^{2} \cdot \color{blue}{\left(\frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)} + \frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \left(u2 \cdot u2\right) \cdot \left(\color{blue}{\left(u2 \cdot u2\right)} \cdot \left(\frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right) + \frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \left(u2 \cdot u2\right) \cdot \left(\color{blue}{u2 \cdot \left(u2 \cdot \left(\frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)} + \frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) \]

      13. accelerator-lowering-fma.f32 N/A

          \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \left(u2 \cdot u2\right) \cdot \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(\frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right), \frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right)}\right)\right) \]

   8. Simplified 64.5%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f32 N/A

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

   10. Applied egg-rr 64.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\pi, 2, u2 \cdot \left(u2 \cdot \mathsf{fma}\left(\pi, \pi \cdot \left(\pi \cdot -1.3333333333333333\right), \left(u2 \cdot \left(u2 \cdot 0.26666666666666666\right)\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \sqrt{\pi}\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{\pi}\right)\right)\right)\right)\right) \cdot \left(u2 \cdot \sqrt{u1}\right)} \]

   11. Taylor expanded in u2 around 0

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

       1. associate-*r* N/A

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

       2. unpow3 N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. distribute-rgt-out N/A

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

       6. *-lowering-*.f32 N/A

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

       7. PI-lowering-PI.f32 N/A

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

       8. accelerator-lowering-fma.f32 N/A

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

       9. *-lowering-*.f32 N/A

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

       10. unpow2 N/A

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

       11. *-lowering-*.f32 N/A

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

       12. unpow2 N/A

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

       13. *-lowering-*.f32 N/A

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

       14. PI-lowering-PI.f32 N/A

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

       15. PI-lowering-PI.f32 60.9

           \[\leadsto \left(\pi \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \color{blue}{\pi}, 2\right)\right) \cdot \left(u2 \cdot \sqrt{u1}\right) \]

   13. Simplified 60.9%

       \[\leadsto \color{blue}{\left(\pi \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 2\right)\right)} \cdot \left(u2 \cdot \sqrt{u1}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.00800000037997961:\\ \;\;\;\;\left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{\left(-u1\right) \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 2\right)\right) \cdot \left(u2 \cdot \sqrt{u1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 78.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.00800000037997961:\\ \;\;\;\;\left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{-u1 \cdot \mathsf{fma}\left(u1, -0.5, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 2\right)\right) \cdot \left(u2 \cdot \sqrt{u1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* (* 2.0 PI) u2) 0.00800000037997961)
   (* (* 2.0 (* PI u2)) (sqrt (- (* u1 (fma u1 -0.5 -1.0)))))
   (*
    (* PI (fma (* -1.3333333333333333 (* u2 u2)) (* PI PI) 2.0))
    (* u2 (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.00800000037997961f) {
		tmp = (2.0f * (((float) M_PI) * u2)) * sqrtf(-(u1 * fmaf(u1, -0.5f, -1.0f)));
	} else {
		tmp = (((float) M_PI) * fmaf((-1.3333333333333333f * (u2 * u2)), (((float) M_PI) * ((float) M_PI)), 2.0f)) * (u2 * sqrtf(u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.00800000037997961))
		tmp = Float32(Float32(Float32(2.0) * Float32(Float32(pi) * u2)) * sqrt(Float32(-Float32(u1 * fma(u1, Float32(-0.5), Float32(-1.0))))));
	else
		tmp = Float32(Float32(Float32(pi) * fma(Float32(Float32(-1.3333333333333333) * Float32(u2 * u2)), Float32(Float32(pi) * Float32(pi)), Float32(2.0))) * Float32(u2 * sqrt(u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00800000038

   1. Initial program 58.1%

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

   3. Taylor expanded in u1 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f32 87.2

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

   5. Simplified 87.2%

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

   6. Taylor expanded in u2 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. *-lowering-*.f32 N/A

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

      3. PI-lowering-PI.f32 86.3

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

   8. Simplified 86.3%

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

   if 0.00800000038 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 52.0%

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

   3. Taylor expanded in u1 around 0

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

   5. Simplified 79.6%

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

   6. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(u2 \cdot \left(2 \cdot \mathsf{PI}\left(\right) + {u2}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{4}{15} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f32 N/A

