Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.6% → 99.0%

Time: 14.2s

Alternatives: 16

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: 57.6% 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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.9%

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

4. Applied egg-rr 99.1%

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

Alternative 2: 97.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot \sqrt{-\log \left(1 - u1\right)} \leq 0.12999999523162842:\\ \;\;\;\;t\_0 \cdot \sqrt{-\mathsf{fma}\left(u1, u1 \cdot \left(u1 \cdot \mathsf{fma}\left(u1, \left(u1 \cdot u1\right) \cdot -0.5, -1\right)\right) - u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, -0.6666666666666666\right), 0.5\right), -u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\pi \cdot \pi, -2 \cdot \left(u2 \cdot u2\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 PI) u2))))
   (if (<= (* t_0 (sqrt (- (log (- 1.0 u1))))) 0.12999999523162842)
     (*
      t_0
      (sqrt
       (-
        (fma
         u1
         (-
          (* u1 (* u1 (fma u1 (* (* u1 u1) -0.5) -1.0)))
          (* u1 (fma u1 (fma u1 0.25 -0.6666666666666666) 0.5)))
         (- u1)))))
     (* (sqrt (- (log1p (- u1)))) (fma (* PI PI) (* -2.0 (* u2 u2)) 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 * sqrtf(-logf((1.0f - u1)))) <= 0.12999999523162842f) {
		tmp = t_0 * sqrtf(-fmaf(u1, ((u1 * (u1 * fmaf(u1, ((u1 * u1) * -0.5f), -1.0f))) - (u1 * fmaf(u1, fmaf(u1, 0.25f, -0.6666666666666666f), 0.5f))), -u1));
	} else {
		tmp = sqrtf(-log1pf(-u1)) * fmaf((((float) M_PI) * ((float) M_PI)), (-2.0f * (u2 * u2)), 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 (Float32(t_0 * sqrt(Float32(-log(Float32(Float32(1.0) - u1))))) <= Float32(0.12999999523162842))
		tmp = Float32(t_0 * sqrt(Float32(-fma(u1, Float32(Float32(u1 * Float32(u1 * fma(u1, Float32(Float32(u1 * u1) * Float32(-0.5)), Float32(-1.0)))) - Float32(u1 * fma(u1, fma(u1, Float32(0.25), Float32(-0.6666666666666666)), Float32(0.5)))), Float32(-u1)))));
	else
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(Float32(pi) * Float32(pi)), Float32(Float32(-2.0) * Float32(u2 * u2)), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.129999995

   1. Initial program 48.5%

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

   4. Applied egg-rr 45.8%

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

   5. Taylor expanded in u1 around 0

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

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

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

      11. cube-mult N/A

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

      12. unpow2 N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f32 46.9

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

   7. Simplified 46.9%

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

   8. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. accelerator-lowering-fma.f32 98.0

          \[\leadsto \sqrt{-\left(\left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \mathsf{fma}\left(u1 \cdot \left(u1 \cdot u1\right), -0.5, -1\right) - \mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \color{blue}{\mathsf{fma}\left(u1, 0.25, -0.6666666666666666\right)}, 0.5\right), u1\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   10. Simplified 98.0%

       \[\leadsto \sqrt{-\left(\left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \mathsf{fma}\left(u1 \cdot \left(u1 \cdot u1\right), -0.5, -1\right) - \color{blue}{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, -0.6666666666666666\right), 0.5\right), u1\right)}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   11. Step-by-step derivation

       1. associate--r+ N/A

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

       2. sub-neg N/A

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

       3. associate-*l* N/A

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

       4. associate-*l* N/A

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

       5. distribute-lft-out-- N/A

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

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

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

   12. Applied egg-rr 98.0%

       \[\leadsto \sqrt{-\color{blue}{\mathsf{fma}\left(u1, u1 \cdot \left(u1 \cdot \mathsf{fma}\left(u1, \left(u1 \cdot u1\right) \cdot -0.5, -1\right)\right) - u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, -0.6666666666666666\right), 0.5\right), -u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   if 0.129999995 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 97.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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 99.2

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

   4. Applied egg-rr 99.2%

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

   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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

