Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.7% → 99.0%

Time: 14.1s

Alternatives: 18

Speedup: 21.0×

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.5%

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

   1. sub-neg N/A

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

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

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

   3. neg-lowering-neg.f32 99.1

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

4. Applied egg-rr 99.1%

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

Alternative 2: 97.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq 0.9995999932289124:\\ \;\;\;\;t\_0 \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1 \cdot u1, u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 PI) u2))))
   (if (<= t_0 0.9995999932289124)
     (*
      t_0
      (sqrt (fma (fma u1 (fma u1 0.25 0.3333333333333333) 0.5) (* u1 u1) u1)))
     (* (sqrt (- (log1p (- u1)))) (fma u2 (* u2 (* -2.0 (* PI PI))) 1.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf(((2.0f * ((float) M_PI)) * u2));
	float tmp;
	if (t_0 <= 0.9995999932289124f) {
		tmp = t_0 * sqrtf(fmaf(fmaf(u1, fmaf(u1, 0.25f, 0.3333333333333333f), 0.5f), (u1 * u1), u1));
	} else {
		tmp = sqrtf(-log1pf(-u1)) * fmaf(u2, (u2 * (-2.0f * (((float) M_PI) * ((float) M_PI)))), 1.0f);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9995999932289124))
		tmp = Float32(t_0 * sqrt(fma(fma(u1, fma(u1, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), Float32(u1 * u1), u1)));
	else
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(u2, Float32(u2 * Float32(Float32(-2.0) * Float32(Float32(pi) * Float32(pi)))), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)) < 0.999599993

   1. Initial program 54.6%

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

   3. Taylor expanded in u1 around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. +-commutative N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f32 92.7

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

   5. Simplified 92.7%

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

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

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

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

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

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

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

      8. *-lowering-*.f32 93.0

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

   7. Applied egg-rr 93.0%

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

   if 0.999599993 < (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   1. Initial program 57.1%

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

      1. sub-neg N/A

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

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

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

      3. neg-lowering-neg.f32 99.5

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(1 + -2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(-2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(\color{blue}{\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot -2} + 1\right) \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(\color{blue}{{u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot -2\right)} + 1\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left({u2}^{2} \cdot \color{blue}{\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)} + 1\right) \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(\color{blue}{\left(u2 \cdot u2\right)} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right) \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(\color{blue}{u2 \cdot \left(u2 \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + 1\right) \]

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

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right), 1\right)} \]

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

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \mathsf{fma}\left(u2, \color{blue}{u2 \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, 1\right) \]

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

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \color{blue}{\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, 1\right) \]

      10. unpow2 N/A

          \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), 1\right) \]

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

          \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), 1\right) \]

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

          \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right), 1\right) \]

      13. PI-lowering-PI.f32 99.5

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

   7. Simplified 99.5%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.9995999932289124:\\ \;\;\;\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1 \cdot u1, u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq 0.9995999932289124:\\ \;\;\;\;t\_0 \cdot \sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 PI) u2))))
   (if (<= t_0 0.9995999932289124)
     (*
      t_0
      (sqrt (* u1 (fma u1 (fma u1 (fma u1 0.25 0.3333333333333333) 0.5) 1.0))))
     (* (sqrt (- (log1p (- u1)))) (fma u2 (* u2 (* -2.0 (* PI PI))) 1.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf(((2.0f * ((float) M_PI)) * u2));
	float tmp;
	if (t_0 <= 0.9995999932289124f) {
		tmp = t_0 * sqrtf((u1 * fmaf(u1, fmaf(u1, fmaf(u1, 0.25f, 0.3333333333333333f), 0.5f), 1.0f)));
	} else {
		tmp = sqrtf(-log1pf(-u1)) * fmaf(u2, (u2 * (-2.0f * (((float) M_PI) * ((float) M_PI)))), 1.0f);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9995999932289124))
		tmp = Float32(t_0 * sqrt(Float32(u1 * fma(u1, fma(u1, fma(u1, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), Float32(1.0)))));
	else
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(u2, Float32(u2 * Float32(Float32(-2.0) * Float32(Float32(pi) * Float32(pi)))), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)) < 0.999599993