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

      2. accelerator-lowering-fma.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left(2, \mathsf{PI}\left(\right), {u2}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{4}{15} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)}\right) \]

      3. PI-lowering-PI.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \color{blue}{\mathsf{PI}\left(\right)}, {u2}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{4}{15} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \color{blue}{{u2}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{4}{15} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \color{blue}{\left(u2 \cdot u2\right)} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{4}{15} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \color{blue}{\left(u2 \cdot u2\right)} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{4}{15} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \left(u2 \cdot u2\right) \cdot \color{blue}{\left(\frac{4}{15} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right) + \frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \left(u2 \cdot u2\right) \cdot \left(\color{blue}{\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right) \cdot \frac{4}{15}} + \frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \left(u2 \cdot u2\right) \cdot \left(\color{blue}{{u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{5} \cdot \frac{4}{15}\right)} + \frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \left(u2 \cdot u2\right) \cdot \left({u2}^{2} \cdot \color{blue}{\left(\frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)} + \frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \left(u2 \cdot u2\right) \cdot \left(\color{blue}{\left(u2 \cdot u2\right)} \cdot \left(\frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right) + \frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \left(u2 \cdot u2\right) \cdot \left(\color{blue}{u2 \cdot \left(u2 \cdot \left(\frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)} + \frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) \]

      13. accelerator-lowering-fma.f32 N/A

          \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \left(u2 \cdot u2\right) \cdot \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(\frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right), \frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right)}\right)\right) \]

   8. Simplified 64.5%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f32 N/A

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

   10. Applied egg-rr 64.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\pi, 2, u2 \cdot \left(u2 \cdot \mathsf{fma}\left(\pi, \pi \cdot \left(\pi \cdot -1.3333333333333333\right), \left(u2 \cdot \left(u2 \cdot 0.26666666666666666\right)\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \sqrt{\pi}\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{\pi}\right)\right)\right)\right)\right) \cdot \left(u2 \cdot \sqrt{u1}\right)} \]

   11. Taylor expanded in u2 around 0

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

       1. associate-*r* N/A

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

       2. unpow3 N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. distribute-rgt-out N/A

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

       6. *-lowering-*.f32 N/A

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

       7. PI-lowering-PI.f32 N/A

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

       8. accelerator-lowering-fma.f32 N/A

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

       9. *-lowering-*.f32 N/A

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

       10. unpow2 N/A

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

       11. *-lowering-*.f32 N/A

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

       12. unpow2 N/A

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

       13. *-lowering-*.f32 N/A

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

       14. PI-lowering-PI.f32 N/A

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

       15. PI-lowering-PI.f32 60.9

           \[\leadsto \left(\pi \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \color{blue}{\pi}, 2\right)\right) \cdot \left(u2 \cdot \sqrt{u1}\right) \]

   13. Simplified 60.9%

       \[\leadsto \color{blue}{\left(\pi \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 2\right)\right)} \cdot \left(u2 \cdot \sqrt{u1}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.00800000037997961:\\ \;\;\;\;\left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{-u1 \cdot \mathsf{fma}\left(u1, -0.5, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 2\right)\right) \cdot \left(u2 \cdot \sqrt{u1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 78.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.00800000037997961:\\ \;\;\;\;\left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{-u1 \cdot \mathsf{fma}\left(u1, -0.5, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \left(\pi \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* (* 2.0 PI) u2) 0.00800000037997961)
   (* (* 2.0 (* PI u2)) (sqrt (- (* u1 (fma u1 -0.5 -1.0)))))
   (*
    (sqrt u1)
    (* u2 (* PI (fma (* -1.3333333333333333 (* u2 u2)) (* PI PI) 2.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.00800000037997961f) {
		tmp = (2.0f * (((float) M_PI) * u2)) * sqrtf(-(u1 * fmaf(u1, -0.5f, -1.0f)));
	} else {
		tmp = sqrtf(u1) * (u2 * (((float) M_PI) * fmaf((-1.3333333333333333f * (u2 * u2)), (((float) M_PI) * ((float) M_PI)), 2.0f)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.00800000037997961))
		tmp = Float32(Float32(Float32(2.0) * Float32(Float32(pi) * u2)) * sqrt(Float32(-Float32(u1 * fma(u1, Float32(-0.5), Float32(-1.0))))));
	else
		tmp = Float32(sqrt(u1) * Float32(u2 * Float32(Float32(pi) * fma(Float32(Float32(-1.3333333333333333) * Float32(u2 * u2)), Float32(Float32(pi) * Float32(pi)), Float32(2.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00800000038

