normal distribution

Percentage Accurate: 99.4% → 99.5%

Time: 4.7s

Alternatives: 6

Speedup: 1.4×

Specification

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

Math

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

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

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

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, 1.0× speedup

Math

\[\frac{0.16666666666666666}{{\left(-2 \cdot \log u1\right)}^{-0.5}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

FPCore

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

C

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

Python

def code(u1, u2):
	return ((0.16666666666666666 / 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(0.16666666666666666 / (Float64(-2.0 * log(u1)) ^ -0.5)) * cos(Float64(Float64(2.0 * pi) * u2))) + 0.5)
end

MATLAB

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

Wolfram

code[u1_, u2_] := N[(N[(N[(0.16666666666666666 / 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]

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. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. remove-double-neg N/A

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

   3. pow-neg N/A

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

   4. lower-unsound-/.f64 N/A

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

   5. lower-unsound-pow.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. metadata-eval 99.3

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

3. Applied rewrites 99.3%

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

4. Taylor expanded in u1 around 0

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

   1. metadata-eval N/A

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

   2. lower-/.f64 N/A

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

   3. metadata-eval N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-log.f64 99.4

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

6. Applied rewrites 99.4%

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

Alternative 2: 99.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[u1_, u2_] := N[(N[(N[(1.0 / N[(6.0 / N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(2.0 * Pi), $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. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. remove-double-neg N/A

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

   3. pow-neg N/A

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

   4. lower-unsound-/.f64 N/A

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

   5. lower-unsound-pow.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. metadata-eval 99.3

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

3. Applied rewrites 99.3%

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

4. Taylor expanded in u1 around 0

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

   1. metadata-eval N/A

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

   2. lower-/.f64 N/A

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

   3. metadata-eval N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-log.f64 99.4

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

6. Applied rewrites 99.4%

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

   1. metadata-eval N/A

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

   2. lift-/.f64 N/A

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

   3. div-flip N/A

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

   4. lower-unsound-/.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lift-pow.f64 N/A

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

   7. metadata-eval N/A

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

   8. pow-flip N/A

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

   9. pow1/2 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. associate-/r* N/A

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

   12. lift-*.f64 N/A

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

   13. lower-unsound-/.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. associate-/r* N/A

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

   19. lower-/.f64 99.5

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

8. Applied rewrites 99.5%

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

Alternative 3: 99.4% accurate, 1.4× speedup

Math

\[\mathsf{fma}\left(\cos \left(u2 \cdot 6.283185307179586\right) \cdot 0.16666666666666666, \sqrt{\log u1 \cdot -2}, 0.5\right) \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (* (cos (* u2 6.283185307179586)) 0.16666666666666666)
  (sqrt (* (log u1) -2.0))
  0.5))

C

double code(double u1, double u2) {
	return fma((cos((u2 * 6.283185307179586)) * 0.16666666666666666), sqrt((log(u1) * -2.0)), 0.5);
}

Julia

function code(u1, u2)
	return fma(Float64(cos(Float64(u2 * 6.283185307179586)) * 0.16666666666666666), sqrt(Float64(log(u1) * -2.0)), 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(N[Cos[N[(u2 * 6.283185307179586), $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. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

3. Applied rewrites 99.4%

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

4. Evaluated real constant 99.4%

   \[\leadsto \mathsf{fma}\left(\cos \left(u2 \cdot \color{blue}{\frac{884279719003555}{140737488355328}}\right) \cdot \frac{1}{6}, \sqrt{\log u1 \cdot -2}, \frac{1}{2}\right) \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 3.8× speedup

Math

\[\mathsf{fma}\left(\frac{\sqrt{\log u1 \cdot -2}}{6}, 1, 0.5\right) \]

FPCore

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

C

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

Julia

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

Wolfram

code[u1_, u2_] := N[(N[(N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] / 6.0), $MachinePrecision] * 1.0 + 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. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\frac{1}{2} + \frac{1}{6} \cdot \sqrt{-2 \cdot \log u1}} \]
3. Step-by-step derivation

   1. metadata-eval N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. metadata-eval N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-log.f64 98.0

      \[\leadsto 0.5 + 0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1} \]

4. Applied rewrites 98.0%

   \[\leadsto \color{blue}{0.5 + 0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-sqrt.f64 N/A

