Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.7% → 98.3%

Time: 15.0s

Alternatives: 22

Speedup: 4.7×

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: 57.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.7%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Add Preprocessing
3. 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. lower-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. lower-neg.f32 98.5

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

4. Applied rewrites 98.5%

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

   1. lift-PI.f32 N/A

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

   2. associate-*l* N/A

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

   3. sin-2 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-sin.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-cos.f32 N/A

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

   9. lower-*.f32 98.3

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

6. Applied rewrites 98.3%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right)} \]
7. Step-by-step derivation

   1. lift-neg.f32 N/A

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

   2. lift-log1p.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-sqrt.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-sin.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-cos.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 98.3

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

8. Applied rewrites 98.5%

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

Alternative 2: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right)\\ t_1 := u2 \cdot \left(\pi \cdot 2\right)\\ t_2 := \mathsf{fma}\left(u1, t\_0, -u1\right)\\ \mathbf{if}\;t\_1 \leq 0.05000000074505806:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(u2 + u2, \pi, \left(u2 \cdot \left(u2 \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right) \cdot \left(u2 \cdot -1.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{t\_2}{t\_2 \cdot \mathsf{fma}\left(u1, t\_0, u1\right)}}} \cdot \sin t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u1 (fma u1 (fma u1 0.25 0.3333333333333333) 0.5)))
        (t_1 (* u2 (* PI 2.0)))
        (t_2 (fma u1 t_0 (- u1))))
   (if (<= t_1 0.05000000074505806)
     (*
      (sqrt (- (log1p (- u1))))
      (fma
       (+ u2 u2)
       PI
       (* (* u2 (* u2 (* PI (* PI PI)))) (* u2 -1.3333333333333333))))
     (* (/ 1.0 (sqrt (/ t_2 (* t_2 (fma u1 t_0 u1))))) (sin t_1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u1 * fmaf(u1, fmaf(u1, 0.25f, 0.3333333333333333f), 0.5f);
	float t_1 = u2 * (((float) M_PI) * 2.0f);
	float t_2 = fmaf(u1, t_0, -u1);
	float tmp;
	if (t_1 <= 0.05000000074505806f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf((u2 + u2), ((float) M_PI), ((u2 * (u2 * (((float) M_PI) * (((float) M_PI) * ((float) M_PI))))) * (u2 * -1.3333333333333333f)));
	} else {
		tmp = (1.0f / sqrtf((t_2 / (t_2 * fmaf(u1, t_0, u1))))) * sinf(t_1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u1 * fma(u1, fma(u1, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)))
	t_1 = Float32(u2 * Float32(Float32(pi) * Float32(2.0)))
	t_2 = fma(u1, t_0, Float32(-u1))
	tmp = Float32(0.0)
	if (t_1 <= Float32(0.05000000074505806))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(u2 + u2), Float32(pi), Float32(Float32(u2 * Float32(u2 * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))))) * Float32(u2 * Float32(-1.3333333333333333)))));
	else
		tmp = Float32(Float32(Float32(1.0) / sqrt(Float32(t_2 / Float32(t_2 * fma(u1, t_0, u1))))) * sin(t_1));
	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 56.7%

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

      1. 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. lower-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. lower-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 rewrites 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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f32 N/A

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

   7. Applied rewrites 98.6%

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

      1. lift-*.f32 N/A

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

      2. lift-PI.f32 N/A

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

      3. lift-PI.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. +-commutative N/A

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

      11. distribute-lft-in N/A

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

   9. Applied rewrites 98.7%

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

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

   1. Initial program 50.4%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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. lower-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. lower-*.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. lower-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. lower-fma.f32 94.8

