normal distribution

Percentage Accurate: 99.4% → 99.6%

Time: 12.1s

Alternatives: 7

Speedup: 1.5×

Specification

\[\left(0 \leq u1 \land u1 \leq 1\right) \land \left(0 \leq u2 \land u2 \leq 1\right)\]

Math

\[\begin{array}{l} \\ \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 PI) u2)))
  0.5))

C

double code(double u1, double u2) {
	return (((1.0 / 6.0) * pow((-2.0 * log(u1)), 0.5)) * cos(((2.0 * ((double) M_PI)) * u2))) + 0.5;
}

Java

public static double code(double u1, double u2) {
	return (((1.0 / 6.0) * Math.pow((-2.0 * Math.log(u1)), 0.5)) * Math.cos(((2.0 * Math.PI) * u2))) + 0.5;
}

Python

def code(u1, u2):
	return (((1.0 / 6.0) * math.pow((-2.0 * math.log(u1)), 0.5)) * math.cos(((2.0 * math.pi) * u2))) + 0.5

Julia

function code(u1, u2)
	return Float64(Float64(Float64(Float64(1.0 / 6.0) * (Float64(-2.0 * log(u1)) ^ 0.5)) * cos(Float64(Float64(2.0 * pi) * u2))) + 0.5)
end

MATLAB

function tmp = code(u1, u2)
	tmp = (((1.0 / 6.0) * ((-2.0 * log(u1)) ^ 0.5)) * cos(((2.0 * pi) * u2))) + 0.5;
end

Wolfram

code[u1_, u2_] := N[(N[(N[(N[(1.0 / 6.0), $MachinePrecision] * N[Power[N[(-2.0 * N[Log[u1], $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * u2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]

Initial Program: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 PI) u2)))
  0.5))

C

double code(double u1, double u2) {
	return (((1.0 / 6.0) * pow((-2.0 * log(u1)), 0.5)) * cos(((2.0 * ((double) M_PI)) * u2))) + 0.5;
}

Java

public static double code(double u1, double u2) {
	return (((1.0 / 6.0) * Math.pow((-2.0 * Math.log(u1)), 0.5)) * Math.cos(((2.0 * Math.PI) * u2))) + 0.5;
}

Python

def code(u1, u2):
	return (((1.0 / 6.0) * math.pow((-2.0 * math.log(u1)), 0.5)) * math.cos(((2.0 * math.pi) * u2))) + 0.5

Julia

function code(u1, u2)
	return Float64(Float64(Float64(Float64(1.0 / 6.0) * (Float64(-2.0 * log(u1)) ^ 0.5)) * cos(Float64(Float64(2.0 * pi) * u2))) + 0.5)
end

MATLAB

function tmp = code(u1, u2)
	tmp = (((1.0 / 6.0) * ((-2.0 * log(u1)) ^ 0.5)) * cos(((2.0 * pi) * u2))) + 0.5;
end

Wolfram

code[u1_, u2_] := N[(N[(N[(N[(1.0 / 6.0), $MachinePrecision] * N[Power[N[(-2.0 * N[Log[u1], $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * u2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]

Alternative 1: 99.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(\left(0.16666666666666666 \cdot \sqrt{2}\right) \cdot \sqrt{-\log u1}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  (*
   (* (* 0.16666666666666666 (sqrt 2.0)) (sqrt (- (log u1))))
   (cos (* (* 2.0 PI) u2)))
  0.5))

C

double code(double u1, double u2) {
	return (((0.16666666666666666 * sqrt(2.0)) * sqrt(-log(u1))) * cos(((2.0 * ((double) M_PI)) * u2))) + 0.5;
}

Java

public static double code(double u1, double u2) {
	return (((0.16666666666666666 * Math.sqrt(2.0)) * Math.sqrt(-Math.log(u1))) * Math.cos(((2.0 * Math.PI) * u2))) + 0.5;
}

Python

def code(u1, u2):
	return (((0.16666666666666666 * math.sqrt(2.0)) * math.sqrt(-math.log(u1))) * math.cos(((2.0 * math.pi) * u2))) + 0.5

Julia

function code(u1, u2)
	return Float64(Float64(Float64(Float64(0.16666666666666666 * sqrt(2.0)) * sqrt(Float64(-log(u1)))) * cos(Float64(Float64(2.0 * pi) * u2))) + 0.5)
end

MATLAB

function tmp = code(u1, u2)
	tmp = (((0.16666666666666666 * sqrt(2.0)) * sqrt(-log(u1))) * cos(((2.0 * pi) * u2))) + 0.5;
end

Wolfram

code[u1_, u2_] := N[(N[(N[(N[(0.16666666666666666 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-N[Log[u1], $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * u2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
2. Add Preprocessing

