Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.4% → 98.4%

Time: 12.4s

Alternatives: 15

Speedup: 4.5×

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: 57.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.2%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-neg N/A

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

   4. lower-log1p.f32 N/A

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

   5. 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. Add Preprocessing

Alternative 2: 97.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right)\\ t_1 := u1 \cdot t\_0\\ \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.035999998450279236:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{-\frac{u1 \cdot \mathsf{fma}\left(t\_1, t\_1, -1\right)}{\mathsf{fma}\left(u1, t\_0, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, \left(u2 \cdot u2\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (fma u1 (fma u1 -0.25 -0.3333333333333333) -0.5))
        (t_1 (* u1 t_0)))
   (if (<= (- (log (- 1.0 u1))) 0.035999998450279236)
     (*
      (sin (* (* 2.0 PI) u2))
      (sqrt (- (/ (* u1 (fma t_1 t_1 -1.0)) (fma u1 t_0 1.0)))))
     (*
      (sqrt (- (log1p (- u1))))
      (*
       u2
       (fma -1.3333333333333333 (* (* u2 u2) (* PI (* PI PI))) (* 2.0 PI)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = fmaf(u1, fmaf(u1, -0.25f, -0.3333333333333333f), -0.5f);
	float t_1 = u1 * t_0;
	float tmp;
	if (-logf((1.0f - u1)) <= 0.035999998450279236f) {
		tmp = sinf(((2.0f * ((float) M_PI)) * u2)) * sqrtf(-((u1 * fmaf(t_1, t_1, -1.0f)) / fmaf(u1, t_0, 1.0f)));
	} else {
		tmp = sqrtf(-log1pf(-u1)) * (u2 * fmaf(-1.3333333333333333f, ((u2 * u2) * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))), (2.0f * ((float) M_PI))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = fma(u1, fma(u1, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5))
	t_1 = Float32(u1 * t_0)
	tmp = Float32(0.0)
	if (Float32(-log(Float32(Float32(1.0) - u1))) <= Float32(0.035999998450279236))
		tmp = Float32(sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)) * sqrt(Float32(-Float32(Float32(u1 * fma(t_1, t_1, Float32(-1.0))) / fma(u1, t_0, Float32(1.0))))));
	else
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(u2 * fma(Float32(-1.3333333333333333), Float32(Float32(u2 * u2) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(2.0) * Float32(pi)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 0.0359999985

   1. Initial program 49.9%

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

   3. Taylor expanded in u1 around 0

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

      1. 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 98.3

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

   5. Applied rewrites 98.3%

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

   7. Applied rewrites 98.4%

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

   if 0.0359999985 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 98.0%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-neg N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.8

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   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 95.1%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.035999998450279236:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{-\frac{u1 \cdot \mathsf{fma}\left(u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, \left(u2 \cdot u2\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.035999998450279236:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{u1}, u1 \cdot u1, \left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, \left(u2 \cdot u2\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- (log (- 1.0 u1))) 0.035999998450279236)
   (*
    (sin (* (* 2.0 PI) u2))
    (sqrt
     (fma
      (/ 1.0 u1)
      (* u1 u1)
      (* (* u1 u1) (fma u1 (fma u1 0.25 0.3333333333333333) 0.5)))))
   (*
    (sqrt (- (log1p (- u1))))
    (*
     u2
     (fma -1.3333333333333333 (* (* u2 u2) (* PI (* PI PI))) (* 2.0 PI))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (-logf((1.0f - u1)) <= 0.035999998450279236f) {
		tmp = sinf(((2.0f * ((float) M_PI)) * u2)) * sqrtf(fmaf((1.0f / u1), (u1 * u1), ((u1 * u1) * fmaf(u1, fmaf(u1, 0.25f, 0.3333333333333333f), 0.5f))));
	} else {
		tmp = sqrtf(-log1pf(-u1)) * (u2 * fmaf(-1.3333333333333333f, ((u2 * u2) * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))), (2.0f * ((float) M_PI))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(-log(Float32(Float32(1.0) - u1))) <= Float32(0.035999998450279236))
		tmp = Float32(sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)) * sqrt(fma(Float32(Float32(1.0) / u1), Float32(u1 * u1), Float32(Float32(u1 * u1) * fma(u1, fma(u1, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5))))));
	else
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(u2 * fma(Float32(-1.3333333333333333), Float32(Float32(u2 * u2) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(2.0) * Float32(pi)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 0.0359999985

   1. Initial program 49.9%

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-*.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f32 98.3

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

   7. Applied rewrites 98.3%

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

   9. Applied rewrites 98.4%

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

   if 0.0359999985 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 98.0%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-neg N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.8