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

   5. Applied rewrites 94.8%

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

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. flip-+ N/A

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

      5. clear-num N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f32 N/A

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

   7. Applied rewrites 95.1%

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

4. Final simplification 98.1%

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

Alternative 3: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right)\\ t_1 := u1 \cdot t\_0\\ t_2 := u2 \cdot \left(\pi \cdot 2\right)\\ \mathbf{if}\;t\_2 \leq 0.05000000074505806:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(u2 + u2, \pi, \left(u2 \cdot \left(u2 \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right) \cdot \left(u2 \cdot -1.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t\_2 \cdot \frac{1}{\sqrt{\frac{\mathsf{fma}\left(u1, t\_1, -u1\right)}{u1 \cdot \left(t\_1 \cdot \left(t\_0 \cdot \left(u1 \cdot u1\right)\right) - u1\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (fma u1 (fma u1 0.25 0.3333333333333333) 0.5))
        (t_1 (* u1 t_0))
        (t_2 (* u2 (* PI 2.0))))
   (if (<= t_2 0.05000000074505806)
     (*
      (sqrt (- (log1p (- u1))))
      (fma
       (+ u2 u2)
       PI
       (* (* u2 (* u2 (* PI (* PI PI)))) (* u2 -1.3333333333333333))))
     (*
      (sin t_2)
      (/
       1.0
       (sqrt
        (/ (fma u1 t_1 (- u1)) (* u1 (- (* t_1 (* t_0 (* u1 u1))) u1)))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = fmaf(u1, fmaf(u1, 0.25f, 0.3333333333333333f), 0.5f);
	float t_1 = u1 * t_0;
	float t_2 = u2 * (((float) M_PI) * 2.0f);
	float tmp;
	if (t_2 <= 0.05000000074505806f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf((u2 + u2), ((float) M_PI), ((u2 * (u2 * (((float) M_PI) * (((float) M_PI) * ((float) M_PI))))) * (u2 * -1.3333333333333333f)));
	} else {
		tmp = sinf(t_2) * (1.0f / sqrtf((fmaf(u1, t_1, -u1) / (u1 * ((t_1 * (t_0 * (u1 * u1))) - u1)))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = fma(u1, fma(u1, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5))
	t_1 = Float32(u1 * t_0)
	t_2 = Float32(u2 * Float32(Float32(pi) * Float32(2.0)))
	tmp = Float32(0.0)
	if (t_2 <= Float32(0.05000000074505806))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(u2 + u2), Float32(pi), Float32(Float32(u2 * Float32(u2 * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))))) * Float32(u2 * Float32(-1.3333333333333333)))));
	else
		tmp = Float32(sin(t_2) * Float32(Float32(1.0) / sqrt(Float32(fma(u1, t_1, Float32(-u1)) / Float32(u1 * Float32(Float32(t_1 * Float32(t_0 * Float32(u1 * u1))) - u1))))));
	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 56.7%

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

      1. 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. lower-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. lower-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 rewrites 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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f32 N/A

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

   7. Applied rewrites 98.6%

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

      1. lift-*.f32 N/A

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

      2. lift-PI.f32 N/A

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

      3. lift-PI.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. +-commutative N/A

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

      11. distribute-lft-in N/A

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

   9. Applied rewrites 98.7%

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

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

   1. Initial program 50.4%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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. lower-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. lower-*.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. lower-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. lower-fma.f32 94.8

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

   5. Applied rewrites 94.8%

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

      1. lift-*.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. +-commutative N/A

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

      5. flip-+ N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f32 N/A

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

   7. Applied rewrites 94.4%

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

   9. Applied rewrites 95.0%

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

4. Final simplification 98.1%

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

Alternative 4: 97.4% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* PI 2.0))))
   (if (<= t_0 0.05000000074505806)
     (*
      (sqrt (- (log1p (- u1))))
      (fma
       (+ u2 u2)
       PI
       (* (* u2 (* u2 (* PI (* PI PI)))) (* u2 -1.3333333333333333))))
     (*
      (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 = u2 * (((float) M_PI) * 2.0f);
	float tmp;
	if (t_0 <= 0.05000000074505806f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf((u2 + u2), ((float) M_PI), ((u2 * (u2 * (((float) M_PI) * (((float) M_PI) * ((float) M_PI))))) * (u2 * -1.3333333333333333f)));
	} 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(u2 * Float32(Float32(pi) * Float32(2.0)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.05000000074505806))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(u2 + u2), Float32(pi), Float32(Float32(u2 * Float32(u2 * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))))) * Float32(u2 * Float32(-1.3333333333333333)))));
	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 56.7%