   1. Initial program 54.6%

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

   3. Taylor expanded in u1 around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. +-commutative N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f32 92.7

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

   5. Simplified 92.7%

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

   if 0.999599993 < (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   1. Initial program 57.1%

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

      1. sub-neg N/A

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

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

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

      3. neg-lowering-neg.f32 99.5

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(1 + -2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(-2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(\color{blue}{\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot -2} + 1\right) \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(\color{blue}{{u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot -2\right)} + 1\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left({u2}^{2} \cdot \color{blue}{\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)} + 1\right) \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(\color{blue}{\left(u2 \cdot u2\right)} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right) \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(\color{blue}{u2 \cdot \left(u2 \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + 1\right) \]

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

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right), 1\right)} \]

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

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \mathsf{fma}\left(u2, \color{blue}{u2 \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, 1\right) \]

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

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \color{blue}{\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, 1\right) \]

      10. unpow2 N/A

          \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), 1\right) \]

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

          \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), 1\right) \]

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

          \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right), 1\right) \]

      13. PI-lowering-PI.f32 99.5

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

   7. Simplified 99.5%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.9995999932289124:\\ \;\;\;\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq 0.9992600083351135:\\ \;\;\;\;t\_0 \cdot \sqrt{\mathsf{fma}\left(0.5, u1 \cdot u1, u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 PI) u2))))
   (if (<= t_0 0.9992600083351135)
     (* t_0 (sqrt (fma 0.5 (* u1 u1) u1)))
     (* (sqrt (- (log1p (- u1)))) (fma u2 (* u2 (* -2.0 (* PI PI))) 1.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf(((2.0f * ((float) M_PI)) * u2));
	float tmp;
	if (t_0 <= 0.9992600083351135f) {
		tmp = t_0 * sqrtf(fmaf(0.5f, (u1 * u1), u1));
	} else {
		tmp = sqrtf(-log1pf(-u1)) * fmaf(u2, (u2 * (-2.0f * (((float) M_PI) * ((float) M_PI)))), 1.0f);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9992600083351135))
		tmp = Float32(t_0 * sqrt(fma(Float32(0.5), Float32(u1 * u1), u1)));
	else
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(u2, Float32(u2 * Float32(Float32(-2.0) * Float32(Float32(pi) * Float32(pi)))), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)) < 0.999260008

   1. Initial program 55.7%

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

   3. Taylor expanded in u1 around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. +-commutative N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f32 92.2

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

   5. Simplified 92.2%

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

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

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

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

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

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

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

      8. *-lowering-*.f32 92.5

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

   7. Applied egg-rr 92.5%

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

   8. Taylor expanded in u1 around 0

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

   10. Simplified 88.5%

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

   if 0.999260008 < (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   1. Initial program 56.7%

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

      1. sub-neg N/A

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

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

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

      3. neg-lowering-neg.f32 99.5

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(1 + -2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(-2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(\color{blue}{\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot -2} + 1\right) \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(\color{blue}{{u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot -2\right)} + 1\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left({u2}^{2} \cdot \color{blue}{\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)} + 1\right) \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(\color{blue}{\left(u2 \cdot u2\right)} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right) \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(\color{blue}{u2 \cdot \left(u2 \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + 1\right) \]

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

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right), 1\right)} \]

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

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \mathsf{fma}\left(u2, \color{blue}{u2 \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, 1\right) \]

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

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \color{blue}{\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, 1\right) \]

      10. unpow2 N/A

          \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), 1\right) \]

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

          \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), 1\right) \]

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

          \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right), 1\right) \]

