Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.9% → 98.4%

Time: 12.9s

Alternatives: 16

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: 57.9% 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: 98.4% 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 57.4%

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

   1. 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: 97.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right)\\ \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.07199999690055847:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\frac{\left(-u1\right) \cdot \mathsf{fma}\left(t\_0, t\_0 \cdot \left(u1 \cdot u1\right), -1\right)}{\mathsf{fma}\left(u1, t\_0, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (fma u1 (fma u1 -0.25 -0.3333333333333333) -0.5)))
   (if (<= (- (log (- 1.0 u1))) 0.07199999690055847)
     (*
      (sin (* (* 2.0 PI) u2))
      (sqrt (/ (* (- u1) (fma t_0 (* t_0 (* u1 u1)) -1.0)) (fma u1 t_0 1.0))))
     (*
      (sqrt (- (log1p (- u1))))
      (*
       u2
       (fma (* -1.3333333333333333 (* u2 u2)) (* PI (* PI PI)) (* 2.0 PI)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = fmaf(u1, fmaf(u1, -0.25f, -0.3333333333333333f), -0.5f);
	float tmp;
	if (-logf((1.0f - u1)) <= 0.07199999690055847f) {
		tmp = sinf(((2.0f * ((float) M_PI)) * u2)) * sqrtf(((-u1 * fmaf(t_0, (t_0 * (u1 * u1)), -1.0f)) / fmaf(u1, t_0, 1.0f)));
	} else {
		tmp = sqrtf(-log1pf(-u1)) * (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)
	t_0 = fma(u1, fma(u1, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5))
	tmp = Float32(0.0)
	if (Float32(-log(Float32(Float32(1.0) - u1))) <= Float32(0.07199999690055847))
		tmp = Float32(sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)) * sqrt(Float32(Float32(Float32(-u1) * fma(t_0, Float32(t_0 * Float32(u1 * u1)), Float32(-1.0))) / fma(u1, t_0, Float32(1.0)))));
	else
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * 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 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 0.0719999969

   1. Initial program 50.8%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{u1 \cdot \left(u1 \cdot \left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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 98.1

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

   5. Simplified 98.1%

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

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

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

      2. flip-+ N/A

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

      3. associate-*r/ N/A

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

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

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

   7. Applied egg-rr 98.2%

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

   if 0.0719999969 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 97.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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 98.6

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

   4. Applied egg-rr 98.6%

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

   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. *-lowering-*.f32 N/A

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

      2. associate-*r* N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. cube-mult N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. PI-lowering-PI.f32 96.7

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

   7. Simplified 96.7%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.07199999690055847:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\frac{\left(-u1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right) \cdot \left(u1 \cdot u1\right), -1\right)}{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.07199999690055847:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\left(-u1\right) \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- (log (- 1.0 u1))) 0.07199999690055847)
   (*
    (sin (* (* 2.0 PI) u2))
    (sqrt
     (*
      (- u1)
      (fma u1 (fma u1 (fma u1 -0.25 -0.3333333333333333) -0.5) -1.0))))
   (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (-logf((1.0f - u1)) <= 0.07199999690055847f) {
		tmp = sinf(((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(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(-log(Float32(Float32(1.0) - u1))) <= Float32(0.07199999690055847))
		tmp = Float32(sin(Float32(Float32(Float32(2.0) * 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(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 0.0719999969

   1. Initial program 50.8%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{u1 \cdot \left(u1 \cdot \left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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 98.1

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

   5. Simplified 98.1%

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

   if 0.0719999969 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 97.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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 98.6

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

   4. Applied egg-rr 98.6%

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

   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 89.1

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

   7. Simplified 89.1%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.07199999690055847:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\left(-u1\right) \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.07199999690055847:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- (log (- 1.0 u1))) 0.07199999690055847)
   (*
    (sin (* (* 2.0 PI) u2))
    (sqrt (fma (* u1 u1) (fma u1 (fma u1 0.25 0.3333333333333333) 0.5) u1)))
   (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (-logf((1.0f - u1)) <= 0.07199999690055847f) {
		tmp = sinf(((2.0f * ((float) M_PI)) * u2)) * sqrtf(fmaf((u1 * u1), fmaf(u1, fmaf(u1, 0.25f, 0.3333333333333333f), 0.5f), u1));
	} else {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(-log(Float32(Float32(1.0) - u1))) <= Float32(0.07199999690055847))
		tmp = Float32(sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)) * sqrt(fma(Float32(u1 * u1), fma(u1, fma(u1, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), u1)));
	else
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 0.0719999969