      7. unpow2 N/A

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

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

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

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

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 97.8

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

   7. Simplified 97.8%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)} \leq 0.12999999523162842:\\ \;\;\;\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{-\mathsf{fma}\left(u1, u1 \cdot \left(u1 \cdot \mathsf{fma}\left(u1, \left(u1 \cdot u1\right) \cdot -0.5, -1\right)\right) - u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, -0.6666666666666666\right), 0.5\right), -u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\pi \cdot \pi, -2 \cdot \left(u2 \cdot u2\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot \sqrt{-\log \left(1 - u1\right)} \leq 0.12999999523162842:\\ \;\;\;\;t\_0 \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\pi \cdot \pi, -2 \cdot \left(u2 \cdot u2\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 PI) u2))))
   (if (<= (* t_0 (sqrt (- (log (- 1.0 u1))))) 0.12999999523162842)
     (*
      t_0
      (sqrt (fma (* u1 u1) (fma u1 (fma u1 0.25 0.3333333333333333) 0.5) u1)))
     (* (sqrt (- (log1p (- u1)))) (fma (* PI PI) (* -2.0 (* u2 u2)) 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 * sqrtf(-logf((1.0f - u1)))) <= 0.12999999523162842f) {
		tmp = t_0 * sqrtf(fmaf((u1 * u1), fmaf(u1, fmaf(u1, 0.25f, 0.3333333333333333f), 0.5f), u1));
	} else {
		tmp = sqrtf(-log1pf(-u1)) * fmaf((((float) M_PI) * ((float) M_PI)), (-2.0f * (u2 * u2)), 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 (Float32(t_0 * sqrt(Float32(-log(Float32(Float32(1.0) - u1))))) <= Float32(0.12999999523162842))
		tmp = Float32(t_0 * sqrt(fma(Float32(u1 * u1), fma(u1, fma(u1, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), u1)));
	else
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(Float32(pi) * Float32(pi)), Float32(Float32(-2.0) * Float32(u2 * u2)), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.129999995

   1. Initial program 48.5%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f32 97.9

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

   5. Simplified 97.9%

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

   if 0.129999995 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 97.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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 99.2

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

   4. Applied egg-rr 99.2%

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

   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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

      7. unpow2 N/A

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

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

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

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

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 97.8

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

   7. Simplified 97.8%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)} \leq 0.12999999523162842:\\ \;\;\;\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\pi \cdot \pi, -2 \cdot \left(u2 \cdot u2\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \cdot \sqrt{-\log \left(1 - u1\right)} \leq 0.12999999523162842:\\ \;\;\;\;t\_0 \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\pi \cdot \pi, -2 \cdot \left(u2 \cdot u2\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 PI) u2))))
   (if (<= (* t_0 (sqrt (- (log (- 1.0 u1))))) 0.12999999523162842)
     (* t_0 (sqrt (fma (* u1 u1) (fma u1 0.3333333333333333 0.5) u1)))
     (* (sqrt (- (log1p (- u1)))) (fma (* PI PI) (* -2.0 (* u2 u2)) 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 * sqrtf(-logf((1.0f - u1)))) <= 0.12999999523162842f) {
		tmp = t_0 * sqrtf(fmaf((u1 * u1), fmaf(u1, 0.3333333333333333f, 0.5f), u1));
	} else {
		tmp = sqrtf(-log1pf(-u1)) * fmaf((((float) M_PI) * ((float) M_PI)), (-2.0f * (u2 * u2)), 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 (Float32(t_0 * sqrt(Float32(-log(Float32(Float32(1.0) - u1))))) <= Float32(0.12999999523162842))
		tmp = Float32(t_0 * sqrt(fma(Float32(u1 * u1), fma(u1, Float32(0.3333333333333333), Float32(0.5)), u1)));
	else
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(Float32(pi) * Float32(pi)), Float32(Float32(-2.0) * Float32(u2 * u2)), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.129999995

   1. Initial program 48.5%

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

   3. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. accelerator-lowering-fma.f32 97.4

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

   5. Simplified 97.4%

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

   if 0.129999995 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 97.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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 99.2

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

   4. Applied egg-rr 99.2%

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

   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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