3. Taylor expanded in u1 around inf

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{\log \left(\frac{1}{u1}\right)} \cdot \sqrt{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\color{blue}{\sqrt{\log \left(\frac{1}{u1}\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   3. log-rec N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   4. neg-lowering-neg.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   5. log-lowering-log.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\color{blue}{\log u1}\right)} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   6. sqrt-lowering-sqrt.f64 99.5

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{-\log u1} \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

5. Simplified 99.5%

   \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{-\log u1} \cdot \sqrt{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{6} \cdot \sqrt{2}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   3. pow1/2 N/A

      \[\leadsto \left(\left(\frac{1}{6} \cdot \color{blue}{{2}^{\frac{1}{2}}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   4. metadata-eval N/A

      \[\leadsto \left(\left(\frac{1}{6} \cdot {2}^{\color{blue}{\left(\frac{1}{4} + \frac{1}{4}\right)}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   5. pow-prod-up N/A

      \[\leadsto \left(\left(\frac{1}{6} \cdot \color{blue}{\left({2}^{\frac{1}{4}} \cdot {2}^{\frac{1}{4}}\right)}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   6. pow-prod-down N/A

      \[\leadsto \left(\left(\frac{1}{6} \cdot \color{blue}{{\left(2 \cdot 2\right)}^{\frac{1}{4}}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   7. metadata-eval N/A

      \[\leadsto \left(\left(\frac{1}{6} \cdot {\color{blue}{4}}^{\frac{1}{4}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   8. metadata-eval N/A

      \[\leadsto \left(\left(\frac{1}{6} \cdot {\color{blue}{\left(-2 \cdot -2\right)}}^{\frac{1}{4}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   9. pow-prod-down N/A

      \[\leadsto \left(\left(\frac{1}{6} \cdot \color{blue}{\left({-2}^{\frac{1}{4}} \cdot {-2}^{\frac{1}{4}}\right)}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   10. pow-prod-up N/A

       \[\leadsto \left(\left(\frac{1}{6} \cdot \color{blue}{{-2}^{\left(\frac{1}{4} + \frac{1}{4}\right)}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   11. metadata-eval N/A

       \[\leadsto \left(\left(\frac{1}{6} \cdot {-2}^{\color{blue}{\frac{1}{2}}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   12. pow1/2 N/A

       \[\leadsto \left(\left(\frac{1}{6} \cdot \color{blue}{\sqrt{-2}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   13. *-commutative N/A

       \[\leadsto \left(\color{blue}{\left(\sqrt{-2} \cdot \frac{1}{6}\right)} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   14. metadata-eval N/A

       \[\leadsto \left(\left(\sqrt{-2} \cdot \color{blue}{\frac{1}{6}}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \color{blue}{\left(\left(\sqrt{-2} \cdot \frac{1}{6}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

7. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\left(\left(0.16666666666666666 \cdot \sqrt{2}\right) \cdot \sqrt{-\log u1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
8. Add Preprocessing

Alternative 2: 99.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.16666666666666666 \cdot \left(\sqrt{-\log u1} \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right)\right), \sqrt{2}, 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (* 0.16666666666666666 (* (sqrt (- (log u1))) (cos (* PI (+ u2 u2)))))
  (sqrt 2.0)
  0.5))

C

double code(double u1, double u2) {
	return fma((0.16666666666666666 * (sqrt(-log(u1)) * cos((((double) M_PI) * (u2 + u2))))), sqrt(2.0), 0.5);
}

Julia

function code(u1, u2)
	return fma(Float64(0.16666666666666666 * Float64(sqrt(Float64(-log(u1))) * cos(Float64(pi * Float64(u2 + u2))))), sqrt(2.0), 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(0.16666666666666666 * N[(N[Sqrt[(-N[Log[u1], $MachinePrecision])], $MachinePrecision] * N[Cos[N[(Pi * N[(u2 + u2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
2. Add Preprocessing

3. Taylor expanded in u1 around inf

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{\log \left(\frac{1}{u1}\right)} \cdot \sqrt{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\color{blue}{\sqrt{\log \left(\frac{1}{u1}\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   3. log-rec N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   4. neg-lowering-neg.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   5. log-lowering-log.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\color{blue}{\log u1}\right)} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   6. sqrt-lowering-sqrt.f64 99.5

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{-\log u1} \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

5. Simplified 99.5%

   \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{-\log u1} \cdot \sqrt{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right)} + \frac{1}{2} \]

   2. associate-*r* N/A

      \[\leadsto \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \sqrt{2}\right)} + \frac{1}{2} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)\right) \cdot \sqrt{2}} + \frac{1}{2} \]

   4. pow1/2 N/A

      \[\leadsto \left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)\right) \cdot \color{blue}{{2}^{\frac{1}{2}}} + \frac{1}{2} \]

   5. metadata-eval N/A

      \[\leadsto \left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)\right) \cdot {2}^{\color{blue}{\left(\frac{1}{4} + \frac{1}{4}\right)}} + \frac{1}{2} \]