      13. PI-lowering-PI.f32 99.5

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

   7. Simplified 99.5%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.9992600083351135:\\ \;\;\;\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(0.5, u1 \cdot u1, u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq 0.9999974966049194:\\ \;\;\;\;t\_0 \cdot \sqrt{\mathsf{fma}\left(0.5, u1 \cdot u1, u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 PI) u2))))
   (if (<= t_0 0.9999974966049194)
     (* t_0 (sqrt (fma 0.5 (* u1 u1) u1)))
     (sqrt (- (log1p (- u1)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf(((2.0f * ((float) M_PI)) * u2));
	float tmp;
	if (t_0 <= 0.9999974966049194f) {
		tmp = t_0 * sqrtf(fmaf(0.5f, (u1 * u1), u1));
	} else {
		tmp = sqrtf(-log1pf(-u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999974966049194))
		tmp = Float32(t_0 * sqrt(fma(Float32(0.5), Float32(u1 * u1), u1)));
	else
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)) < 0.999997497

   1. Initial program 55.7%

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

   3. Taylor expanded in u1 around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. +-commutative N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f32 93.6

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

   5. Simplified 93.6%

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

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

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

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

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

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

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

      8. *-lowering-*.f32 93.8

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

   7. Applied egg-rr 93.8%

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

   8. Taylor expanded in u1 around 0

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

   10. Simplified 89.7%

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

   if 0.999997497 < (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   1. Initial program 57.0%

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

   3. Taylor expanded in u2 around 0

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

   5. Simplified 56.7%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. sub-neg N/A

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

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

         \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)}\right)} \cdot 1 \]

      3. neg-lowering-neg.f32 98.7

         \[\leadsto \sqrt{-\mathsf{log1p}\left(\color{blue}{-u1}\right)} \cdot 1 \]

   7. Applied egg-rr 98.7%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot 1 \]
   8. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \color{blue}{\sqrt{\mathsf{neg}\left(\log \left(1 + \left(\mathsf{neg}\left(u1\right)\right)\right)\right)}} \]

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

         \[\leadsto \color{blue}{\sqrt{\mathsf{neg}\left(\log \left(1 + \left(\mathsf{neg}\left(u1\right)\right)\right)\right)}} \]

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

         \[\leadsto \sqrt{\color{blue}{\mathsf{neg}\left(\log \left(1 + \left(\mathsf{neg}\left(u1\right)\right)\right)\right)}} \]

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

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

      5. neg-lowering-neg.f32 98.7

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

   9. Applied egg-rr 98.7%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.9999974966049194:\\ \;\;\;\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(0.5, u1 \cdot u1, u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq 0.9999974966049194:\\ \;\;\;\;t\_0 \cdot \sqrt{u1 \cdot \mathsf{fma}\left(u1, 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 PI) u2))))
   (if (<= t_0 0.9999974966049194)
     (* t_0 (sqrt (* u1 (fma u1 0.5 1.0))))
     (sqrt (- (log1p (- u1)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf(((2.0f * ((float) M_PI)) * u2));
	float tmp;
	if (t_0 <= 0.9999974966049194f) {
		tmp = t_0 * sqrtf((u1 * fmaf(u1, 0.5f, 1.0f)));
	} else {
		tmp = sqrtf(-log1pf(-u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999974966049194))
		tmp = Float32(t_0 * sqrt(Float32(u1 * fma(u1, Float32(0.5), Float32(1.0)))));
	else
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)) < 0.999997497

   1. Initial program 55.7%

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

   3. Taylor expanded in u1 around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f32 89.5

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

   5. Simplified 89.5%

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

   if 0.999997497 < (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   1. Initial program 57.0%

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

   3. Taylor expanded in u2 around 0

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

   5. Simplified 56.7%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. sub-neg N/A

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

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

         \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)}\right)} \cdot 1 \]

      3. neg-lowering-neg.f32 98.7

         \[\leadsto \sqrt{-\mathsf{log1p}\left(\color{blue}{-u1}\right)} \cdot 1 \]