   1. Initial program 50.8%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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 98.1

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

   5. Simplified 98.1%

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

   if 0.0719999969 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 97.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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 98.6

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

   4. Applied egg-rr 98.6%

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

   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 89.1

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

   7. Simplified 89.1%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.07199999690055847:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.05000000074505806:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{\left(-u1\right) \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.05000000074505806)
     (*
      (sqrt (- (log1p (- u1))))
      (*
       u2
       (fma (* -1.3333333333333333 (* u2 u2)) (* PI (* PI PI)) (* 2.0 PI))))
     (*
      (sin t_0)
      (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) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.05000000074505806f) {
		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((-u1 * fmaf(u1, fmaf(u1, fmaf(u1, -0.25f, -0.3333333333333333f), -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(0.05000000074505806))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(u2 * fma(Float32(Float32(-1.3333333333333333) * Float32(u2 * u2)), Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))), Float32(Float32(2.0) * Float32(pi)))));
	else
		tmp = Float32(sin(t_0) * sqrt(Float32(Float32(-u1) * fma(u1, fma(u1, fma(u1, Float32(-0.25), Float32(-0.3333333333333333)), 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) < 0.0500000007

   1. Initial program 59.2%

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

      1. 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 98.6

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

   4. Applied egg-rr 98.6%

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

   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. *-lowering-*.f32 N/A

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

      2. associate-*r* N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. cube-mult N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. PI-lowering-PI.f32 98.7

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

   7. Simplified 98.7%

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

   if 0.0500000007 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 49.9%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{u1 \cdot \left(u1 \cdot \left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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 94.8

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

   5. Simplified 94.8%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.05000000074505806:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\left(-u1\right) \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9300000071525574:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \left(\pi \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\left(-u1\right) \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9300000071525574)
   (*
    (sqrt (- (log (- 1.0 u1))))
    (* u2 (* PI (fma (* -1.3333333333333333 (* u2 u2)) (* PI PI) 2.0))))
   (*
    (sin (* (* 2.0 PI) u2))
    (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) {
	float tmp;
	if ((1.0f - u1) <= 0.9300000071525574f) {
		tmp = sqrtf(-logf((1.0f - u1))) * (u2 * (((float) M_PI) * fmaf((-1.3333333333333333f * (u2 * u2)), (((float) M_PI) * ((float) M_PI)), 2.0f)));
	} else {
		tmp = sinf(((2.0f * ((float) M_PI)) * u2)) * sqrtf((-u1 * fmaf(u1, fmaf(u1, fmaf(u1, -0.25f, -0.3333333333333333f), -0.5f), -1.0f)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.9300000071525574))
		tmp = Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * Float32(u2 * Float32(Float32(pi) * fma(Float32(Float32(-1.3333333333333333) * Float32(u2 * u2)), Float32(Float32(pi) * Float32(pi)), Float32(2.0)))));
	else
		tmp = Float32(sin(Float32(Float32(Float32(2.0) * 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)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.930000007

   1. Initial program 97.8%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. unpow3 N/A

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

      12. associate-*r* N/A

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

      13. distribute-rgt-out N/A

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

   5. Simplified 95.9%

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

   if 0.930000007 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 50.8%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{u1 \cdot \left(u1 \cdot \left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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 98.1

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

   5. Simplified 98.1%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9300000071525574:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(u2 \cdot \left(\pi \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\left(-u1\right) \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9800000190734863:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\left(-u1\right) \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right), -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9800000190734863)
   (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))
   (*
    (sin (* (* 2.0 PI) u2))
    (sqrt (* (- u1) (fma u1 (fma u1 -0.3333333333333333 -0.5) -1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((1.0f - u1) <= 0.9800000190734863f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	} else {
		tmp = sinf(((2.0f * ((float) M_PI)) * u2)) * sqrtf((-u1 * fmaf(u1, fmaf(u1, -0.3333333333333333f, -0.5f), -1.0f)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.9800000190734863))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)) * sqrt(Float32(Float32(-u1) * fma(u1, fma(u1, Float32(-0.3333333333333333), Float32(-0.5)), Float32(-1.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.980000019