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

      1. 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. lower-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. lower-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 rewrites 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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f32 N/A

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

   7. Applied rewrites 98.6%

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

      1. lift-*.f32 N/A

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

      2. lift-PI.f32 N/A

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

      3. lift-PI.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. +-commutative N/A

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

      11. distribute-lft-in N/A

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

   9. Applied rewrites 98.7%

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

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

   1. Initial program 50.4%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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. lower-*.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. lower-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. lower-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. lower-fma.f32 94.9

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

   5. Applied rewrites 94.9%

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

4. Final simplification 98.1%

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

Alternative 5: 97.4% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* PI 2.0))))
   (if (<= t_0 0.05000000074505806)
     (*
      (sqrt (- (log1p (- u1))))
      (*
       u2
       (fma -1.3333333333333333 (* (* PI (* PI PI)) (* u2 u2)) (* PI 2.0))))
     (*
      (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 = u2 * (((float) M_PI) * 2.0f);
	float tmp;
	if (t_0 <= 0.05000000074505806f) {
		tmp = sqrtf(-log1pf(-u1)) * (u2 * fmaf(-1.3333333333333333f, ((((float) M_PI) * (((float) M_PI) * ((float) M_PI))) * (u2 * u2)), (((float) M_PI) * 2.0f)));
	} 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(u2 * Float32(Float32(pi) * Float32(2.0)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.05000000074505806))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(u2 * fma(Float32(-1.3333333333333333), Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))) * Float32(u2 * u2)), Float32(Float32(pi) * Float32(2.0)))));
	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 56.7%

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

      1. 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. lower-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. lower-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 rewrites 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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f32 N/A

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

   7. Applied rewrites 98.6%

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

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

   1. Initial program 50.4%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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. lower-*.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. lower-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. lower-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. lower-fma.f32 94.9

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

   5. Applied rewrites 94.9%

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

4. Final simplification 98.0%

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

Alternative 6: 94.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.0020000000949949026:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{-u1 \cdot \mathsf{fma}\left(u1, -0.5, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 (* PI 2.0)) 0.0020000000949949026)
   (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))
   (* (sin (* PI (+ u2 u2))) (sqrt (- (* u1 (fma u1 -0.5 -1.0)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * (((float) M_PI) * 2.0f)) <= 0.0020000000949949026f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	} else {
		tmp = sinf((((float) M_PI) * (u2 + u2))) * sqrtf(-(u1 * fmaf(u1, -0.5f, -1.0f)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(Float32(pi) * Float32(2.0))) <= Float32(0.0020000000949949026))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(sin(Float32(Float32(pi) * Float32(u2 + u2))) * sqrt(Float32(-Float32(u1 * fma(u1, Float32(-0.5), Float32(-1.0))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 57.6%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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. lower-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. lower-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 rewrites 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. lower-*.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. lower-*.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. lower-PI.f32 98.4

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

   7. Applied rewrites 98.4%

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

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

   1. Initial program 51.9%

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

   3. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 90.7

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

   5. Applied rewrites 90.7%

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

      1. lift-fma.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-sqrt.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. lift-sin.f32 N/A

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

      9. lift-*.f32 90.7

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

      10. lift-*.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f32 N/A

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

      14. count-2 N/A

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

      15. lift-*.f32 N/A

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

      16. lift-*.f32 N/A

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

      17. distribute-lft-out N/A

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

      18. lower-*.f32 N/A

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

      19. lower-+.f32 90.7

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

   7. Applied rewrites 90.7%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.0020000000949949026:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{-u1 \cdot \mathsf{fma}\left(u1, -0.5, -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9818999767303467:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{\left(-u1\right) \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right), -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9818999767303467)
   (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))
   (*
    (sin (* PI (+ u2 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) <= 0.9818999767303467f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	} else {
		tmp = sinf((((float) M_PI) * (u2 + 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(0.9818999767303467))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(sin(Float32(Float32(pi) * Float32(u2 + 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) < 0.981899977