   1. Initial program 58.1%

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

   3. Taylor expanded in u1 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f32 87.2

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

   5. Simplified 87.2%

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

   6. Taylor expanded in u2 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. *-lowering-*.f32 N/A

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

      3. PI-lowering-PI.f32 86.3

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

   8. Simplified 86.3%

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

   if 0.00800000038 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 52.0%

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

   3. Taylor expanded in u1 around 0

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

   5. Simplified 79.6%

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

      1. add-log-exp N/A

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

      2. *-un-lft-identity N/A

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

      3. exp-prod N/A

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

      4. log-pow N/A

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

      5. *-lowering-*.f32 N/A

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

      6. PI-lowering-PI.f32 N/A

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

      7. log-lowering-log.f32 N/A

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

      8. exp-1-e N/A

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

      9. E-lowering-E.f32 79.1

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

   7. Applied egg-rr 79.1%

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

   8. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. log-E N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. log-E N/A

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

      7. metadata-eval N/A

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

      8. *-rgt-identity N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-rgt-identity N/A

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

   10. Simplified 60.9%

       \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(u2 \cdot \left(\pi \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 2\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.00800000037997961:\\ \;\;\;\;\left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{-u1 \cdot \mathsf{fma}\left(u1, -0.5, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(u2 \cdot \left(\pi \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 78.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.00800000037997961:\\ \;\;\;\;\left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{-u1 \cdot \mathsf{fma}\left(u1, -0.5, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;u2 \cdot \left(\sqrt{u1} \cdot \left(\pi \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* (* 2.0 PI) u2) 0.00800000037997961)
   (* (* 2.0 (* PI u2)) (sqrt (- (* u1 (fma u1 -0.5 -1.0)))))
   (*
    u2
    (*
     (sqrt u1)
     (* PI (fma (* -1.3333333333333333 (* u2 u2)) (* PI PI) 2.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.00800000037997961f) {
		tmp = (2.0f * (((float) M_PI) * u2)) * sqrtf(-(u1 * fmaf(u1, -0.5f, -1.0f)));
	} else {
		tmp = u2 * (sqrtf(u1) * (((float) M_PI) * fmaf((-1.3333333333333333f * (u2 * u2)), (((float) M_PI) * ((float) M_PI)), 2.0f)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.00800000037997961))
		tmp = Float32(Float32(Float32(2.0) * Float32(Float32(pi) * u2)) * sqrt(Float32(-Float32(u1 * fma(u1, Float32(-0.5), Float32(-1.0))))));
	else
		tmp = Float32(u2 * Float32(sqrt(u1) * Float32(Float32(pi) * fma(Float32(Float32(-1.3333333333333333) * Float32(u2 * u2)), Float32(Float32(pi) * Float32(pi)), Float32(2.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00800000038

   1. Initial program 58.1%

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

   3. Taylor expanded in u1 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f32 87.2

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

   5. Simplified 87.2%

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

   6. Taylor expanded in u2 around 0

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

      1. *-lowering-*.f32 N/A

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

      2. *-lowering-*.f32 N/A

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

      3. PI-lowering-PI.f32 86.3

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

   8. Simplified 86.3%

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

   if 0.00800000038 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 52.0%

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

   3. Taylor expanded in u1 around 0

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

   5. Simplified 79.6%

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

   6. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(u2 \cdot \left(2 \cdot \mathsf{PI}\left(\right) + {u2}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{4}{15} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f32 N/A

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

      2. accelerator-lowering-fma.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \color{blue}{\mathsf{fma}\left(2, \mathsf{PI}\left(\right), {u2}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{4}{15} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)}\right) \]