   6. pow-prod-up N/A

      \[\leadsto \left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)\right) \cdot \color{blue}{\left({2}^{\frac{1}{4}} \cdot {2}^{\frac{1}{4}}\right)} + \frac{1}{2} \]

   7. pow-prod-down N/A

      \[\leadsto \left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)\right) \cdot \color{blue}{{\left(2 \cdot 2\right)}^{\frac{1}{4}}} + \frac{1}{2} \]

   8. metadata-eval N/A

      \[\leadsto \left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)\right) \cdot {\color{blue}{4}}^{\frac{1}{4}} + \frac{1}{2} \]

   9. metadata-eval N/A

      \[\leadsto \left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)\right) \cdot {\color{blue}{\left(-2 \cdot -2\right)}}^{\frac{1}{4}} + \frac{1}{2} \]

   10. pow-prod-down N/A

       \[\leadsto \left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)\right) \cdot \color{blue}{\left({-2}^{\frac{1}{4}} \cdot {-2}^{\frac{1}{4}}\right)} + \frac{1}{2} \]

   11. pow-prod-up N/A

       \[\leadsto \left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)\right) \cdot \color{blue}{{-2}^{\left(\frac{1}{4} + \frac{1}{4}\right)}} + \frac{1}{2} \]

   12. metadata-eval N/A

       \[\leadsto \left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)\right) \cdot {-2}^{\color{blue}{\frac{1}{2}}} + \frac{1}{2} \]

   13. pow1/2 N/A

       \[\leadsto \left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)\right) \cdot \color{blue}{\sqrt{-2}} + \frac{1}{2} \]

   14. accelerator-lowering-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right), \sqrt{-2}, \frac{1}{2}\right)} \]

7. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \left(0.16666666666666666 \cdot \sqrt{-\log u1}\right), \sqrt{2}, 0.5\right)} \]
8. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right) \cdot \frac{1}{6}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}}, \sqrt{2}, \frac{1}{2}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \left(\cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right) \cdot \frac{1}{6}\right)}, \sqrt{2}, \frac{1}{2}\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)\right) \cdot \frac{1}{6}}, \sqrt{2}, \frac{1}{2}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)\right) \cdot \frac{1}{6}}, \sqrt{2}, \frac{1}{2}\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)\right)} \cdot \frac{1}{6}, \sqrt{2}, \frac{1}{2}\right) \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\sqrt{\mathsf{neg}\left(\log u1\right)}} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)\right) \cdot \frac{1}{6}, \sqrt{2}, \frac{1}{2}\right) \]

   7. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)\right) \cdot \frac{1}{6}, \sqrt{2}, \frac{1}{2}\right) \]

   8. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(\sqrt{\mathsf{neg}\left(\color{blue}{\log u1}\right)} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)\right) \cdot \frac{1}{6}, \sqrt{2}, \frac{1}{2}\right) \]

   9. cos-2 N/A

      \[\leadsto \mathsf{fma}\left(\left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \color{blue}{\left(\cos \left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot u2\right) - \sin \left(\mathsf{PI}\left(\right) \cdot u2\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)}\right) \cdot \frac{1}{6}, \sqrt{2}, \frac{1}{2}\right) \]

   10. cos-sum N/A

       \[\leadsto \mathsf{fma}\left(\left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot u2 + \mathsf{PI}\left(\right) \cdot u2\right)}\right) \cdot \frac{1}{6}, \sqrt{2}, \frac{1}{2}\right) \]

   11. cos-lowering-cos.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \color{blue}{\cos \left(\mathsf{PI}\left(\right) \cdot u2 + \mathsf{PI}\left(\right) \cdot u2\right)}\right) \cdot \frac{1}{6}, \sqrt{2}, \frac{1}{2}\right) \]

   12. distribute-lft-out N/A

       \[\leadsto \mathsf{fma}\left(\left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)}\right) \cdot \frac{1}{6}, \sqrt{2}, \frac{1}{2}\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)}\right) \cdot \frac{1}{6}, \sqrt{2}, \frac{1}{2}\right) \]

   14. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(u2 + u2\right)\right)\right) \cdot \frac{1}{6}, \sqrt{2}, \frac{1}{2}\right) \]

   15. +-lowering-+.f64 99.4

       \[\leadsto \mathsf{fma}\left(\left(\sqrt{-\log u1} \cdot \cos \left(\pi \cdot \color{blue}{\left(u2 + u2\right)}\right)\right) \cdot 0.16666666666666666, \sqrt{2}, 0.5\right) \]

9. Applied egg-rr 99.4%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\sqrt{-\log u1} \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right)\right) \cdot 0.16666666666666666}, \sqrt{2}, 0.5\right) \]