   1. Initial program 97.1%

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

      1. sub-neg N/A

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

      2. lower-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. lower-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 rewrites 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. lower-*.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. lower-*.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. lower-PI.f32 89.1

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

   7. Applied rewrites 89.1%

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

   if 0.981899977 < (-.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. 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. lower-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. lower-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 rewrites 98.4%

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

      1. lift-PI.f32 N/A

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

      2. associate-*l* N/A

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

      3. sin-2 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-sin.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-cos.f32 N/A

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

      9. lower-*.f32 98.2

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

   6. Applied rewrites 98.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lift-neg.f32 N/A

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

      2. lift-log1p.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-sqrt.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-sin.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. lift-cos.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. lift-*.f32 N/A

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

      13. lift-*.f32 98.2

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

   8. Applied rewrites 98.4%

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

   9. Taylor expanded in u1 around 0

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

       1. lower-*.f32 N/A

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

       2. sub-neg N/A

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

       3. metadata-eval N/A

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

       4. lower-fma.f32 N/A

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

       5. sub-neg N/A

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

       6. *-commutative N/A

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

       7. metadata-eval N/A

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

       8. lower-fma.f32 98.3

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

   11. Applied rewrites 98.3%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9818999767303467:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{\left(-u1\right) \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right), -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.0020000000949949026:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1 \cdot 0.5, u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 (* PI 2.0)) 0.0020000000949949026)
   (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))
   (* (sin (* PI (+ u2 u2))) (sqrt (fma u1 (* u1 0.5) u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * (((float) M_PI) * 2.0f)) <= 0.0020000000949949026f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	} else {
		tmp = sinf((((float) M_PI) * (u2 + u2))) * sqrtf(fmaf(u1, (u1 * 0.5f), u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(Float32(pi) * Float32(2.0))) <= Float32(0.0020000000949949026))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(sin(Float32(Float32(pi) * Float32(u2 + u2))) * sqrt(fma(u1, Float32(u1 * Float32(0.5)), u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 57.6%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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. lower-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. lower-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 rewrites 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. lower-*.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. lower-*.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. lower-PI.f32 98.4

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

   7. Applied rewrites 98.4%

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

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

   1. Initial program 51.9%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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. lower-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. lower-neg.f32 98.3

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

   4. Applied rewrites 98.3%

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

      1. lift-PI.f32 N/A

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

      2. associate-*l* N/A

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

      3. sin-2 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-sin.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-cos.f32 N/A

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

      9. lower-*.f32 97.8

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

   6. Applied rewrites 97.8%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lift-neg.f32 N/A

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

      2. lift-log1p.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-sqrt.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-sin.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. lift-cos.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. lift-*.f32 N/A

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

      13. lift-*.f32 97.8

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

   8. Applied rewrites 98.3%

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

   9. Taylor expanded in u1 around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f32 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f32 90.7

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

   11. Applied rewrites 90.7%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.0020000000949949026:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1 \cdot 0.5, u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.2% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sin (* u2 (* PI 2.0)))
  (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) {
	return sinf((u2 * (((float) M_PI) * 2.0f))) * 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(sin(Float32(u2 * Float32(Float32(pi) * Float32(2.0)))) * 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 55.7%

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

3. Taylor expanded in u1 around 0

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

   1. lower-*.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. lower-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. lower-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. lower-fma.f32 94.6

