normal distribution

Percentage Accurate: 99.4% → 99.6%

Time: 10.3s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift-*.f64 N/A

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

   3. 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. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lift-pow.f64 N/A

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

   7. sqr-pow N/A

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

   8. associate-*l* N/A

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

   9. lower-fma.f64 N/A

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in u1 around inf

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

7. Applied rewrites 99.6%

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

8. Final simplification 99.6%

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

Alternative 2: 99.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.4

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

   4. lift-pow.f64 N/A

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

   5. unpow1/2 N/A

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

   6. lower-sqrt.f64 99.4

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.4

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

   10. lift-/.f64 N/A

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

   11. metadata-eval 99.4

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

4. Applied rewrites 99.4%

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

5. Final simplification 99.4%

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

Alternative 3: 99.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift-*.f64 N/A

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

   3. 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. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

4. Applied rewrites 99.4%

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

5. Final simplification 99.4%

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

Alternative 4: 99.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt{-0.05555555555555555 \cdot \log u1}, \left|\cos \left(\left(\pi \cdot 2\right) \cdot u2\right)\right|, 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (sqrt (* -0.05555555555555555 (log u1)))
  (fabs (cos (* (* PI 2.0) u2)))
  0.5))

C

double code(double u1, double u2) {
	return fma(sqrt((-0.05555555555555555 * log(u1))), fabs(cos(((((double) M_PI) * 2.0) * u2))), 0.5);
}

Julia

function code(u1, u2)
	return fma(sqrt(Float64(-0.05555555555555555 * log(u1))), abs(cos(Float64(Float64(pi * 2.0) * u2))), 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[Sqrt[N[(-0.05555555555555555 * N[Log[u1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[N[Cos[N[(N[(Pi * 2.0), $MachinePrecision] * u2), $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

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.4

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

   4. lift-pow.f64 N/A

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

   5. unpow1/2 N/A

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

   6. lower-sqrt.f64 99.4

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.4

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

   10. lift-/.f64 N/A

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

   11. metadata-eval 99.4

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

4. Applied rewrites 99.4%

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

6. Applied rewrites 99.4%

   \[\leadsto \color{blue}{{\left(\left(0.027777777777777776 \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right)\right)\right) \cdot \left(\log u1 \cdot -2\right)\right)}^{0.5}} + 0.5 \]
7. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \color{blue}{{\left(\left(\frac{1}{36} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\right)\right)\right) \cdot \left(\log u1 \cdot -2\right)\right)}^{\frac{1}{2}}} + \frac{1}{2} \]

   2. unpow1/2 N/A

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

   3. lower-sqrt.f64 99.4

      \[\leadsto \color{blue}{\sqrt{\left(0.027777777777777776 \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right)\right)\right) \cdot \left(\log u1 \cdot -2\right)}} + 0.5 \]

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. rem-square-sqrt N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. metadata-eval N/A

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

   12. swap-sqr N/A

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

8. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\sqrt{\left(-0.05555555555555555 \cdot \log u1\right) \cdot \mathsf{fma}\left(\cos \left(4 \cdot \left(\pi \cdot u2\right)\right), 0.5, 0.5\right)}} + 0.5 \]
9. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\left(\frac{-1}{18} \cdot \log u1\right) \cdot \mathsf{fma}\left(\cos \left(4 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right), \frac{1}{2}, \frac{1}{2}\right)} + \frac{1}{2}} \]

   2. lift-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\left(\frac{-1}{18} \cdot \log u1\right) \cdot \mathsf{fma}\left(\cos \left(4 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right), \frac{1}{2}, \frac{1}{2}\right)}} + \frac{1}{2} \]

   3. lift-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(\frac{-1}{18} \cdot \log u1\right) \cdot \mathsf{fma}\left(\cos \left(4 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right), \frac{1}{2}, \frac{1}{2}\right)}} + \frac{1}{2} \]

   4. sqrt-prod N/A

      \[\leadsto \color{blue}{\sqrt{\frac{-1}{18} \cdot \log u1} \cdot \sqrt{\mathsf{fma}\left(\cos \left(4 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right), \frac{1}{2}, \frac{1}{2}\right)}} + \frac{1}{2} \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{-1}{18} \cdot \log u1}, \sqrt{\mathsf{fma}\left(\cos \left(4 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right), \frac{1}{2}, \frac{1}{2}\right)}, \frac{1}{2}\right)} \]