10. Final simplification 99.4%

    \[\leadsto \mathsf{fma}\left(0.16666666666666666 \cdot \left(\sqrt{-\log u1} \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right)\right), \sqrt{2}, 0.5\right) \]
11. Add Preprocessing

Alternative 3: 99.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.16666666666666666 \cdot \sqrt{\log u1 \cdot -2}, \cos \left(\pi \cdot \left(u2 + u2\right)\right), 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (* 0.16666666666666666 (sqrt (* (log u1) -2.0)))
  (cos (* PI (+ u2 u2)))
  0.5))

C

double code(double u1, double u2) {
	return fma((0.16666666666666666 * sqrt((log(u1) * -2.0))), cos((((double) M_PI) * (u2 + u2))), 0.5);
}

Julia

function code(u1, u2)
	return fma(Float64(0.16666666666666666 * sqrt(Float64(log(u1) * -2.0))), cos(Float64(pi * Float64(u2 + u2))), 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(0.16666666666666666 * N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(Pi * N[(u2 + u2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
2. Add Preprocessing

3. Taylor expanded in u1 around inf

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{\log \left(\frac{1}{u1}\right)} \cdot \sqrt{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\color{blue}{\sqrt{\log \left(\frac{1}{u1}\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   3. log-rec N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   4. neg-lowering-neg.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   5. log-lowering-log.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\color{blue}{\log u1}\right)} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   6. sqrt-lowering-sqrt.f64 99.5

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{-\log u1} \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

5. Simplified 99.5%

   \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{-\log u1} \cdot \sqrt{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right)} + \frac{1}{2} \]

   2. associate-*r* N/A

      \[\leadsto \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \sqrt{2}\right)} + \frac{1}{2} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)\right) \cdot \sqrt{2}} + \frac{1}{2} \]

   4. pow1/2 N/A

      \[\leadsto \left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)\right) \cdot \color{blue}{{2}^{\frac{1}{2}}} + \frac{1}{2} \]

   5. metadata-eval N/A

      \[\leadsto \left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)\right) \cdot {2}^{\color{blue}{\left(\frac{1}{4} + \frac{1}{4}\right)}} + \frac{1}{2} \]

   6. pow-prod-up N/A

      \[\leadsto \left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)\right) \cdot \color{blue}{\left({2}^{\frac{1}{4}} \cdot {2}^{\frac{1}{4}}\right)} + \frac{1}{2} \]

   7. pow-prod-down N/A

      \[\leadsto \left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)\right) \cdot \color{blue}{{\left(2 \cdot 2\right)}^{\frac{1}{4}}} + \frac{1}{2} \]

   8. metadata-eval N/A

      \[\leadsto \left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)\right) \cdot {\color{blue}{4}}^{\frac{1}{4}} + \frac{1}{2} \]

   9. metadata-eval N/A

      \[\leadsto \left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)\right) \cdot {\color{blue}{\left(-2 \cdot -2\right)}}^{\frac{1}{4}} + \frac{1}{2} \]

   10. pow-prod-down N/A

       \[\leadsto \left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)\right) \cdot \color{blue}{\left({-2}^{\frac{1}{4}} \cdot {-2}^{\frac{1}{4}}\right)} + \frac{1}{2} \]

   11. pow-prod-up N/A

       \[\leadsto \left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)\right) \cdot \color{blue}{{-2}^{\left(\frac{1}{4} + \frac{1}{4}\right)}} + \frac{1}{2} \]

   12. metadata-eval N/A

       \[\leadsto \left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)\right) \cdot {-2}^{\color{blue}{\frac{1}{2}}} + \frac{1}{2} \]

   13. pow1/2 N/A

       \[\leadsto \left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)\right) \cdot \color{blue}{\sqrt{-2}} + \frac{1}{2} \]

   14. accelerator-lowering-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right), \sqrt{-2}, \frac{1}{2}\right)} \]

7. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \left(0.16666666666666666 \cdot \sqrt{-\log u1}\right), \sqrt{2}, 0.5\right)} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)\right)} + \frac{1}{2} \]

   2. *-commutative N/A

      \[\leadsto \sqrt{2} \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)\right)} + \frac{1}{2} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)\right) \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)} + \frac{1}{2} \]

   4. associate-*l* N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \frac{1}{6}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right) + \frac{1}{2} \]

   5. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\frac{1}{6} \cdot \sqrt{2}\right)} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right) + \frac{1}{2} \]

   6. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{1}{6} \cdot \sqrt{2}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}, \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right), \frac{1}{2}\right)} \]

9. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666 \cdot \sqrt{\log u1 \cdot -2}, \cos \left(\pi \cdot \left(u2 + u2\right)\right), 0.5\right)} \]
10. Add Preprocessing

Alternative 4: 98.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 + \left(\frac{1}{6} \cdot \left(\sqrt{2} \cdot \sqrt{-\log u1}\right)\right) \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  0.5
  (*
   (* (/ 1.0 6.0) (* (sqrt 2.0) (sqrt (- (log u1)))))
   (fma u2 (* u2 (* -2.0 (* PI PI))) 1.0))))