      3. PI-lowering-PI.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \color{blue}{\mathsf{PI}\left(\right)}, {u2}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{4}{15} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \color{blue}{{u2}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{4}{15} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \color{blue}{\left(u2 \cdot u2\right)} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{4}{15} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \color{blue}{\left(u2 \cdot u2\right)} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{4}{15} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \left(u2 \cdot u2\right) \cdot \color{blue}{\left(\frac{4}{15} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right) + \frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \left(u2 \cdot u2\right) \cdot \left(\color{blue}{\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right) \cdot \frac{4}{15}} + \frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \left(u2 \cdot u2\right) \cdot \left(\color{blue}{{u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{5} \cdot \frac{4}{15}\right)} + \frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \left(u2 \cdot u2\right) \cdot \left({u2}^{2} \cdot \color{blue}{\left(\frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)} + \frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \left(u2 \cdot u2\right) \cdot \left(\color{blue}{\left(u2 \cdot u2\right)} \cdot \left(\frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right) + \frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \left(u2 \cdot u2\right) \cdot \left(\color{blue}{u2 \cdot \left(u2 \cdot \left(\frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)} + \frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right) \]

      13. accelerator-lowering-fma.f32 N/A

          \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \left(u2 \cdot u2\right) \cdot \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(\frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right), \frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right)}\right)\right) \]

   8. Simplified 64.5%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f32 N/A

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

   10. Applied egg-rr 64.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\pi, 2, u2 \cdot \left(u2 \cdot \mathsf{fma}\left(\pi, \pi \cdot \left(\pi \cdot -1.3333333333333333\right), \left(u2 \cdot \left(u2 \cdot 0.26666666666666666\right)\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \sqrt{\pi}\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{\pi}\right)\right)\right)\right)\right) \cdot \left(u2 \cdot \sqrt{u1}\right)} \]

   11. Taylor expanded in u2 around 0

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

   13. Simplified 60.8%

       \[\leadsto \color{blue}{u2 \cdot \left(\sqrt{u1} \cdot \left(\pi \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 2\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.00800000037997961:\\ \;\;\;\;\left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{-u1 \cdot \mathsf{fma}\left(u1, -0.5, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;u2 \cdot \left(\sqrt{u1} \cdot \left(\pi \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 74.5% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{-u1 \cdot \mathsf{fma}\left(u1, -0.5, -1\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (* 2.0 (* PI u2)) (sqrt (- (* u1 (fma u1 -0.5 -1.0))))))

C

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

Julia

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

Derivation

1. Initial program 56.1%

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

3. Taylor expanded in u1 around 0

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

   1. *-lowering-*.f32 N/A

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

   2. sub-neg N/A

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

   3. *-commutative N/A

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

   4. metadata-eval N/A

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

   5. accelerator-lowering-fma.f32 88.0

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

5. Simplified 88.0%

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

6. Taylor expanded in u2 around 0

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

   1. *-lowering-*.f32 N/A

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

   2. *-lowering-*.f32 N/A

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

   3. PI-lowering-PI.f32 73.0

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

8. Simplified 73.0%

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

9. Final simplification 73.0%

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

Alternative 19: 66.4% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.1%

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

3. Taylor expanded in u1 around 0

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

5. Simplified 77.2%

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

6. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. *-lowering-*.f32 N/A

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

   6. sqrt-lowering-sqrt.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. *-lowering-*.f32 N/A

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

   11. *-lowering-*.f32 N/A

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

   12. PI-lowering-PI.f32 65.6

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

8. Simplified 65.6%

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. *-lowering-*.f32 N/A

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

   4. *-lowering-*.f32 N/A

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

   5. *-commutative N/A

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

   6. *-lowering-*.f32 N/A

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

   7. sqrt-lowering-sqrt.f32 N/A

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

   8. PI-lowering-PI.f32 65.7

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

10. Applied egg-rr 65.7%

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

11. Final simplification 65.7%

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

Alternative 20: 66.4% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.1%

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

3. Taylor expanded in u1 around 0

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

5. Simplified 77.2%

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

6. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. *-lowering-*.f32 N/A

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

   6. sqrt-lowering-sqrt.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. *-lowering-*.f32 N/A

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

   11. *-lowering-*.f32 N/A

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

   12. PI-lowering-PI.f32 65.6

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

8. Simplified 65.6%

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

9. Final simplification 65.6%

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

Reproduce

herbie shell --seed 2024198 
(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))))