   1. Initial program 96.6%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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 98.2

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

   4. Applied egg-rr 98.2%

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

   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 88.6

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

   7. Simplified 88.6%

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

   if 0.980000019 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 47.5%

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

   3. Taylor expanded in u1 around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. accelerator-lowering-fma.f32 98.2

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

   5. Simplified 98.2%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9800000190734863:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\left(-u1\right) \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right), -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(\pi \cdot u2\right)\\ \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.0011599999852478504:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \left(\mathsf{fma}\left(u1, 0.25, 1\right) \cdot \sqrt{u1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* 2.0 (* PI u2))))
   (if (<= (* (* 2.0 PI) u2) 0.0011599999852478504)
     (* (sqrt (- (log1p (- u1)))) t_0)
     (* (sin t_0) (* (fma u1 0.25 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 (((2.0f * ((float) M_PI)) * u2) <= 0.0011599999852478504f) {
		tmp = sqrtf(-log1pf(-u1)) * t_0;
	} else {
		tmp = sinf(t_0) * (fmaf(u1, 0.25f, 1.0f) * sqrtf(u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(2.0) * Float32(Float32(pi) * u2))
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.0011599999852478504))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * t_0);
	else
		tmp = Float32(sin(t_0) * Float32(fma(u1, Float32(0.25), 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) < 0.00115999999

   1. Initial program 60.9%

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

      1. 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 98.6

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

   4. Applied egg-rr 98.6%

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

   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 98.5

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

   7. Simplified 98.5%

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

   if 0.00115999999 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 51.1%

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

   4. Applied egg-rr 48.1%

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

   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. sin-lowering-sin.f32 N/A

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

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

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

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

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

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

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

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

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

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

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

      10. cube-mult N/A

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

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

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

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

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

      15. sqrt-lowering-sqrt.f32 89.3

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

   7. Simplified 89.3%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

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

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

   9. Applied egg-rr 89.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(u1 \cdot \left(u1 \cdot u1\right), 0.0625, -u1\right) \cdot \sin \left(\pi \cdot \left(2 \cdot u2\right)\right)}{\sqrt{u1} \cdot \left(u1 \cdot 0.25\right) - \sqrt{u1}}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

   11. Applied egg-rr 89.0%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.0011599999852478504:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \left(\mathsf{fma}\left(u1, 0.25, 1\right) \cdot \sqrt{u1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 95.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9800000190734863:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9800000190734863)
   (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))
   (*
    (sin (* (* 2.0 PI) u2))
    (sqrt (fma (* u1 u1) (fma u1 0.3333333333333333 0.5) u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((1.0f - u1) <= 0.9800000190734863f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	} else {
		tmp = sinf(((2.0f * ((float) M_PI)) * u2)) * sqrtf(fmaf((u1 * u1), fmaf(u1, 0.3333333333333333f, 0.5f), u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - u1) <= Float32(0.9800000190734863))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)) * 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 #s(literal 1 binary32) u1) < 0.980000019

   1. Initial program 96.6%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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 98.2

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

   4. Applied egg-rr 98.2%

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

   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 88.6

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

   7. Simplified 88.6%

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

   if 0.980000019 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 47.5%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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 98.1

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

   5. Simplified 98.1%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9800000190734863:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.029999999329447746:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.029999999329447746)
     (* (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 <= 0.029999999329447746f) {
		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(0.029999999329447746))
		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) < 0.0299999993

   1. Initial program 59.6%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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 98.6

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

   4. Applied egg-rr 98.6%

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

   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 95.5

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

   7. Simplified 95.5%

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

   if 0.0299999993 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 49.6%

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

   3. Taylor expanded in u1 around 0

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

   5. Simplified 80.1%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.029999999329447746:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 89.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.0949999988079071:\\ \;\;\;\;\left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \pi\right)\right) \cdot \sqrt{\left(-u1\right) \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.0949999988079071)
     (*
      (*
       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 <= 0.0949999988079071f) {
		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(0.0949999988079071))
		tmp = Float32(Float32(u2 * fma(Float32(Float32(-1.3333333333333333) * 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) < 0.0949999988