C

double code(double u1, double u2) {
	return 0.5 + (((1.0 / 6.0) * (sqrt(2.0) * sqrt(-log(u1)))) * fma(u2, (u2 * (-2.0 * (((double) M_PI) * ((double) M_PI)))), 1.0));
}

Julia

function code(u1, u2)
	return Float64(0.5 + Float64(Float64(Float64(1.0 / 6.0) * Float64(sqrt(2.0) * sqrt(Float64(-log(u1))))) * fma(u2, Float64(u2 * Float64(-2.0 * Float64(pi * pi))), 1.0)))
end

Wolfram

code[u1_, u2_] := N[(0.5 + N[(N[(N[(1.0 / 6.0), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[(-N[Log[u1], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(u2 * N[(u2 * N[(-2.0 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
2. Add Preprocessing

3. Taylor expanded in u1 around inf

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{\log \left(\frac{1}{u1}\right)} \cdot \sqrt{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\color{blue}{\sqrt{\log \left(\frac{1}{u1}\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   3. log-rec N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   4. neg-lowering-neg.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   5. log-lowering-log.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\color{blue}{\log u1}\right)} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   6. sqrt-lowering-sqrt.f64 99.5

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{-\log u1} \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

5. Simplified 99.5%

   \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{-\log u1} \cdot \sqrt{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

6. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\left(-2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} + \frac{1}{2} \]

   2. *-commutative N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\color{blue}{\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot -2} + 1\right) + \frac{1}{2} \]

   3. rem-square-sqrt N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \color{blue}{\left(\sqrt{-2} \cdot \sqrt{-2}\right)} + 1\right) + \frac{1}{2} \]

   4. unpow2 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \color{blue}{{\left(\sqrt{-2}\right)}^{2}} + 1\right) + \frac{1}{2} \]

   5. associate-*r* N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\color{blue}{{u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\left(\sqrt{-2}\right)}^{2}\right)} + 1\right) + \frac{1}{2} \]

   6. unpow2 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\color{blue}{\left(u2 \cdot u2\right)} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\left(\sqrt{-2}\right)}^{2}\right) + 1\right) + \frac{1}{2} \]

   7. unpow2 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\left(u2 \cdot u2\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \color{blue}{\left(\sqrt{-2} \cdot \sqrt{-2}\right)}\right) + 1\right) + \frac{1}{2} \]

   8. rem-square-sqrt N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\left(u2 \cdot u2\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \color{blue}{-2}\right) + 1\right) + \frac{1}{2} \]

   9. *-commutative N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\left(u2 \cdot u2\right) \cdot \color{blue}{\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)} + 1\right) + \frac{1}{2} \]

   10. associate-*l* N/A

       \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\color{blue}{u2 \cdot \left(u2 \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + 1\right) + \frac{1}{2} \]

   11. accelerator-lowering-fma.f64 N/A

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

8. Simplified 99.1%

   \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{-\log u1} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(\left(\pi \cdot \pi\right) \cdot -2\right), 1\right)} + 0.5 \]

9. Final simplification 99.1%

   \[\leadsto 0.5 + \left(\frac{1}{6} \cdot \left(\sqrt{2} \cdot \sqrt{-\log u1}\right)\right) \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right) \]
10. Add Preprocessing

Alternative 5: 98.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\pi \cdot \pi, -2 \cdot \left(u2 \cdot u2\right), 1\right) \cdot \left(0.16666666666666666 \cdot \sqrt{-\log u1}\right), \sqrt{2}, 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (*
   (fma (* PI PI) (* -2.0 (* u2 u2)) 1.0)
   (* 0.16666666666666666 (sqrt (- (log u1)))))
  (sqrt 2.0)
  0.5))

C

double code(double u1, double u2) {
	return fma((fma((((double) M_PI) * ((double) M_PI)), (-2.0 * (u2 * u2)), 1.0) * (0.16666666666666666 * sqrt(-log(u1)))), sqrt(2.0), 0.5);
}

Julia

function code(u1, u2)
	return fma(Float64(fma(Float64(pi * pi), Float64(-2.0 * Float64(u2 * u2)), 1.0) * Float64(0.16666666666666666 * sqrt(Float64(-log(u1))))), sqrt(2.0), 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * N[(-2.0 * N[(u2 * u2), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(0.16666666666666666 * N[Sqrt[(-N[Log[u1], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
2. Add Preprocessing

3. Taylor expanded in u1 around inf

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{\log \left(\frac{1}{u1}\right)} \cdot \sqrt{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\color{blue}{\sqrt{\log \left(\frac{1}{u1}\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   3. log-rec N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   4. neg-lowering-neg.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   5. log-lowering-log.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\color{blue}{\log u1}\right)} \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   6. sqrt-lowering-sqrt.f64 99.5