      7. unpow2 N/A

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

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

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

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

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 97.8

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

   7. Simplified 97.8%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)} \leq 0.12999999523162842:\\ \;\;\;\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\pi \cdot \pi, -2 \cdot \left(u2 \cdot u2\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq 0.9999949932098389:\\ \;\;\;\;t\_0 \cdot \sqrt{\mathsf{fma}\left(u1, u1 \cdot 0.5, u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 PI) u2))))
   (if (<= t_0 0.9999949932098389)
     (* t_0 (sqrt (fma u1 (* u1 0.5) 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 <= 0.9999949932098389f) {
		tmp = t_0 * sqrtf(fmaf(u1, (u1 * 0.5f), 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(0.9999949932098389))
		tmp = Float32(t_0 * sqrt(fma(u1, Float32(u1 * Float32(0.5)), 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)) < 0.999994993

   1. Initial program 53.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f32 89.4

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

   5. Simplified 89.4%

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

   if 0.999994993 < (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   1. Initial program 57.6%

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

   3. Taylor expanded in 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 57.3%

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

      1. *-rgt-identity N/A

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

      2. neg-log N/A

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

      3. flip-- N/A

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

      4. associate-/r/ N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. sum-log N/A

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

      9. neg-log N/A

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

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

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

      15. neg-log N/A

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

   7. Applied egg-rr 98.7%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.9999949932098389:\\ \;\;\;\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1 \cdot 0.5, u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq 0.999970018863678:\\ \;\;\;\;t\_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 PI) u2))))
   (if (<= t_0 0.999970018863678)
     (* 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 <= 0.999970018863678f) {
		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(0.999970018863678))
		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)) < 0.999970019

   1. Initial program 52.1%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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 79.9%

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

   if 0.999970019 < (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   1. Initial program 57.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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 57.1%

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

      1. *-rgt-identity N/A

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

      2. neg-log N/A

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

      3. flip-- N/A

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

      4. associate-/r/ N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. sum-log N/A

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

      9. neg-log N/A

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

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

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

      15. neg-log N/A

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

   7. Applied egg-rr 97.2%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.999970018863678:\\ \;\;\;\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.9%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
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 99.0

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

4. Applied egg-rr 99.0%

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

5. Final simplification 99.0%

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

Alternative 8: 96.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 56.1%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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 99.4

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

   4. Applied egg-rr 99.4%

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

   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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

      7. unpow2 N/A

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

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

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

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

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 99.0

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

   7. Simplified 99.0%

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

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

   1. Initial program 55.2%

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

   3. Taylor expanded in u1 around 0

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

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

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f32 87.0

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

   5. Simplified 87.0%

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

4. Final simplification 96.7%

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

Alternative 9: 94.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 57.6%

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

   3. Taylor expanded in 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 57.3%

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

      1. *-rgt-identity N/A

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

      2. neg-log N/A

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

      3. flip-- N/A

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

      4. associate-/r/ N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. sum-log N/A

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

      9. neg-log N/A

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

      10. +-commutative N/A

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

      11. sub-neg N/A

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

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

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

      15. neg-log N/A

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

   7. Applied egg-rr 98.7%

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

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

   1. Initial program 53.2%

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

   3. Taylor expanded in u1 around 0

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

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

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f32 89.4

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

   5. Simplified 89.4%

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

4. Final simplification 95.2%

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

Alternative 10: 80.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.999970018863678:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot \pi, -2 \cdot \left(u2 \cdot u2\right), 1\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (cos (* (* 2.0 PI) u2)) 0.999970018863678)
   (* (fma (* PI PI) (* -2.0 (* u2 u2)) 1.0) (sqrt u1))
   (sqrt (fma (* u1 u1) (fma u1 (fma u1 0.25 0.3333333333333333) 0.5) u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (cosf(((2.0f * ((float) M_PI)) * u2)) <= 0.999970018863678f) {
		tmp = fmaf((((float) M_PI) * ((float) M_PI)), (-2.0f * (u2 * u2)), 1.0f) * sqrtf(u1);
	} else {
		tmp = sqrtf(fmaf((u1 * u1), fmaf(u1, fmaf(u1, 0.25f, 0.3333333333333333f), 0.5f), u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)) <= Float32(0.999970018863678))
		tmp = Float32(fma(Float32(Float32(pi) * Float32(pi)), Float32(Float32(-2.0) * Float32(u2 * u2)), Float32(1.0)) * sqrt(u1));
	else
		tmp = sqrt(fma(Float32(u1 * u1), fma(u1, fma(u1, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)) < 0.999970019