   7. Applied egg-rr 98.7%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot 1 \]
   8. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \color{blue}{\sqrt{\mathsf{neg}\left(\log \left(1 + \left(\mathsf{neg}\left(u1\right)\right)\right)\right)}} \]

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

         \[\leadsto \color{blue}{\sqrt{\mathsf{neg}\left(\log \left(1 + \left(\mathsf{neg}\left(u1\right)\right)\right)\right)}} \]

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

         \[\leadsto \sqrt{\color{blue}{\mathsf{neg}\left(\log \left(1 + \left(\mathsf{neg}\left(u1\right)\right)\right)\right)}} \]

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

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

      5. neg-lowering-neg.f32 98.7

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

   9. Applied egg-rr 98.7%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.9999974966049194:\\ \;\;\;\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1 \cdot \mathsf{fma}\left(u1, 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq 0.9999899864196777:\\ \;\;\;\;t\_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 PI) u2))))
   (if (<= t_0 0.9999899864196777)
     (* t_0 (sqrt u1))
     (sqrt (- (log1p (- u1)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf(((2.0f * ((float) M_PI)) * u2));
	float tmp;
	if (t_0 <= 0.9999899864196777f) {
		tmp = t_0 * sqrtf(u1);
	} else {
		tmp = sqrtf(-log1pf(-u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999899864196777))
		tmp = Float32(t_0 * sqrt(u1));
	else
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)) < 0.999989986

   1. Initial program 54.5%

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

   3. Taylor expanded in u1 around 0

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

   5. Simplified 80.5%

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

   if 0.999989986 < (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   1. Initial program 57.5%

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

   3. Taylor expanded in u2 around 0

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

   5. Simplified 56.9%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{1} \]
   6. Step-by-step derivation

      1. sub-neg N/A

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

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

         \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)}\right)} \cdot 1 \]

      3. neg-lowering-neg.f32 98.2

         \[\leadsto \sqrt{-\mathsf{log1p}\left(\color{blue}{-u1}\right)} \cdot 1 \]

   7. Applied egg-rr 98.2%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot 1 \]
   8. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \color{blue}{\sqrt{\mathsf{neg}\left(\log \left(1 + \left(\mathsf{neg}\left(u1\right)\right)\right)\right)}} \]

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

         \[\leadsto \color{blue}{\sqrt{\mathsf{neg}\left(\log \left(1 + \left(\mathsf{neg}\left(u1\right)\right)\right)\right)}} \]

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

         \[\leadsto \sqrt{\color{blue}{\mathsf{neg}\left(\log \left(1 + \left(\mathsf{neg}\left(u1\right)\right)\right)\right)}} \]

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

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

      5. neg-lowering-neg.f32 98.2

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

   9. Applied egg-rr 98.2%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.9999899864196777:\\ \;\;\;\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right)\\ \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot \mathsf{fma}\left(t\_0, \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right) \cdot \left(u1 \cdot u1\right), -1\right)}{\mathsf{fma}\left(u1, t\_0, -1\right)}} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (fma u1 (fma u1 0.25 0.3333333333333333) 0.5)))
   (*
    (cos (* (* 2.0 PI) u2))
    (sqrt
     (/
      (* u1 (fma t_0 (* (fma u1 0.3333333333333333 0.5) (* u1 u1)) -1.0))
      (fma u1 t_0 -1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = fmaf(u1, fmaf(u1, 0.25f, 0.3333333333333333f), 0.5f);
	return cosf(((2.0f * ((float) M_PI)) * u2)) * sqrtf(((u1 * fmaf(t_0, (fmaf(u1, 0.3333333333333333f, 0.5f) * (u1 * u1)), -1.0f)) / fmaf(u1, t_0, -1.0f)));
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = fma(u1, fma(u1, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5))
	return Float32(cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)) * sqrt(Float32(Float32(u1 * fma(t_0, Float32(fma(u1, Float32(0.3333333333333333), Float32(0.5)) * Float32(u1 * u1)), Float32(-1.0))) / fma(u1, t_0, Float32(-1.0)))))
end