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

5. Applied rewrites 94.6%

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

6. Final simplification 94.6%

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

Alternative 10: 95.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9818999767303467:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1 \cdot \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9818999767303467)
   (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))
   (*
    (sin (* PI (+ u2 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) <= 0.9818999767303467f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	} else {
		tmp = sinf((((float) M_PI) * (u2 + 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(0.9818999767303467))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(sin(Float32(Float32(pi) * Float32(u2 + u2))) * sqrt(fma(u1, Float32(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) < 0.981899977

   1. Initial program 97.1%

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

      1. sub-neg N/A

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

      2. lower-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. lower-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 rewrites 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. lower-*.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. lower-*.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. lower-PI.f32 89.1

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

   7. Applied rewrites 89.1%

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

   if 0.981899977 < (-.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. 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. lower-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. lower-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 rewrites 98.4%

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

      1. lift-PI.f32 N/A

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

      2. associate-*l* N/A

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

      3. sin-2 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-sin.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-cos.f32 N/A

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

      9. lower-*.f32 98.2

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

   6. Applied rewrites 98.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(2 \cdot \left(\sin \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lift-neg.f32 N/A

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

      2. lift-log1p.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-sqrt.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-sin.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. lift-cos.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. lift-*.f32 N/A

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

      13. lift-*.f32 98.2

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

   8. Applied rewrites 98.4%

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

   9. Taylor expanded in u1 around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. lower-fma.f32 N/A

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

       5. lower-*.f32 N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f32 98.3

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

   11. Applied rewrites 98.3%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9818999767303467:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1 \cdot \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 93.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.7%

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

3. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
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. lower-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. lower-*.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. lower-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. lower-fma.f32 94.6

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

5. Applied rewrites 94.6%

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

   1. lift-*.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-fma.f32 N/A

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

   4. +-commutative N/A

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

   5. flip-+ N/A

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

   6. sqrt-div N/A

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

   7. lower-/.f32 N/A

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

7. Applied rewrites 94.2%

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

9. Applied rewrites 94.6%

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

10. Final simplification 94.6%

    \[\leadsto \sin \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)} \]
11. Add Preprocessing

Alternative 12: 87.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ u2 \cdot \left(\sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.26666666666666666\right), u2 \cdot u2, \pi \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.3333333333333333\right)\right), u2 \cdot u2, \pi \cdot 2\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  u2
  (*
   (sqrt (fma (* u1 u1) (fma u1 (fma u1 0.25 0.3333333333333333) 0.5) u1))
   (fma
    (fma
     (* (* PI PI) (* (* PI (* PI PI)) 0.26666666666666666))
     (* u2 u2)
     (* PI (* (* PI PI) -1.3333333333333333)))
    (* u2 u2)
    (* PI 2.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return u2 * (sqrtf(fmaf((u1 * u1), fmaf(u1, fmaf(u1, 0.25f, 0.3333333333333333f), 0.5f), u1)) * fmaf(fmaf(((((float) M_PI) * ((float) M_PI)) * ((((float) M_PI) * (((float) M_PI) * ((float) M_PI))) * 0.26666666666666666f)), (u2 * u2), (((float) M_PI) * ((((float) M_PI) * ((float) M_PI)) * -1.3333333333333333f))), (u2 * u2), (((float) M_PI) * 2.0f)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(u2 * Float32(sqrt(fma(Float32(u1 * u1), fma(u1, fma(u1, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), u1)) * fma(fma(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))) * Float32(0.26666666666666666))), Float32(u2 * u2), Float32(Float32(pi) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-1.3333333333333333)))), Float32(u2 * u2), Float32(Float32(pi) * Float32(2.0)))))
end

Derivation

1. Initial program 55.7%

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

3. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
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. lower-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. lower-*.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. lower-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. lower-fma.f32 94.6

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

5. Applied rewrites 94.6%

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

6. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{1}{4}, \frac{1}{3}\right), \frac{1}{2}\right), u1\right)} \cdot \color{blue}{\left(u2 \cdot \left(2 \cdot \mathsf{PI}\left(\right) + {u2}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{4}{15} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right)} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