10. Applied rewrites 99.4%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\log u1 \cdot -0.05555555555555555}, \left|\cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\right|, 0.5\right)} \]

11. Final simplification 99.4%

    \[\leadsto \mathsf{fma}\left(\sqrt{-0.05555555555555555 \cdot \log u1}, \left|\cos \left(\left(\pi \cdot 2\right) \cdot u2\right)\right|, 0.5\right) \]
12. Add Preprocessing

Alternative 5: 99.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 0.16666666666666666\right) \cdot \sqrt{2}, \sqrt{-\log u1}, 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (*
   (fma (* -0.3333333333333333 (* u2 u2)) (* PI PI) 0.16666666666666666)
   (sqrt 2.0))
  (sqrt (- (log u1)))
  0.5))

C

double code(double u1, double u2) {
	return fma((fma((-0.3333333333333333 * (u2 * u2)), (((double) M_PI) * ((double) M_PI)), 0.16666666666666666) * sqrt(2.0)), sqrt(-log(u1)), 0.5);
}

Julia

function code(u1, u2)
	return fma(Float64(fma(Float64(-0.3333333333333333 * Float64(u2 * u2)), Float64(pi * pi), 0.16666666666666666) * sqrt(2.0)), sqrt(Float64(-log(u1))), 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(N[(N[(-0.3333333333333333 * N[(u2 * u2), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-N[Log[u1], $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

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

   2. lift-*.f64 N/A

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

   3. 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. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lift-pow.f64 N/A

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

   7. sqr-pow N/A

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

   8. associate-*l* N/A

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

   9. lower-fma.f64 N/A

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in u1 around inf

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in u2 around 0

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

10. Applied rewrites 98.2%

    \[\leadsto \mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -0.3333333333333333, \pi \cdot \pi, 0.16666666666666666\right), \sqrt{\color{blue}{-\log u1}}, 0.5\right) \]

11. Final simplification 98.2%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 0.16666666666666666\right) \cdot \sqrt{2}, \sqrt{-\log u1}, 0.5\right) \]
12. Add Preprocessing

Alternative 6: 99.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(u2 \cdot u2, 0.2222222222222222 \cdot \left(\pi \cdot \pi\right), -0.05555555555555555\right) \cdot \log u1} + 0.5 \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  (sqrt
   (*
    (fma (* u2 u2) (* 0.2222222222222222 (* PI PI)) -0.05555555555555555)
    (log u1)))
  0.5))

C

double code(double u1, double u2) {
	return sqrt((fma((u2 * u2), (0.2222222222222222 * (((double) M_PI) * ((double) M_PI))), -0.05555555555555555) * log(u1))) + 0.5;
}

Julia

function code(u1, u2)
	return Float64(sqrt(Float64(fma(Float64(u2 * u2), Float64(0.2222222222222222 * Float64(pi * pi)), -0.05555555555555555) * log(u1))) + 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[Sqrt[N[(N[(N[(u2 * u2), $MachinePrecision] * N[(0.2222222222222222 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] + -0.05555555555555555), $MachinePrecision] * N[Log[u1], $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

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.4

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

   4. lift-pow.f64 N/A

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

   5. unpow1/2 N/A

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

   6. lower-sqrt.f64 99.4

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.4

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

   10. lift-/.f64 N/A

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

   11. metadata-eval 99.4

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

4. Applied rewrites 99.4%

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

6. Applied rewrites 99.4%

   \[\leadsto \color{blue}{{\left(\left(0.027777777777777776 \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right)\right)\right) \cdot \left(\log u1 \cdot -2\right)\right)}^{0.5}} + 0.5 \]
7. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \color{blue}{{\left(\left(\frac{1}{36} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\right)\right)\right) \cdot \left(\log u1 \cdot -2\right)\right)}^{\frac{1}{2}}} + \frac{1}{2} \]

   2. unpow1/2 N/A

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

   3. lower-sqrt.f64 99.4

      \[\leadsto \color{blue}{\sqrt{\left(0.027777777777777776 \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right)\right)\right) \cdot \left(\log u1 \cdot -2\right)}} + 0.5 \]

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. rem-square-sqrt N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. metadata-eval N/A