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{-\log u1} \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

5. Simplified 99.5%

   \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{-\log u1} \cdot \sqrt{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right)} + \frac{1}{2} \]

   2. associate-*r* N/A

      \[\leadsto \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \color{blue}{\left(\left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right) \cdot \sqrt{2}\right)} + \frac{1}{2} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)\right) \cdot \sqrt{2}} + \frac{1}{2} \]

   4. pow1/2 N/A

      \[\leadsto \left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)\right) \cdot \color{blue}{{2}^{\frac{1}{2}}} + \frac{1}{2} \]

   5. metadata-eval N/A

      \[\leadsto \left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)\right) \cdot {2}^{\color{blue}{\left(\frac{1}{4} + \frac{1}{4}\right)}} + \frac{1}{2} \]

   6. pow-prod-up N/A

      \[\leadsto \left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)\right) \cdot \color{blue}{\left({2}^{\frac{1}{4}} \cdot {2}^{\frac{1}{4}}\right)} + \frac{1}{2} \]

   7. pow-prod-down N/A

      \[\leadsto \left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)\right) \cdot \color{blue}{{\left(2 \cdot 2\right)}^{\frac{1}{4}}} + \frac{1}{2} \]

   8. metadata-eval N/A

      \[\leadsto \left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)\right) \cdot {\color{blue}{4}}^{\frac{1}{4}} + \frac{1}{2} \]

   9. metadata-eval N/A

      \[\leadsto \left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)\right) \cdot {\color{blue}{\left(-2 \cdot -2\right)}}^{\frac{1}{4}} + \frac{1}{2} \]

   10. pow-prod-down N/A

       \[\leadsto \left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)\right) \cdot \color{blue}{\left({-2}^{\frac{1}{4}} \cdot {-2}^{\frac{1}{4}}\right)} + \frac{1}{2} \]

   11. pow-prod-up N/A

       \[\leadsto \left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)\right) \cdot \color{blue}{{-2}^{\left(\frac{1}{4} + \frac{1}{4}\right)}} + \frac{1}{2} \]

   12. metadata-eval N/A

       \[\leadsto \left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)\right) \cdot {-2}^{\color{blue}{\frac{1}{2}}} + \frac{1}{2} \]

   13. pow1/2 N/A

       \[\leadsto \left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)\right) \cdot \color{blue}{\sqrt{-2}} + \frac{1}{2} \]

   14. accelerator-lowering-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right), \sqrt{-2}, \frac{1}{2}\right)} \]

7. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \left(0.16666666666666666 \cdot \sqrt{-\log u1}\right), \sqrt{2}, 0.5\right)} \]

8. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({\mathsf{PI}\left(\right)}^{2}, -2 \cdot {u2}^{2}, 1\right)} \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right), \sqrt{2}, \frac{1}{2}\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, -2 \cdot {u2}^{2}, 1\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right), \sqrt{2}, \frac{1}{2}\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, -2 \cdot {u2}^{2}, 1\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right), \sqrt{2}, \frac{1}{2}\right) \]

   7. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right), -2 \cdot {u2}^{2}, 1\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right), \sqrt{2}, \frac{1}{2}\right) \]

   8. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}, -2 \cdot {u2}^{2}, 1\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right), \sqrt{2}, \frac{1}{2}\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \color{blue}{{u2}^{2} \cdot -2}, 1\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right), \sqrt{2}, \frac{1}{2}\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \color{blue}{{u2}^{2} \cdot -2}, 1\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right), \sqrt{2}, \frac{1}{2}\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \color{blue}{\left(u2 \cdot u2\right)} \cdot -2, 1\right) \cdot \left(\frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right), \sqrt{2}, \frac{1}{2}\right) \]

   12. *-lowering-*.f64 99.0

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\pi \cdot \pi, \color{blue}{\left(u2 \cdot u2\right)} \cdot -2, 1\right) \cdot \left(0.16666666666666666 \cdot \sqrt{-\log u1}\right), \sqrt{2}, 0.5\right) \]

10. Simplified 99.0%

    \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\pi \cdot \pi, \left(u2 \cdot u2\right) \cdot -2, 1\right)} \cdot \left(0.16666666666666666 \cdot \sqrt{-\log u1}\right), \sqrt{2}, 0.5\right) \]

11. Final simplification 99.0%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\pi \cdot \pi, -2 \cdot \left(u2 \cdot u2\right), 1\right) \cdot \left(0.16666666666666666 \cdot \sqrt{-\log u1}\right), \sqrt{2}, 0.5\right) \]
12. Add Preprocessing

Alternative 6: 98.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.16666666666666666 \cdot \sqrt{2}, \sqrt{-\log u1}, 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma (* 0.16666666666666666 (sqrt 2.0)) (sqrt (- (log u1))) 0.5))