8. Applied rewrites 90.1%

   \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)} \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, 0.26666666666666666 \cdot {\pi}^{5}, -1.3333333333333333 \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right), 2 \cdot \pi\right)\right)} \]

10. Applied rewrites 90.1%

    \[\leadsto \color{blue}{\mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot \left(0.26666666666666666 \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \sqrt{\pi}\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{\pi}\right)\right)\right), \pi \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.3333333333333333\right)\right), 2 \cdot \pi\right) \cdot \left(u2 \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)}\right)} \]

12. Applied rewrites 90.1%

    \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.26666666666666666\right), u2 \cdot u2, \pi \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.3333333333333333\right)\right), u2 \cdot u2, \pi \cdot 2\right) \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)}\right) \cdot u2} \]

13. Final simplification 90.1%

    \[\leadsto u2 \cdot \left(\sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.26666666666666666\right), u2 \cdot u2, \pi \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.3333333333333333\right)\right), u2 \cdot u2, \pi \cdot 2\right)\right) \]
14. Add Preprocessing

Alternative 13: 87.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.26666666666666666\right), u2 \cdot u2, \pi \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.3333333333333333\right)\right), u2 \cdot u2, \pi \cdot 2\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (fma (* u1 u1) (fma u1 (fma u1 0.25 0.3333333333333333) 0.5) u1))
  (*
   u2
   (fma
    (fma
     (* (* PI PI) (* (* PI (* PI PI)) 0.26666666666666666))
     (* u2 u2)
     (* PI (* (* PI PI) -1.3333333333333333)))
    (* u2 u2)
    (* PI 2.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(fmaf((u1 * u1), fmaf(u1, fmaf(u1, 0.25f, 0.3333333333333333f), 0.5f), u1)) * (u2 * fmaf(fmaf(((((float) M_PI) * ((float) M_PI)) * ((((float) M_PI) * (((float) M_PI) * ((float) M_PI))) * 0.26666666666666666f)), (u2 * u2), (((float) M_PI) * ((((float) M_PI) * ((float) M_PI)) * -1.3333333333333333f))), (u2 * u2), (((float) M_PI) * 2.0f)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(fma(Float32(u1 * u1), fma(u1, fma(u1, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), u1)) * Float32(u2 * fma(fma(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))) * Float32(0.26666666666666666))), Float32(u2 * u2), Float32(Float32(pi) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-1.3333333333333333)))), Float32(u2 * u2), Float32(Float32(pi) * Float32(2.0)))))
end

Derivation

1. Initial program 55.7%

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

3. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
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. lower-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. lower-*.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. lower-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. lower-fma.f32 94.6

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

5. Applied rewrites 94.6%

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

6. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{1}{4}, \frac{1}{3}\right), \frac{1}{2}\right), u1\right)} \cdot \color{blue}{\left(u2 \cdot \left(2 \cdot \mathsf{PI}\left(\right) + {u2}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{4}{15} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right)} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

8. Applied rewrites 90.1%

   \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)} \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, 0.26666666666666666 \cdot {\pi}^{5}, -1.3333333333333333 \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right), 2 \cdot \pi\right)\right)} \]

10. Applied rewrites 90.1%

    \[\leadsto \color{blue}{\mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot \left(0.26666666666666666 \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \sqrt{\pi}\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{\pi}\right)\right)\right), \pi \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.3333333333333333\right)\right), 2 \cdot \pi\right) \cdot \left(u2 \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)}\right)} \]

12. Applied rewrites 90.1%

    \[\leadsto \color{blue}{\left(u2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.26666666666666666\right), u2 \cdot u2, \pi \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.3333333333333333\right)\right), u2 \cdot u2, \pi \cdot 2\right)\right) \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)}} \]

13. Final simplification 90.1%

    \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.26666666666666666\right), u2 \cdot u2, \pi \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.3333333333333333\right)\right), u2 \cdot u2, \pi \cdot 2\right)\right) \]
14. Add Preprocessing