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

   12. swap-sqr N/A

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

8. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\sqrt{\left(-0.05555555555555555 \cdot \log u1\right) \cdot \mathsf{fma}\left(\cos \left(4 \cdot \left(\pi \cdot u2\right)\right), 0.5, 0.5\right)}} + 0.5 \]

9. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\color{blue}{\frac{-1}{18} \cdot \log u1 + \frac{2}{9} \cdot \left({u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \log u1\right)\right)}} + \frac{1}{2} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \sqrt{\color{blue}{\frac{2}{9} \cdot \left({u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \log u1\right)\right) + \frac{-1}{18} \cdot \log u1}} + \frac{1}{2} \]

    2. *-commutative N/A

       \[\leadsto \sqrt{\color{blue}{\left({u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \log u1\right)\right) \cdot \frac{2}{9}} + \frac{-1}{18} \cdot \log u1} + \frac{1}{2} \]

    3. associate-*r* N/A

       \[\leadsto \sqrt{\color{blue}{{u2}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{2} \cdot \log u1\right) \cdot \frac{2}{9}\right)} + \frac{-1}{18} \cdot \log u1} + \frac{1}{2} \]

    4. *-commutative N/A

       \[\leadsto \sqrt{{u2}^{2} \cdot \color{blue}{\left(\frac{2}{9} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \log u1\right)\right)} + \frac{-1}{18} \cdot \log u1} + \frac{1}{2} \]

    5. associate-*r* N/A

       \[\leadsto \sqrt{{u2}^{2} \cdot \color{blue}{\left(\left(\frac{2}{9} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \log u1\right)} + \frac{-1}{18} \cdot \log u1} + \frac{1}{2} \]

    6. associate-*r* N/A

       \[\leadsto \sqrt{\color{blue}{\left({u2}^{2} \cdot \left(\frac{2}{9} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \log u1} + \frac{-1}{18} \cdot \log u1} + \frac{1}{2} \]

    7. distribute-rgt-out N/A

       \[\leadsto \sqrt{\color{blue}{\log u1 \cdot \left({u2}^{2} \cdot \left(\frac{2}{9} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{18}\right)}} + \frac{1}{2} \]

    8. lower-*.f64 N/A

       \[\leadsto \sqrt{\color{blue}{\log u1 \cdot \left({u2}^{2} \cdot \left(\frac{2}{9} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{18}\right)}} + \frac{1}{2} \]

    9. lower-log.f64 N/A

       \[\leadsto \sqrt{\color{blue}{\log u1} \cdot \left({u2}^{2} \cdot \left(\frac{2}{9} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{18}\right)} + \frac{1}{2} \]

    10. lower-fma.f64 N/A

        \[\leadsto \sqrt{\log u1 \cdot \color{blue}{\mathsf{fma}\left({u2}^{2}, \frac{2}{9} \cdot {\mathsf{PI}\left(\right)}^{2}, \frac{-1}{18}\right)}} + \frac{1}{2} \]

    11. unpow2 N/A

        \[\leadsto \sqrt{\log u1 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{2}{9} \cdot {\mathsf{PI}\left(\right)}^{2}, \frac{-1}{18}\right)} + \frac{1}{2} \]

    12. lower-*.f64 N/A

        \[\leadsto \sqrt{\log u1 \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{2}{9} \cdot {\mathsf{PI}\left(\right)}^{2}, \frac{-1}{18}\right)} + \frac{1}{2} \]

    13. *-commutative N/A

        \[\leadsto \sqrt{\log u1 \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{{\mathsf{PI}\left(\right)}^{2} \cdot \frac{2}{9}}, \frac{-1}{18}\right)} + \frac{1}{2} \]

    14. lower-*.f64 N/A

        \[\leadsto \sqrt{\log u1 \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{{\mathsf{PI}\left(\right)}^{2} \cdot \frac{2}{9}}, \frac{-1}{18}\right)} + \frac{1}{2} \]

    15. unpow2 N/A

        \[\leadsto \sqrt{\log u1 \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{2}{9}, \frac{-1}{18}\right)} + \frac{1}{2} \]

    16. lower-*.f64 N/A

        \[\leadsto \sqrt{\log u1 \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{2}{9}, \frac{-1}{18}\right)} + \frac{1}{2} \]

    17. lower-PI.f64 N/A

        \[\leadsto \sqrt{\log u1 \cdot \mathsf{fma}\left(u2 \cdot u2, \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{2}{9}, \frac{-1}{18}\right)} + \frac{1}{2} \]