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in u1 around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. +-commutative N/A

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

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

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. accelerator-lowering-fma.f32 94.6

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

5. Simplified 94.6%

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

   1. *-commutative N/A

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

   2. flip-+ N/A

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

   3. associate-*l/ N/A

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

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

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

7. Applied egg-rr 94.7%

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

8. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

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

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

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

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

   6. unpow2 N/A

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

   7. *-lowering-*.f32 95.5

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

10. Simplified 95.5%

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

11. Final simplification 95.5%

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

Alternative 9: 94.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot \mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.3611111111111111, 0.3333333333333333\right), 0.25\right), -1\right)}{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), -1\right)}} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (cos (* (* 2.0 PI) u2))
  (sqrt
   (/
    (*
     u1
     (fma
      (* u1 u1)
      (fma u1 (fma u1 0.3611111111111111 0.3333333333333333) 0.25)
      -1.0))
    (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 cosf(((2.0f * ((float) M_PI)) * u2)) * sqrtf(((u1 * fmaf((u1 * u1), fmaf(u1, fmaf(u1, 0.3611111111111111f, 0.3333333333333333f), 0.25f), -1.0f)) / fmaf(u1, fmaf(u1, fmaf(u1, 0.25f, 0.3333333333333333f), 0.5f), -1.0f)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)) * sqrt(Float32(Float32(u1 * fma(Float32(u1 * u1), fma(u1, fma(u1, Float32(0.3611111111111111), Float32(0.3333333333333333)), Float32(0.25)), Float32(-1.0))) / fma(u1, fma(u1, fma(u1, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), Float32(-1.0)))))
end

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in u1 around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. +-commutative N/A

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

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

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. accelerator-lowering-fma.f32 94.6

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

5. Simplified 94.6%

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

   1. *-commutative N/A

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

   2. flip-+ N/A

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

   3. associate-*l/ N/A

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

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

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

7. Applied egg-rr 94.7%

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

8. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{\frac{\color{blue}{\left({u1}^{2} \cdot \left(\frac{1}{4} + u1 \cdot \left(\frac{1}{3} + \frac{13}{36} \cdot u1\right)\right) - 1\right)} \cdot u1}{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{1}{4}, \frac{1}{3}\right), \frac{1}{2}\right), -1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
9. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \sqrt{\frac{\color{blue}{\left({u1}^{2} \cdot \left(\frac{1}{4} + u1 \cdot \left(\frac{1}{3} + \frac{13}{36} \cdot u1\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot u1}{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{1}{4}, \frac{1}{3}\right), \frac{1}{2}\right), -1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   2. metadata-eval N/A

      \[\leadsto \sqrt{\frac{\left({u1}^{2} \cdot \left(\frac{1}{4} + u1 \cdot \left(\frac{1}{3} + \frac{13}{36} \cdot u1\right)\right) + \color{blue}{-1}\right) \cdot u1}{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{1}{4}, \frac{1}{3}\right), \frac{1}{2}\right), -1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

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

      \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{fma}\left({u1}^{2}, \frac{1}{4} + u1 \cdot \left(\frac{1}{3} + \frac{13}{36} \cdot u1\right), -1\right)} \cdot u1}{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{1}{4}, \frac{1}{3}\right), \frac{1}{2}\right), -1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. unpow2 N/A

      \[\leadsto \sqrt{\frac{\mathsf{fma}\left(\color{blue}{u1 \cdot u1}, \frac{1}{4} + u1 \cdot \left(\frac{1}{3} + \frac{13}{36} \cdot u1\right), -1\right) \cdot u1}{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{1}{4}, \frac{1}{3}\right), \frac{1}{2}\right), -1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