Alternative 14: 83.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.0006699999794363976:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(u2 \cdot u2\right), \pi \cdot 2\right)\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1 \cdot 0.5, u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 (* PI 2.0)) 0.0006699999794363976)
   (*
    (sqrt (fma (* u1 u1) (fma u1 (fma u1 0.25 0.3333333333333333) 0.5) u1))
    (* 2.0 (* PI u2)))
   (*
    (* u2 (fma -1.3333333333333333 (* (* PI (* PI PI)) (* u2 u2)) (* PI 2.0)))
    (sqrt (fma u1 (* u1 0.5) u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * (((float) M_PI) * 2.0f)) <= 0.0006699999794363976f) {
		tmp = sqrtf(fmaf((u1 * u1), fmaf(u1, fmaf(u1, 0.25f, 0.3333333333333333f), 0.5f), u1)) * (2.0f * (((float) M_PI) * u2));
	} else {
		tmp = (u2 * fmaf(-1.3333333333333333f, ((((float) M_PI) * (((float) M_PI) * ((float) M_PI))) * (u2 * u2)), (((float) M_PI) * 2.0f))) * sqrtf(fmaf(u1, (u1 * 0.5f), u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(Float32(pi) * Float32(2.0))) <= Float32(0.0006699999794363976))
		tmp = Float32(sqrt(fma(Float32(u1 * u1), fma(u1, fma(u1, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), u1)) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(Float32(u2 * fma(Float32(-1.3333333333333333), Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))) * Float32(u2 * u2)), Float32(Float32(pi) * Float32(2.0)))) * sqrt(fma(u1, Float32(u1 * Float32(0.5)), u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 6.6999998e-4

   1. Initial program 57.6%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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. lower-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. lower-*.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. lower-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. lower-fma.f32 94.6

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

   5. Applied rewrites 94.6%

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

   6. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f32 N/A

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

   8. Applied rewrites 94.6%

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

   if 6.6999998e-4 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 52.0%

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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f32 N/A

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

   7. Applied rewrites 79.0%

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

   8. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 74.0

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

   10. Applied rewrites 74.0%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.0006699999794363976:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(u2 \cdot u2\right), \pi \cdot 2\right)\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1 \cdot 0.5, u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 84.9% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.7%

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

3. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
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. lower-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. lower-*.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. lower-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. lower-fma.f32 94.6

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

5. Applied rewrites 94.6%

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

6. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{1}{4}, \frac{1}{3}\right), \frac{1}{2}\right), u1\right)} \cdot \color{blue}{\left(u2 \cdot \left(2 \cdot \mathsf{PI}\left(\right) + {u2}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + \frac{4}{15} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right)} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

8. Applied rewrites 90.1%

   \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)} \cdot \color{blue}{\left(u2 \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2 \cdot u2, 0.26666666666666666 \cdot {\pi}^{5}, -1.3333333333333333 \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right), 2 \cdot \pi\right)\right)} \]

10. Applied rewrites 90.1%

    \[\leadsto \color{blue}{\mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(u2, u2 \cdot \left(0.26666666666666666 \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \sqrt{\pi}\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sqrt{\pi}\right)\right)\right), \pi \cdot \left(\left(\pi \cdot \pi\right) \cdot -1.3333333333333333\right)\right), 2 \cdot \pi\right) \cdot \left(u2 \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)}\right)} \]

11. Taylor expanded in u2 around 0

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

    1. lower-*.f32 N/A

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

    2. associate-*r* N/A

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

    3. associate-*r* N/A

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

    4. distribute-rgt-out N/A

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

    5. lower-*.f32 N/A

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

13. Applied rewrites 88.3%

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

14. Final simplification 88.3%

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

Alternative 16: 84.9% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.7%

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

3. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
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. lower-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. lower-*.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. lower-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. lower-fma.f32 94.6

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

5. Applied rewrites 94.6%

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

6. Taylor expanded in u2 around 0

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

8. Applied rewrites 88.3%

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

9. Final simplification 88.3%

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

Alternative 17: 83.2% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.7%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Add Preprocessing
3. 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. lower-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. lower-neg.f32 98.5