   1. Initial program 52.1%

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

   4. Applied egg-rr 98.2%

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

   5. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

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

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

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

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. sqrt-lowering-sqrt.f32 90.0

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

   7. Simplified 90.0%

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

   8. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

      7. unpow2 N/A

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

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

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

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

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 62.7

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

   10. Simplified 62.7%

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

   11. Taylor expanded in u1 around 0

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

       1. sqrt-lowering-sqrt.f32 58.4

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

   13. Simplified 58.4%

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

   if 0.999970019 < (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   1. Initial program 57.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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 57.1%

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

   6. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f32 91.9

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

   8. Simplified 91.9%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.999970018863678:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot \pi, -2 \cdot \left(u2 \cdot u2\right), 1\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.999970018863678:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot \pi, -2 \cdot \left(u2 \cdot u2\right), 1\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1 \cdot \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (cos (* (* 2.0 PI) u2)) 0.999970018863678)
   (* (fma (* PI PI) (* -2.0 (* u2 u2)) 1.0) (sqrt u1))
   (sqrt (fma u1 (* u1 (fma u1 0.3333333333333333 0.5)) u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (cosf(((2.0f * ((float) M_PI)) * u2)) <= 0.999970018863678f) {
		tmp = fmaf((((float) M_PI) * ((float) M_PI)), (-2.0f * (u2 * u2)), 1.0f) * sqrtf(u1);
	} else {
		tmp = sqrtf(fmaf(u1, (u1 * fmaf(u1, 0.3333333333333333f, 0.5f)), u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)) <= Float32(0.999970018863678))
		tmp = Float32(fma(Float32(Float32(pi) * Float32(pi)), Float32(Float32(-2.0) * Float32(u2 * u2)), Float32(1.0)) * sqrt(u1));
	else
		tmp = sqrt(fma(u1, Float32(u1 * fma(u1, Float32(0.3333333333333333), Float32(0.5))), u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)) < 0.999970019

   1. Initial program 52.1%

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

   4. Applied egg-rr 98.2%

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

   5. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

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

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

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

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. sqrt-lowering-sqrt.f32 90.0

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

   7. Simplified 90.0%

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

   8. Taylor expanded in u2 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

      7. unpow2 N/A

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

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

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

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

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 62.7

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

   10. Simplified 62.7%

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

   11. Taylor expanded in u1 around 0

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

       1. sqrt-lowering-sqrt.f32 58.4

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

   13. Simplified 58.4%

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

   if 0.999970019 < (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   1. Initial program 57.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. *-rgt-identity N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. accelerator-lowering-fma.f32 91.8

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

   5. Simplified 91.8%

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

   6. Taylor expanded in u2 around 0

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

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

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. accelerator-lowering-fma.f32 90.1

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

   8. Simplified 90.1%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.999970018863678:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot \pi, -2 \cdot \left(u2 \cdot u2\right), 1\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(u1, u1 \cdot \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(u2 \cdot u2, \left(\pi \cdot \pi\right) \cdot -2, 1\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, -0.6666666666666666\right), 0.5\right), u1 - \left(u1 \cdot u1\right) \cdot \left(u1 \cdot \mathsf{fma}\left(u1 \cdot u1, u1 \cdot -0.5, -1\right)\right)\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (fma (* u2 u2) (* (* PI PI) -2.0) 1.0)
  (sqrt
   (fma
    u1
    (* u1 (fma u1 (fma u1 0.25 -0.6666666666666666) 0.5))
    (- u1 (* (* u1 u1) (* u1 (fma (* u1 u1) (* u1 -0.5) -1.0))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return fmaf((u2 * u2), ((((float) M_PI) * ((float) M_PI)) * -2.0f), 1.0f) * sqrtf(fmaf(u1, (u1 * fmaf(u1, fmaf(u1, 0.25f, -0.6666666666666666f), 0.5f)), (u1 - ((u1 * u1) * (u1 * fmaf((u1 * u1), (u1 * -0.5f), -1.0f))))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(fma(Float32(u2 * u2), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-2.0)), Float32(1.0)) * sqrt(fma(u1, Float32(u1 * fma(u1, fma(u1, Float32(0.25), Float32(-0.6666666666666666)), Float32(0.5))), Float32(u1 - Float32(Float32(u1 * u1) * Float32(u1 * fma(Float32(u1 * u1), Float32(u1 * Float32(-0.5)), Float32(-1.0))))))))
end

