normal distribution

Percentage Accurate: 99.4% → 99.5%

Time: 9.1s

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: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
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. lower-*.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. lower-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. lower-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. lower-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. lower-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. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-cos.f64 99.5

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. lift-*.f64 N/A

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

   9. count-2 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. distribute-lft-out N/A

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

   13. lower-*.f64 N/A

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

   14. lower-+.f64 99.5

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

7. Applied egg-rr 99.5%

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

   1. metadata-eval N/A

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

   2. lift-log.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.5

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

9. Applied egg-rr 99.5%

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

    1. lift-log.f64 N/A

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

    2. lift-neg.f64 N/A

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

    3. lift-sqrt.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-sqrt.f64 N/A

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

    6. *-commutative N/A

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

    7. lift-*.f64 N/A

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

    8. associate-*r* N/A

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

    9. lower-*.f64 N/A

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

    10. lower-*.f64 99.5

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

11. Applied egg-rr 99.5%

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

12. Final simplification 99.5%

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

Alternative 2: 99.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-/.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. *-commutative N/A

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

4. Applied egg-rr 99.4%

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

   1. lift-log.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-fma.f64 99.4

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

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

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

Alternative 3: 98.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double u1, double u2) {
	return 0.5 + (((1.0 / 6.0) * (sqrt(2.0) * sqrt(-log(u1)))) * fma((((double) M_PI) * ((double) M_PI)), (-2.0 * (u2 * u2)), 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(Float64(pi * pi), Float64(-2.0 * Float64(u2 * u2)), 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[(N[(Pi * Pi), $MachinePrecision] * N[(-2.0 * N[(u2 * u2), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
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. lower-*.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. lower-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. lower-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. lower-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. lower-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. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. rem-square-sqrt N/A

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

   5. unpow2 N/A

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

   6. unpow2 N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-PI.f64 N/A

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

   11. lower-PI.f64 N/A

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

   12. unpow2 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 N/A

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

   17. unpow2 N/A

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

   18. rem-square-sqrt 98.8

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

8. Simplified 98.8%

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

9. Final simplification 98.8%

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

Alternative 4: 98.8% accurate, 2.3× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-/.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-*.f64 N/A

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

4. Applied egg-rr 99.3%

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

5. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. rem-square-sqrt N/A

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

   5. unpow2 N/A

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

   6. unpow2 N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-PI.f64 N/A

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

   11. lower-PI.f64 N/A

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

   12. unpow2 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 N/A

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

   17. unpow2 N/A

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

   18. rem-square-sqrt 98.6

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

7. Simplified 98.6%

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

9. Applied egg-rr 98.7%

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

Alternative 5: 98.9% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-/.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-*.f64 N/A

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

4. Applied egg-rr 99.3%

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

5. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. rem-square-sqrt N/A

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

   5. unpow2 N/A

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

   6. unpow2 N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-PI.f64 N/A

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

   11. lower-PI.f64 N/A

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

   12. unpow2 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 N/A

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

   17. unpow2 N/A

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

   18. rem-square-sqrt 98.6

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

7. Simplified 98.6%

   \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(0.16666666666666666, \sqrt{-2 \cdot \log u1} \cdot \color{blue}{\mathsf{fma}\left(\pi \cdot \pi, \left(u2 \cdot u2\right) \cdot -2, 1\right)}, 0.5\right)}} \]
8. Step-by-step derivation

9. Applied egg-rr 98.7%

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

Alternative 6: 98.5% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. lift-/.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-cos.f64 N/A

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

4. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(-2 \cdot \log u1\right)}^{0.25}, {\left(-2 \cdot \log u1\right)}^{0.25} \cdot \left(0.16666666666666666 \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\right), 0.5\right)} \]

5. Taylor expanded in u2 around 0

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

7. Simplified 97.7%

   \[\leadsto \mathsf{fma}\left({\left(-2 \cdot \log u1\right)}^{0.25}, {\left(-2 \cdot \log u1\right)}^{0.25} \cdot \left(0.16666666666666666 \cdot \color{blue}{1}\right), 0.5\right) \]

8. Taylor expanded in u1 around inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. log-rec N/A

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

   7. lower-neg.f64 N/A

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

   8. lower-log.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 98.1

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

10. Simplified 98.1%

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

Alternative 7: 98.3% 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 99.4%

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

   1. lift-/.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-*.f64 N/A

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

4. Applied egg-rr 99.3%

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

5. Taylor expanded in u2 around 0

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

7. Simplified 97.8%

   \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(0.16666666666666666, \sqrt{-2 \cdot \log u1} \cdot \color{blue}{1}, 0.5\right)}} \]
8. Step-by-step derivation

   1. lift-log.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. remove-double-div N/A

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

   7. lift-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. remove-double-div N/A

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

   10. remove-double-div 97.9

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

   11. lift-fma.f64 N/A

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

9. Applied egg-rr 97.9%

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

Reproduce

herbie shell --seed 2024219 
(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))