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f32 N/A

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

7. Applied rewrites 91.6%

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

8. Taylor expanded in u1 around 0

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

   1. lower-*.f32 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f32 N/A

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f32 86.7

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

10. Applied rewrites 86.7%

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

11. Final simplification 86.7%

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

Alternative 18: 83.3% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.7%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Add Preprocessing
3. 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. lower-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. lower-neg.f32 98.5

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f32 N/A

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

7. Applied rewrites 91.6%

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

8. Taylor expanded in u1 around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 86.6

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

10. Applied rewrites 86.6%

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

11. Final simplification 86.6%

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

Alternative 19: 81.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.009999999776482582:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(u2 \cdot u2\right), \pi \cdot 2\right)\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 (* PI 2.0)) 0.009999999776482582)
   (*
    (sqrt (fma (* u1 u1) (fma u1 (fma u1 0.25 0.3333333333333333) 0.5) u1))
    (* 2.0 (* PI u2)))
   (*
    (* u2 (fma -1.3333333333333333 (* (* PI (* PI PI)) (* u2 u2)) (* PI 2.0)))
    (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * (((float) M_PI) * 2.0f)) <= 0.009999999776482582f) {
		tmp = sqrtf(fmaf((u1 * u1), fmaf(u1, fmaf(u1, 0.25f, 0.3333333333333333f), 0.5f), u1)) * (2.0f * (((float) M_PI) * u2));
	} else {
		tmp = (u2 * fmaf(-1.3333333333333333f, ((((float) M_PI) * (((float) M_PI) * ((float) M_PI))) * (u2 * u2)), (((float) M_PI) * 2.0f))) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(Float32(pi) * Float32(2.0))) <= Float32(0.009999999776482582))
		tmp = Float32(sqrt(fma(Float32(u1 * u1), fma(u1, fma(u1, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), u1)) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(Float32(u2 * fma(Float32(-1.3333333333333333), Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))) * Float32(u2 * u2)), Float32(Float32(pi) * Float32(2.0)))) * sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00999999978

   1. Initial program 57.6%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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. lower-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. lower-*.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. lower-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. lower-fma.f32 94.4

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in u2 around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f32 N/A

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

   8. Applied rewrites 93.1%

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

   if 0.00999999978 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 50.1%

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

      1. sub-neg N/A

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

      2. lower-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. lower-neg.f32 98.0

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f32 N/A

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

   7. Applied rewrites 71.9%

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

   8. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 63.3

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

   10. Applied rewrites 63.3%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(\pi \cdot 2\right) \leq 0.009999999776482582:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(u2 \cdot u2\right), \pi \cdot 2\right)\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 77.7% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.7%

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

3. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
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. lower-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. lower-*.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. lower-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. lower-fma.f32 94.6

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

5. Applied rewrites 94.6%

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

6. Taylor expanded in u2 around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 N/A

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

   12. lower-sqrt.f32 N/A

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

   13. +-commutative N/A

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

   14. lower-fma.f32 N/A

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

8. Applied rewrites 81.5%

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

9. Final simplification 81.5%

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

Alternative 21: 4.7% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.7%

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

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. rem-square-sqrt N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f32 N/A

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

   6. lower-sqrt.f32 4.1

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

5. Applied rewrites 4.1%

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

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. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-sqrt.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 4.4

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

8. Applied rewrites 4.4%

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

   1. lift-sqrt.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 4.4

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

10. Applied rewrites 4.4%

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

11. Final simplification 4.4%

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

Alternative 22: 4.7% 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 55.7%

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

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. rem-square-sqrt N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f32 N/A

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

   6. lower-sqrt.f32 4.1

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

5. Applied rewrites 4.1%

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

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. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-sqrt.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 4.4

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

8. Applied rewrites 4.4%

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

9. Final simplification 4.4%

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

Reproduce

herbie shell --seed 2024219 
(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))))