Derivation

1. Initial program 55.9%

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

4. Applied egg-rr 53.5%

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

5. Taylor expanded in u1 around 0

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

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

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

   2. cube-mult N/A

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

   3. unpow2 N/A

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

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

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

   5. unpow2 N/A

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

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

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

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

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

   11. cube-mult N/A

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

   12. unpow2 N/A

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

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

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

   14. unpow2 N/A

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

   15. *-lowering-*.f32 52.1

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

7. Simplified 52.1%

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

8. Taylor expanded in u1 around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. *-rgt-identity N/A

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

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

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

   7. unpow2 N/A

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

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

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

   9. +-commutative N/A

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

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

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

   11. sub-neg N/A

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

   12. *-commutative N/A

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

   13. metadata-eval N/A

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

   14. accelerator-lowering-fma.f32 94.3

       \[\leadsto \sqrt{-\left(\left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \mathsf{fma}\left(u1 \cdot \left(u1 \cdot u1\right), -0.5, -1\right) - \mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \color{blue}{\mathsf{fma}\left(u1, 0.25, -0.6666666666666666\right)}, 0.5\right), u1\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

10. Simplified 94.3%

    \[\leadsto \sqrt{-\left(\left(u1 \cdot \left(u1 \cdot u1\right)\right) \cdot \mathsf{fma}\left(u1 \cdot \left(u1 \cdot u1\right), -0.5, -1\right) - \color{blue}{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, -0.6666666666666666\right), 0.5\right), u1\right)}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

11. Taylor expanded in u2 around 0

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

13. Simplified 84.3%

    \[\leadsto \color{blue}{\mathsf{fma}\left(u2 \cdot u2, -2 \cdot \left(\pi \cdot \pi\right), 1\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, -0.6666666666666666\right), 0.5\right), u1 - \left(u1 \cdot u1\right) \cdot \left(u1 \cdot \mathsf{fma}\left(u1 \cdot u1, u1 \cdot -0.5, -1\right)\right)\right)}} \]

14. Final simplification 84.3%

    \[\leadsto \mathsf{fma}\left(u2 \cdot u2, \left(\pi \cdot \pi\right) \cdot -2, 1\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, -0.6666666666666666\right), 0.5\right), u1 - \left(u1 \cdot u1\right) \cdot \left(u1 \cdot \mathsf{fma}\left(u1 \cdot u1, u1 \cdot -0.5, -1\right)\right)\right)} \]
15. Add Preprocessing

Alternative 13: 82.8% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.9%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
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. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. *-rgt-identity N/A

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

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

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

   7. unpow2 N/A

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

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

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. accelerator-lowering-fma.f32 92.1

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

5. Simplified 92.1%

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

6. Taylor expanded in u2 around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt1-in N/A

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

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

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

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

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

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

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

   8. unpow2 N/A

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

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

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

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

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

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

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

   12. unpow2 N/A

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

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

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

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

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

8. Simplified 82.6%

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

9. Final simplification 82.6%

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

Alternative 14: 75.7% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.9%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
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. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. unpow2 N/A

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

   5. *-rgt-identity N/A

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

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

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

   7. unpow2 N/A

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

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

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. accelerator-lowering-fma.f32 92.1

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

5. Simplified 92.1%

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

6. Taylor expanded in u2 around 0

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

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

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

   2. +-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

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

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

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

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. accelerator-lowering-fma.f32 73.9

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

8. Simplified 73.9%

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

Alternative 15: 73.2% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.9%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
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 47.6%

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

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. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

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

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

   5. *-commutative N/A

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

   6. *-lowering-*.f32 71.4

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

8. Simplified 71.4%

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

9. Final simplification 71.4%

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

Alternative 16: 65.1% 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 55.9%

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

4. Applied egg-rr 75.4%

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

5. Taylor expanded in u2 around 0

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

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

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

   2. accelerator-lowering-log1p.f32 62.4

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

7. Simplified 62.4%

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

8. Taylor expanded in u1 around 0

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

   1. sqrt-lowering-sqrt.f32 63.8

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

10. Simplified 63.8%

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

Reproduce

herbie shell --seed 2024198 
(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))))