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

      \[\leadsto \sqrt{\frac{\mathsf{fma}\left(\color{blue}{u1 \cdot u1}, \frac{1}{4} + u1 \cdot \left(\frac{1}{3} + \frac{13}{36} \cdot u1\right), -1\right) \cdot u1}{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{1}{4}, \frac{1}{3}\right), \frac{1}{2}\right), -1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   6. +-commutative N/A

      \[\leadsto \sqrt{\frac{\mathsf{fma}\left(u1 \cdot u1, \color{blue}{u1 \cdot \left(\frac{1}{3} + \frac{13}{36} \cdot u1\right) + \frac{1}{4}}, -1\right) \cdot u1}{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{1}{4}, \frac{1}{3}\right), \frac{1}{2}\right), -1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

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

      \[\leadsto \sqrt{\frac{\mathsf{fma}\left(u1 \cdot u1, \color{blue}{\mathsf{fma}\left(u1, \frac{1}{3} + \frac{13}{36} \cdot u1, \frac{1}{4}\right)}, -1\right) \cdot u1}{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{1}{4}, \frac{1}{3}\right), \frac{1}{2}\right), -1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   8. +-commutative N/A

      \[\leadsto \sqrt{\frac{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \color{blue}{\frac{13}{36} \cdot u1 + \frac{1}{3}}, \frac{1}{4}\right), -1\right) \cdot u1}{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{1}{4}, \frac{1}{3}\right), \frac{1}{2}\right), -1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   9. *-commutative N/A

      \[\leadsto \sqrt{\frac{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \color{blue}{u1 \cdot \frac{13}{36}} + \frac{1}{3}, \frac{1}{4}\right), -1\right) \cdot u1}{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{1}{4}, \frac{1}{3}\right), \frac{1}{2}\right), -1\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   10. accelerator-lowering-fma.f32 94.9

       \[\leadsto \sqrt{\frac{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \color{blue}{\mathsf{fma}\left(u1, 0.3611111111111111, 0.3333333333333333\right)}, 0.25\right), -1\right) \cdot u1}{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), -1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

10. Simplified 94.9%

    \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.3611111111111111, 0.3333333333333333\right), 0.25\right), -1\right)} \cdot u1}{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), -1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

11. Final simplification 94.9%

    \[\leadsto \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\frac{u1 \cdot \mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.3611111111111111, 0.3333333333333333\right), 0.25\right), -1\right)}{\mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), -1\right)}} \]
12. Add Preprocessing

Alternative 10: 97.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.03799999877810478:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos t\_0 \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), u1 \cdot u1, u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.03799999877810478)
     (* (sqrt (- (log1p (- u1)))) (fma u2 (* u2 (* -2.0 (* PI PI))) 1.0))
     (* (cos t_0) (sqrt (fma (fma u1 0.3333333333333333 0.5) (* u1 u1) 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 <= 0.03799999877810478f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf(u2, (u2 * (-2.0f * (((float) M_PI) * ((float) M_PI)))), 1.0f);
	} else {
		tmp = cosf(t_0) * sqrtf(fmaf(fmaf(u1, 0.3333333333333333f, 0.5f), (u1 * u1), 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(0.03799999877810478))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(u2, Float32(u2 * Float32(Float32(-2.0) * Float32(Float32(pi) * Float32(pi)))), Float32(1.0)));
	else
		tmp = Float32(cos(t_0) * sqrt(fma(fma(u1, Float32(0.3333333333333333), Float32(0.5)), Float32(u1 * u1), u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.0379999988

   1. Initial program 56.7%

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

      1. sub-neg N/A

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

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

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

      3. neg-lowering-neg.f32 99.5

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(1 + -2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(-2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(\color{blue}{\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot -2} + 1\right) \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(\color{blue}{{u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot -2\right)} + 1\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left({u2}^{2} \cdot \color{blue}{\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)} + 1\right) \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(\color{blue}{\left(u2 \cdot u2\right)} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right) \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(\color{blue}{u2 \cdot \left(u2 \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + 1\right) \]

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

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right), 1\right)} \]

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

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \mathsf{fma}\left(u2, \color{blue}{u2 \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, 1\right) \]

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

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \color{blue}{\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, 1\right) \]