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   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 95.1%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.035999998450279236:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{1}{u1}, u1 \cdot u1, \left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, \left(u2 \cdot u2\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.035999998450279236:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), u1 \cdot \left(u1 \cdot u1\right), \mathsf{fma}\left(u1, u1 \cdot 0.5, u1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, \left(u2 \cdot u2\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- (log (- 1.0 u1))) 0.035999998450279236)
   (*
    (sin (* (* 2.0 PI) u2))
    (sqrt
     (fma
      (fma u1 0.25 0.3333333333333333)
      (* u1 (* u1 u1))
      (fma u1 (* u1 0.5) u1))))
   (*
    (sqrt (- (log1p (- u1))))
    (*
     u2
     (fma -1.3333333333333333 (* (* u2 u2) (* PI (* PI PI))) (* 2.0 PI))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (-logf((1.0f - u1)) <= 0.035999998450279236f) {
		tmp = sinf(((2.0f * ((float) M_PI)) * u2)) * sqrtf(fmaf(fmaf(u1, 0.25f, 0.3333333333333333f), (u1 * (u1 * u1)), fmaf(u1, (u1 * 0.5f), u1)));
	} else {
		tmp = sqrtf(-log1pf(-u1)) * (u2 * fmaf(-1.3333333333333333f, ((u2 * u2) * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))), (2.0f * ((float) M_PI))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(-log(Float32(Float32(1.0) - u1))) <= Float32(0.035999998450279236))
		tmp = Float32(sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)) * sqrt(fma(fma(u1, Float32(0.25), Float32(0.3333333333333333)), Float32(u1 * Float32(u1 * u1)), fma(u1, Float32(u1 * Float32(0.5)), u1))));
	else
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(u2 * fma(Float32(-1.3333333333333333), Float32(Float32(u2 * u2) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(2.0) * Float32(pi)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 0.0359999985

   1. Initial program 49.9%

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

   4. Applied rewrites 48.1%

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

   5. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   6. 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 98.3

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

   7. Applied rewrites 98.3%

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

   9. Applied rewrites 98.4%

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

   if 0.0359999985 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 98.0%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-neg N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.8

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   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 95.1%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.035999998450279236:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), u1 \cdot \left(u1 \cdot u1\right), \mathsf{fma}\left(u1, u1 \cdot 0.5, u1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, \left(u2 \cdot u2\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.035999998450279236:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, \left(u2 \cdot u2\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- (log (- 1.0 u1))) 0.035999998450279236)
   (*
    (sin (* (* 2.0 PI) u2))
    (sqrt (fma (* u1 u1) (fma u1 (fma u1 0.25 0.3333333333333333) 0.5) u1)))
   (*
    (sqrt (- (log1p (- u1))))
    (*
     u2
     (fma -1.3333333333333333 (* (* u2 u2) (* PI (* PI PI))) (* 2.0 PI))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (-logf((1.0f - u1)) <= 0.035999998450279236f) {
		tmp = sinf(((2.0f * ((float) M_PI)) * u2)) * sqrtf(fmaf((u1 * u1), fmaf(u1, fmaf(u1, 0.25f, 0.3333333333333333f), 0.5f), u1));
	} else {
		tmp = sqrtf(-log1pf(-u1)) * (u2 * fmaf(-1.3333333333333333f, ((u2 * u2) * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))), (2.0f * ((float) M_PI))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(-log(Float32(Float32(1.0) - u1))) <= Float32(0.035999998450279236))
		tmp = Float32(sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)) * sqrt(fma(Float32(u1 * u1), fma(u1, fma(u1, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), u1)));
	else
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(u2 * fma(Float32(-1.3333333333333333), Float32(Float32(u2 * u2) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(2.0) * Float32(pi)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 0.0359999985

   1. Initial program 49.9%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

   if 0.0359999985 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 98.0%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-neg N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.8

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(u2 \cdot \left(\frac{-4}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   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 95.1%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.035999998450279236:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(u2 \cdot \mathsf{fma}\left(-1.3333333333333333, \left(u2 \cdot u2\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.035999998450279236:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- (log (- 1.0 u1))) 0.035999998450279236)
   (*
    (sin (* (* 2.0 PI) u2))
    (sqrt (fma (* u1 u1) (fma u1 (fma u1 0.25 0.3333333333333333) 0.5) u1)))
   (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (-logf((1.0f - u1)) <= 0.035999998450279236f) {
		tmp = sinf(((2.0f * ((float) M_PI)) * u2)) * sqrtf(fmaf((u1 * u1), fmaf(u1, fmaf(u1, 0.25f, 0.3333333333333333f), 0.5f), u1));
	} else {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(-log(Float32(Float32(1.0) - u1))) <= Float32(0.035999998450279236))
		tmp = Float32(sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)) * sqrt(fma(Float32(u1 * u1), fma(u1, fma(u1, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), u1)));
	else
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 0.0359999985

   1. Initial program 49.9%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

   if 0.0359999985 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 98.0%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-neg N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 98.8