    18. lower-PI.f64 98.2

        \[\leadsto \sqrt{\log u1 \cdot \mathsf{fma}\left(u2 \cdot u2, \left(\pi \cdot \color{blue}{\pi}\right) \cdot 0.2222222222222222, -0.05555555555555555\right)} + 0.5 \]

11. Applied rewrites 98.2%

    \[\leadsto \sqrt{\color{blue}{\log u1 \cdot \mathsf{fma}\left(u2 \cdot u2, \left(\pi \cdot \pi\right) \cdot 0.2222222222222222, -0.05555555555555555\right)}} + 0.5 \]

12. Final simplification 98.2%

    \[\leadsto \sqrt{\mathsf{fma}\left(u2 \cdot u2, 0.2222222222222222 \cdot \left(\pi \cdot \pi\right), -0.05555555555555555\right) \cdot \log u1} + 0.5 \]
13. Add Preprocessing

Alternative 7: 98.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{-0.05555555555555555 \cdot \log u1} + 0.5 \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+ (sqrt (* -0.05555555555555555 (log u1))) 0.5))

C

double code(double u1, double u2) {
	return sqrt((-0.05555555555555555 * log(u1))) + 0.5;
}

Fortran

real(8) function code(u1, u2)
    real(8), intent (in) :: u1
    real(8), intent (in) :: u2
    code = sqrt(((-0.05555555555555555d0) * log(u1))) + 0.5d0
end function

Java

public static double code(double u1, double u2) {
	return Math.sqrt((-0.05555555555555555 * Math.log(u1))) + 0.5;
}

Python

def code(u1, u2):
	return math.sqrt((-0.05555555555555555 * math.log(u1))) + 0.5

Julia

function code(u1, u2)
	return Float64(sqrt(Float64(-0.05555555555555555 * log(u1))) + 0.5)
end

MATLAB

function tmp = code(u1, u2)
	tmp = sqrt((-0.05555555555555555 * log(u1))) + 0.5;
end

Wolfram

code[u1_, u2_] := N[(N[Sqrt[N[(-0.05555555555555555 * N[Log[u1], $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

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.4

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

   4. lift-pow.f64 N/A

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

   5. unpow1/2 N/A

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

   6. lower-sqrt.f64 99.4

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.4

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

   10. lift-/.f64 N/A

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

   11. metadata-eval 99.4

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

4. Applied rewrites 99.4%

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

6. Applied rewrites 99.4%

   \[\leadsto \color{blue}{{\left(\left(0.027777777777777776 \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(u2 \cdot \left(\pi \cdot 2\right)\right)\right)\right)\right) \cdot \left(\log u1 \cdot -2\right)\right)}^{0.5}} + 0.5 \]

7. Taylor expanded in u2 around 0

   \[\leadsto {\color{blue}{\left(\frac{-1}{18} \cdot \log u1\right)}}^{\frac{1}{2}} + \frac{1}{2} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto {\color{blue}{\left(\frac{-1}{18} \cdot \log u1\right)}}^{\frac{1}{2}} + \frac{1}{2} \]

   2. lower-log.f64 97.4

      \[\leadsto {\left(-0.05555555555555555 \cdot \color{blue}{\log u1}\right)}^{0.5} + 0.5 \]

9. Applied rewrites 97.4%

   \[\leadsto {\color{blue}{\left(-0.05555555555555555 \cdot \log u1\right)}}^{0.5} + 0.5 \]
10. Step-by-step derivation

    1. lift-pow.f64 N/A

       \[\leadsto \color{blue}{{\left(\frac{-1}{18} \cdot \log u1\right)}^{\frac{1}{2}}} + \frac{1}{2} \]

    2. unpow1/2 N/A

       \[\leadsto \color{blue}{\sqrt{\frac{-1}{18} \cdot \log u1}} + \frac{1}{2} \]

    3. lower-sqrt.f64 97.4

       \[\leadsto \color{blue}{\sqrt{-0.05555555555555555 \cdot \log u1}} + 0.5 \]

11. Applied rewrites 97.4%

    \[\leadsto \color{blue}{\sqrt{\log u1 \cdot -0.05555555555555555}} + 0.5 \]

12. Final simplification 97.4%

    \[\leadsto \sqrt{-0.05555555555555555 \cdot \log u1} + 0.5 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024235 
(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))