      10. unpow2 N/A

          \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), 1\right) \]

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

          \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), 1\right) \]

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

          \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right), 1\right) \]

      13. PI-lowering-PI.f32 99.5

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

   7. Simplified 99.5%

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

   if 0.0379999988 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 55.7%

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

   3. Taylor expanded in u1 around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. +-commutative N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f32 92.2

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

   5. Simplified 92.2%

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lft-identity N/A

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

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

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

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

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

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

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

      8. *-lowering-*.f32 92.5

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

   7. Applied egg-rr 92.5%

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

   8. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f32 90.9

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

   10. Simplified 90.9%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.03799999877810478:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), u1 \cdot u1, u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.03799999877810478:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos t\_0 \cdot \sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.03799999877810478)
     (* (sqrt (- (log1p (- u1)))) (fma u2 (* u2 (* -2.0 (* PI PI))) 1.0))
     (*
      (cos t_0)
      (sqrt (* u1 (fma u1 (fma u1 0.3333333333333333 0.5) 1.0)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.03799999877810478f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf(u2, (u2 * (-2.0f * (((float) M_PI) * ((float) M_PI)))), 1.0f);
	} else {
		tmp = cosf(t_0) * sqrtf((u1 * fmaf(u1, fmaf(u1, 0.3333333333333333f, 0.5f), 1.0f)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(2.0) * Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.03799999877810478))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(u2, Float32(u2 * Float32(Float32(-2.0) * Float32(Float32(pi) * Float32(pi)))), Float32(1.0)));
	else
		tmp = Float32(cos(t_0) * sqrt(Float32(u1 * fma(u1, fma(u1, Float32(0.3333333333333333), Float32(0.5)), Float32(1.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.0379999988

   1. Initial program 56.7%

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

      1. sub-neg N/A

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

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

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

      3. neg-lowering-neg.f32 99.5

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(1 + -2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\left(-2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(\color{blue}{\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot -2} + 1\right) \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(\color{blue}{{u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot -2\right)} + 1\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left({u2}^{2} \cdot \color{blue}{\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)} + 1\right) \]

      5. unpow2 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(\color{blue}{\left(u2 \cdot u2\right)} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right) \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \left(\color{blue}{u2 \cdot \left(u2 \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + 1\right) \]

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

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right), 1\right)} \]

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

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \mathsf{fma}\left(u2, \color{blue}{u2 \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, 1\right) \]

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

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \color{blue}{\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, 1\right) \]

      10. unpow2 N/A

          \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), 1\right) \]

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

          \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), 1\right) \]

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

          \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right), 1\right) \]

      13. PI-lowering-PI.f32 99.5

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

   7. Simplified 99.5%

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

   if 0.0379999988 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 55.7%

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

   3. Taylor expanded in u1 around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f32 90.7

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

   5. Simplified 90.7%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.03799999877810478:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.0% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0
         (sqrt
          (* u1 (fma u1 (fma u1 (fma u1 0.25 0.3333333333333333) 0.5) 1.0)))))
   (fma (* u2 u2) (* (* -2.0 (* PI PI)) t_0) t_0)))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf((u1 * fmaf(u1, fmaf(u1, fmaf(u1, 0.25f, 0.3333333333333333f), 0.5f), 1.0f)));
	return fmaf((u2 * u2), ((-2.0f * (((float) M_PI) * ((float) M_PI))) * t_0), t_0);
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(u1 * fma(u1, fma(u1, fma(u1, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), Float32(1.0))))
	return fma(Float32(u2 * u2), Float32(Float32(Float32(-2.0) * Float32(Float32(pi) * Float32(pi))) * t_0), t_0)
end

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in u1 around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. +-commutative N/A

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

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

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. accelerator-lowering-fma.f32 94.6

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

5. Simplified 94.6%

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

6. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

8. Simplified 84.2%

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

9. Final simplification 84.2%

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

Alternative 13: 84.0% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in u1 around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. +-commutative N/A