   1. Initial program 59.3%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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 92.7

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

   5. Simplified 92.7%

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

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

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

      2. associate-*r* N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. cube-mult N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. PI-lowering-PI.f32 92.8

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

   8. Simplified 92.8%

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

   if 0.0949999988 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 48.2%

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

   3. Taylor expanded in u1 around 0

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

   5. Simplified 79.5%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.0949999988079071:\\ \;\;\;\;\left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \pi\right)\right) \cdot \sqrt{\left(-u1\right) \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 85.0% accurate, 3.0× speedup

Math

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

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(Float32(-1.3333333333333333) * 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 57.4%

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

3. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{u1 \cdot \left(u1 \cdot \left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
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 93.0

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

5. Simplified 93.0%

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

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. *-lowering-*.f32 N/A

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

   2. associate-*r* N/A

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

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

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

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

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

   5. unpow2 N/A

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

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

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

   7. cube-mult N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   14. PI-lowering-PI.f32 85.7

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

8. Simplified 85.7%

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

9. Final simplification 85.7%

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

Alternative 13: 81.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.004000000189989805:\\ \;\;\;\;\sqrt{\left(-u1\right) \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \pi\right)\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* (* 2.0 PI) u2) 0.004000000189989805)
   (*
    (sqrt
     (* (- u1) (fma u1 (fma u1 (fma u1 -0.25 -0.3333333333333333) -0.5) -1.0)))
    (* 2.0 (* PI u2)))
   (*
    (* u2 (fma (* -1.3333333333333333 (* u2 u2)) (* PI (* PI PI)) (* 2.0 PI)))
    (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.004000000189989805f) {
		tmp = sqrtf((-u1 * fmaf(u1, fmaf(u1, fmaf(u1, -0.25f, -0.3333333333333333f), -0.5f), -1.0f))) * (2.0f * (((float) M_PI) * u2));
	} else {
		tmp = (u2 * fmaf((-1.3333333333333333f * (u2 * u2)), (((float) M_PI) * (((float) M_PI) * ((float) M_PI))), (2.0f * ((float) M_PI)))) * 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(0.004000000189989805))
		tmp = Float32(sqrt(Float32(Float32(-u1) * fma(u1, fma(u1, fma(u1, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5)), Float32(-1.0)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(Float32(u2 * fma(Float32(Float32(-1.3333333333333333) * Float32(u2 * u2)), Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))), Float32(Float32(2.0) * Float32(pi)))) * sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00400000019

   1. Initial program 60.6%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{u1 \cdot \left(u1 \cdot \left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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 92.6

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

   8. Simplified 92.6%

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

   if 0.00400000019 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 50.6%

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

   3. Taylor expanded in u1 around 0

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

   5. Simplified 79.3%

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

   6. Taylor expanded in u2 around 0

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

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

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

      2. associate-*r* N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. cube-mult N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. PI-lowering-PI.f32 62.8

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

   8. Simplified 62.8%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.004000000189989805:\\ \;\;\;\;\sqrt{\left(-u1\right) \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \pi\right)\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 78.0% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.4%

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

3. 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 93.0

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

5. Simplified 93.0%

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

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 79.4

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

8. Simplified 79.4%

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

9. Final simplification 79.4%

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

Alternative 15: 66.4% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

3. 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 76.3%

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

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. associate-*r* N/A

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

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

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

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

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

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

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

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

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

   6. PI-lowering-PI.f32 66.4

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

8. Simplified 66.4%

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

   1. associate-*r* N/A

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

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

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

   3. associate-*l* N/A

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

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

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

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

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

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

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

   7. PI-lowering-PI.f32 66.4

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

10. Applied egg-rr 66.4%

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

11. Final simplification 66.4%

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

Alternative 16: 66.4% accurate, 8.9× speedup

Math

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

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(Float32(pi) * 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 57.4%

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

3. Taylor expanded in u1 around 0

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

5. Simplified 76.3%

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

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. associate-*r* N/A

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

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

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

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

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

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

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

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

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

   6. PI-lowering-PI.f32 66.4

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

8. Simplified 66.4%

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

9. Final simplification 66.4%

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

Reproduce

herbie shell --seed 2024204 
(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))))