C

double code(double u1, double u2) {
	return fma((0.16666666666666666 * sqrt(2.0)), sqrt(-log(u1)), 0.5);
}

Julia

function code(u1, u2)
	return fma(Float64(0.16666666666666666 * sqrt(2.0)), sqrt(Float64(-log(u1))), 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(0.16666666666666666 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-N[Log[u1], $MachinePrecision])], $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(\sqrt{\log u1} \cdot \sqrt{-2}\right) + \frac{1}{2}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sqrt{\log u1} \cdot \sqrt{-2}\right) \cdot \frac{1}{6}} + \frac{1}{2} \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{\sqrt{\log u1} \cdot \left(\sqrt{-2} \cdot \frac{1}{6}\right)} + \frac{1}{2} \]

   4. *-commutative N/A

      \[\leadsto \sqrt{\log u1} \cdot \color{blue}{\left(\frac{1}{6} \cdot \sqrt{-2}\right)} + \frac{1}{2} \]

   5. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\log u1}, \frac{1}{6} \cdot \sqrt{-2}, \frac{1}{2}\right)} \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\log u1}}, \frac{1}{6} \cdot \sqrt{-2}, \frac{1}{2}\right) \]

   7. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\log u1}}, \frac{1}{6} \cdot \sqrt{-2}, \frac{1}{2}\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\log u1}, \color{blue}{\sqrt{-2} \cdot \frac{1}{6}}, \frac{1}{2}\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\log u1}, \color{blue}{\sqrt{-2} \cdot \frac{1}{6}}, \frac{1}{2}\right) \]

   10. sqrt-lowering-sqrt.f64 0.0

       \[\leadsto \mathsf{fma}\left(\sqrt{\log u1}, \color{blue}{\sqrt{-2}} \cdot 0.16666666666666666, 0.5\right) \]

5. Simplified 0.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\log u1}, \sqrt{-2} \cdot 0.16666666666666666, 0.5\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sqrt{-2} \cdot \frac{1}{6}\right) \cdot \sqrt{\log u1}} + \frac{1}{2} \]

   2. metadata-eval N/A

      \[\leadsto \left(\sqrt{-2} \cdot \color{blue}{\frac{1}{6}}\right) \cdot \sqrt{\log u1} + \frac{1}{2} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot \sqrt{-2}\right)} \cdot \sqrt{\log u1} + \frac{1}{2} \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(\sqrt{-2} \cdot \sqrt{\log u1}\right)} + \frac{1}{2} \]

   5. sqrt-prod N/A

      \[\leadsto \frac{1}{6} \cdot \color{blue}{\sqrt{-2 \cdot \log u1}} + \frac{1}{2} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{-2 \cdot \log u1} \cdot \frac{1}{6}} + \frac{1}{2} \]

   7. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{-2 \cdot \log u1}, \frac{1}{6}, \frac{1}{2}\right)} \]

   8. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{-2 \cdot \log u1}}, \frac{1}{6}, \frac{1}{2}\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\log u1 \cdot -2}}, \frac{1}{6}, \frac{1}{2}\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\log u1 \cdot -2}}, \frac{1}{6}, \frac{1}{2}\right) \]

   11. log-lowering-log.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\log u1} \cdot -2}, \frac{1}{6}, \frac{1}{2}\right) \]

   12. metadata-eval 98.7

       \[\leadsto \mathsf{fma}\left(\sqrt{\log u1 \cdot -2}, \color{blue}{0.16666666666666666}, 0.5\right) \]

7. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\log u1 \cdot -2}, 0.16666666666666666, 0.5\right)} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \sqrt{\log u1 \cdot -2}} + \frac{1}{2} \]

   2. remove-double-neg N/A

      \[\leadsto \frac{1}{6} \cdot \sqrt{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\log u1 \cdot -2\right)\right)\right)}} + \frac{1}{2} \]

   3. distribute-rgt-neg-in N/A

      \[\leadsto \frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\color{blue}{\log u1 \cdot \left(\mathsf{neg}\left(-2\right)\right)}\right)} + \frac{1}{2} \]

   4. metadata-eval N/A

      \[\leadsto \frac{1}{6} \cdot \sqrt{\mathsf{neg}\left(\log u1 \cdot \color{blue}{2}\right)} + \frac{1}{2} \]

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

      \[\leadsto \frac{1}{6} \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left(\log u1\right)\right) \cdot 2}} + \frac{1}{2} \]

   6. sqrt-unprod N/A

      \[\leadsto \frac{1}{6} \cdot \color{blue}{\left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)} + \frac{1}{2} \]

   7. metadata-eval N/A

      \[\leadsto \color{blue}{\frac{1}{6}} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right) + \frac{1}{2} \]

   8. *-rgt-identity N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)\right) \cdot 1} + \frac{1}{2} \]