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in u2 around 0

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

      1. 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 90.6

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

   7. Applied rewrites 90.6%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.035999998450279236:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.01600000075995922:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- (log (- 1.0 u1))) 0.01600000075995922)
   (*
    (sin (* (* 2.0 PI) u2))
    (sqrt (fma (* u1 u1) (fma u1 0.3333333333333333 0.5) u1)))
   (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (-logf((1.0f - u1)) <= 0.01600000075995922f) {
		tmp = sinf(((2.0f * ((float) M_PI)) * u2)) * sqrtf(fmaf((u1 * u1), fmaf(u1, 0.3333333333333333f, 0.5f), u1));
	} else {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(-log(Float32(Float32(1.0) - u1))) <= Float32(0.01600000075995922))
		tmp = Float32(sin(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)) * sqrt(fma(Float32(u1 * u1), fma(u1, Float32(0.3333333333333333), Float32(0.5)), u1)));
	else
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 0.0160000008

   1. Initial program 49.1%

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

   3. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f32 98.3

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

   5. Applied rewrites 98.3%

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

   if 0.0160000008 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 97.7%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-neg N/A

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

      4. lower-log1p.f32 N/A

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

      5. 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 90.3

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

   7. Applied rewrites 90.3%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.01600000075995922:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.0024999999441206455:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{\mathsf{fma}\left(u1, u1 \cdot 0.5, u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.0024999999441206455)
     (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))
     (* (sin t_0) (sqrt (fma u1 (* u1 0.5) u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.0024999999441206455f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	} else {
		tmp = sinf(t_0) * sqrtf(fmaf(u1, (u1 * 0.5f), u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(2.0) * Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0024999999441206455))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(sin(t_0) * sqrt(fma(u1, Float32(u1 * Float32(0.5)), u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00249999994

   1. Initial program 59.2%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-neg N/A

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

      4. lower-log1p.f32 N/A

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

      5. 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

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   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 98.1

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

   7. Applied rewrites 98.1%

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

   if 0.00249999994 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 52.1%

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

   3. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 90.4

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

   5. Applied rewrites 90.4%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.0024999999441206455:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1 \cdot 0.5, u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.0024999999441206455:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.0024999999441206455)
     (* (sqrt (- (log1p (- u1)))) (* 2.0 (* PI u2)))
     (* (sin t_0) (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.0024999999441206455f) {
		tmp = sqrtf(-log1pf(-u1)) * (2.0f * (((float) M_PI) * u2));
	} else {
		tmp = sinf(t_0) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(2.0) * Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0024999999441206455))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(2.0) * Float32(Float32(pi) * u2)));
	else
		tmp = Float32(sin(t_0) * sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00249999994

   1. Initial program 59.2%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-neg N/A

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

      4. lower-log1p.f32 N/A

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

      5. 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

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(2 \cdot \left(u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   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 98.1

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

   7. Applied rewrites 98.1%

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

   if 0.00249999994 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 52.1%

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

   4. Applied rewrites 77.4%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{0.25}\right)}^{2}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 79.4

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

   7. Applied rewrites 79.4%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.0024999999441206455:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(2 \cdot \left(\pi \cdot u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 57.8%

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

   3. Taylor expanded in u1 around 0

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

      1. 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 93.0

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

   5. Applied rewrites 93.0%

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

   6. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f32 N/A

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

   8. Applied rewrites 93.0%

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

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

   1. Initial program 53.6%

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

   4. Applied rewrites 76.7%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{0.25}\right)}^{2}} \cdot \sin \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 78.8

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

   7. Applied rewrites 78.8%

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

4. Final simplification 91.0%

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

Alternative 11: 85.2% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.2%

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

3. Taylor expanded in u1 around 0

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

   1. 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 93.1

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

5. Applied rewrites 93.1%

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

6. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f32 N/A

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

8. Applied rewrites 86.7%

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

9. Final simplification 86.7%

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

Alternative 12: 78.3% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.2%

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

3. Taylor expanded in u1 around 0

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

   1. 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 93.1

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

5. Applied rewrites 93.1%

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

6. Taylor expanded in u2 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 81.6

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

8. Applied rewrites 81.6%

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

9. Final simplification 81.6%

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

Alternative 13: 4.6% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.2%

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

3. Taylor expanded in u1 around 0

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

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 4.3

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

8. Applied rewrites 4.3%

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

10. Applied rewrites 4.3%

    \[\leadsto \frac{-u1}{\color{blue}{0 + \sqrt{u1}}} \cdot \left(2 \cdot \left(u2 \cdot \pi\right)\right) \]
11. Step-by-step derivation

12. Applied rewrites 4.3%

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

13. Final simplification 4.3%

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

Alternative 14: 4.6% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.2%

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

3. Taylor expanded in u1 around 0

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

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 4.3

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

8. Applied rewrites 4.3%

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

10. Applied rewrites 4.3%

    \[\leadsto \frac{-u1}{\color{blue}{0 + \sqrt{u1}}} \cdot \left(2 \cdot \left(u2 \cdot \pi\right)\right) \]
11. Step-by-step derivation

12. Applied rewrites 4.3%

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

13. Final simplification 4.3%

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

Alternative 15: 4.6% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.2%

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

3. Taylor expanded in u1 around 0

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

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 4.3

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

8. Applied rewrites 4.3%

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

9. Final simplification 4.3%

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

Reproduce

herbie shell --seed 2024222 
(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))))