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

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

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. accelerator-lowering-fma.f32 94.6

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

5. Simplified 94.6%

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

   1. distribute-rgt-in N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

   8. *-lowering-*.f32 94.7

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

7. Applied egg-rr 94.7%

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

8. Taylor expanded in u2 around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt1-in N/A

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

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

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

10. Simplified 84.2%

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

11. Final simplification 84.2%

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

Alternative 14: 84.0% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in u1 around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. +-commutative N/A

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

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

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. accelerator-lowering-fma.f32 94.6

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

5. Simplified 94.6%

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

6. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

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

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

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

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

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

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

   10. unpow2 N/A

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

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

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

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

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

   13. PI-lowering-PI.f32 84.2

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

8. Simplified 84.2%

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

9. Final simplification 84.2%

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

Alternative 15: 76.8% accurate, 6.8× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (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 sqrtf((u1 * fmaf(u1, fmaf(u1, fmaf(u1, 0.25f, 0.3333333333333333f), 0.5f), 1.0f)));
}

Julia

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

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in u1 around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. +-commutative N/A

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

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

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. accelerator-lowering-fma.f32 94.6

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

5. Simplified 94.6%

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

6. Taylor expanded in u2 around 0

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

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

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

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

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

   3. +-commutative N/A

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

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

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

   5. +-commutative N/A

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

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

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. accelerator-lowering-fma.f32 76.9

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

8. Simplified 76.9%

   \[\leadsto \color{blue}{\sqrt{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), 1\right)}} \]
9. Add Preprocessing

Alternative 16: 75.6% accurate, 8.3× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (sqrt (* u1 (fma u1 (fma u1 0.3333333333333333 0.5) 1.0))))

C

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

Julia

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

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in u2 around 0

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

5. Simplified 48.5%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{1} \]

6. Taylor expanded in u1 around 0

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

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

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

   2. +-commutative N/A

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

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

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. accelerator-lowering-fma.f32 75.5

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

8. Simplified 75.5%

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

9. Final simplification 75.5%

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

Alternative 17: 73.1% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1 \cdot \mathsf{fma}\left(u1, 0.5, 1\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (sqrt (* u1 (fma u1 0.5 1.0))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 * fmaf(u1, 0.5f, 1.0f)));
}

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(u1 * fma(u1, Float32(0.5), Float32(1.0))))
end

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in u2 around 0

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

5. Simplified 48.5%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{1} \]

6. Taylor expanded in u1 around 0

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

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

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. accelerator-lowering-fma.f32 72.9

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

8. Simplified 72.9%

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

9. Final simplification 72.9%

   \[\leadsto \sqrt{u1 \cdot \mathsf{fma}\left(u1, 0.5, 1\right)} \]
10. Add Preprocessing

Alternative 18: 65.0% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (sqrt u1))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(u1);
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt(u1)
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(u1)
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(u1);
end

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in u2 around 0

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

5. Simplified 48.5%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{1} \]
6. Step-by-step derivation

   1. sub-neg N/A

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

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

      \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)}\right)} \cdot 1 \]

   3. neg-lowering-neg.f32 79.5

      \[\leadsto \sqrt{-\mathsf{log1p}\left(\color{blue}{-u1}\right)} \cdot 1 \]

7. Applied egg-rr 79.5%

   \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot 1 \]
8. Step-by-step derivation

   1. neg-log N/A

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

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

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

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

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

   4. unsub-neg N/A

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

   5. --lowering--.f32 46.8

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

9. Applied egg-rr 46.8%

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

10. Taylor expanded in u1 around 0

    \[\leadsto \color{blue}{\sqrt{u1}} \]
11. Step-by-step derivation

    1. sqrt-lowering-sqrt.f32 65.0

       \[\leadsto \color{blue}{\sqrt{u1}} \]

12. Simplified 65.0%

    \[\leadsto \color{blue}{\sqrt{u1}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024196 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_x"
  :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)))) (cos (* (* 2.0 PI) u2))))