   9. *-rgt-identity N/A

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right)} + \frac{1}{2} \]

   10. metadata-eval N/A

       \[\leadsto \color{blue}{\frac{1}{6}} \cdot \left(\sqrt{\mathsf{neg}\left(\log u1\right)} \cdot \sqrt{2}\right) + \frac{1}{2} \]

   11. *-commutative N/A

       \[\leadsto \frac{1}{6} \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}\right)} + \frac{1}{2} \]

   12. associate-*l* N/A

       \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot \sqrt{2}\right) \cdot \sqrt{\mathsf{neg}\left(\log u1\right)}} + \frac{1}{2} \]

   13. accelerator-lowering-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6} \cdot \sqrt{2}, \sqrt{\mathsf{neg}\left(\log u1\right)}, \frac{1}{2}\right)} \]

9. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666 \cdot \sqrt{2}, \sqrt{-\log u1}, 0.5\right)} \]
10. Add Preprocessing

Alternative 7: 98.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt{\log u1 \cdot -2}, 0.16666666666666666, 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma (sqrt (* (log u1) -2.0)) 0.16666666666666666 0.5))

C

double code(double u1, double u2) {
	return fma(sqrt((log(u1) * -2.0)), 0.16666666666666666, 0.5);
}

Julia

function code(u1, u2)
	return fma(sqrt(Float64(log(u1) * -2.0)), 0.16666666666666666, 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(\sqrt{\log u1} \cdot \sqrt{-2}\right) + \frac{1}{2}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sqrt{\log u1} \cdot \sqrt{-2}\right) \cdot \frac{1}{6}} + \frac{1}{2} \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{\sqrt{\log u1} \cdot \left(\sqrt{-2} \cdot \frac{1}{6}\right)} + \frac{1}{2} \]

   4. *-commutative N/A

      \[\leadsto \sqrt{\log u1} \cdot \color{blue}{\left(\frac{1}{6} \cdot \sqrt{-2}\right)} + \frac{1}{2} \]

   5. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\log u1}, \frac{1}{6} \cdot \sqrt{-2}, \frac{1}{2}\right)} \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\log u1}}, \frac{1}{6} \cdot \sqrt{-2}, \frac{1}{2}\right) \]

   7. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\log u1}}, \frac{1}{6} \cdot \sqrt{-2}, \frac{1}{2}\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\log u1}, \color{blue}{\sqrt{-2} \cdot \frac{1}{6}}, \frac{1}{2}\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\log u1}, \color{blue}{\sqrt{-2} \cdot \frac{1}{6}}, \frac{1}{2}\right) \]

   10. sqrt-lowering-sqrt.f64 0.0

       \[\leadsto \mathsf{fma}\left(\sqrt{\log u1}, \color{blue}{\sqrt{-2}} \cdot 0.16666666666666666, 0.5\right) \]

5. Simplified 0.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\log u1}, \sqrt{-2} \cdot 0.16666666666666666, 0.5\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sqrt{-2} \cdot \frac{1}{6}\right) \cdot \sqrt{\log u1}} + \frac{1}{2} \]

   2. metadata-eval N/A

      \[\leadsto \left(\sqrt{-2} \cdot \color{blue}{\frac{1}{6}}\right) \cdot \sqrt{\log u1} + \frac{1}{2} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot \sqrt{-2}\right)} \cdot \sqrt{\log u1} + \frac{1}{2} \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(\sqrt{-2} \cdot \sqrt{\log u1}\right)} + \frac{1}{2} \]

   5. sqrt-prod N/A

      \[\leadsto \frac{1}{6} \cdot \color{blue}{\sqrt{-2 \cdot \log u1}} + \frac{1}{2} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{-2 \cdot \log u1} \cdot \frac{1}{6}} + \frac{1}{2} \]

   7. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{-2 \cdot \log u1}, \frac{1}{6}, \frac{1}{2}\right)} \]

   8. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{-2 \cdot \log u1}}, \frac{1}{6}, \frac{1}{2}\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\log u1 \cdot -2}}, \frac{1}{6}, \frac{1}{2}\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\log u1 \cdot -2}}, \frac{1}{6}, \frac{1}{2}\right) \]

   11. log-lowering-log.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\log u1} \cdot -2}, \frac{1}{6}, \frac{1}{2}\right) \]

   12. metadata-eval 98.7

       \[\leadsto \mathsf{fma}\left(\sqrt{\log u1 \cdot -2}, \color{blue}{0.16666666666666666}, 0.5\right) \]

7. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\log u1 \cdot -2}, 0.16666666666666666, 0.5\right)} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(FPCore (u1 u2)
  :name "normal distribution"
  :precision binary64
  :pre (and (and (<= 0.0 u1) (<= u1 1.0)) (and (<= 0.0 u2) (<= u2 1.0)))
  (+ (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 PI) u2))) 0.5))