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

   8. pow1/2 N/A

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

   9. metadata-eval N/A

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

   10. pow-flip N/A

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

   11. lift-pow.f64 N/A

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

   12. mult-flip-rev N/A

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

   13. lower-/.f64 98.0

       \[\leadsto 0.5 + \frac{0.16666666666666666}{\color{blue}{{\left(\log u1 \cdot -2\right)}^{-0.5}}} \]

   14. lift-pow.f64 N/A

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

   15. metadata-eval N/A

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

   16. pow-flip N/A

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

   17. pow1/2 N/A

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

   18. lift-sqrt.f64 N/A

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

   19. lower-/.f64 97.9

       \[\leadsto 0.5 + \frac{0.16666666666666666}{\frac{1}{\color{blue}{\sqrt{\log u1 \cdot -2}}}} \]

   20. lift-*.f64 N/A

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

   21. *-commutative N/A

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

   22. lift-*.f64 97.9

       \[\leadsto 0.5 + \frac{0.16666666666666666}{\frac{1}{\sqrt{-2 \cdot \log u1}}} \]

6. Applied rewrites 97.9%

   \[\leadsto 0.5 + \frac{0.16666666666666666}{\color{blue}{\frac{1}{\sqrt{-2 \cdot \log u1}}}} \]
7. Step-by-step derivation

   1. metadata-eval N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

8. Applied rewrites 98.2%

   \[\leadsto \mathsf{fma}\left(\frac{\sqrt{\log u1 \cdot -2}}{6}, \color{blue}{1}, 0.5\right) \]
9. Add Preprocessing

Alternative 5: 98.1% accurate, 3.8× speedup

Math

\[\mathsf{fma}\left(0.027777777777777776, \frac{\sqrt{\log u1 \cdot -2}}{0.16666666666666666}, 0.5\right) \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  0.027777777777777776
  (/ (sqrt (* (log u1) -2.0)) 0.16666666666666666)
  0.5))

C

double code(double u1, double u2) {
	return fma(0.027777777777777776, (sqrt((log(u1) * -2.0)) / 0.16666666666666666), 0.5);
}

Julia

function code(u1, u2)
	return fma(0.027777777777777776, Float64(sqrt(Float64(log(u1) * -2.0)) / 0.16666666666666666), 0.5)
end

Wolfram

code[u1_, u2_] := N[(0.027777777777777776 * N[(N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -2.0), $MachinePrecision]], $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. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\frac{1}{2} + \frac{1}{6} \cdot \sqrt{-2 \cdot \log u1}} \]
3. Step-by-step derivation

   1. metadata-eval N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. metadata-eval N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-log.f64 98.0

      \[\leadsto 0.5 + 0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1} \]

4. Applied rewrites 98.0%

   \[\leadsto \color{blue}{0.5 + 0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-sqrt.f64 N/A

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

   8. pow1/2 N/A

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

   9. metadata-eval N/A

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

   10. pow-flip N/A

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

   11. lift-pow.f64 N/A

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

   12. mult-flip-rev N/A

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

   13. lower-/.f64 98.0

       \[\leadsto 0.5 + \frac{0.16666666666666666}{\color{blue}{{\left(\log u1 \cdot -2\right)}^{-0.5}}} \]

   14. lift-pow.f64 N/A

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

   15. metadata-eval N/A

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

   16. pow-flip N/A

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

   17. pow1/2 N/A

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

   18. lift-sqrt.f64 N/A

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

   19. lower-/.f64 97.9

       \[\leadsto 0.5 + \frac{0.16666666666666666}{\frac{1}{\color{blue}{\sqrt{\log u1 \cdot -2}}}} \]

   20. lift-*.f64 N/A

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

   21. *-commutative N/A

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

   22. lift-*.f64 97.9

       \[\leadsto 0.5 + \frac{0.16666666666666666}{\frac{1}{\sqrt{-2 \cdot \log u1}}} \]

6. Applied rewrites 97.9%

   \[\leadsto 0.5 + \frac{0.16666666666666666}{\color{blue}{\frac{1}{\sqrt{-2 \cdot \log u1}}}} \]
7. Step-by-step derivation

   1. metadata-eval N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

8. Applied rewrites 98.1%

   \[\leadsto \mathsf{fma}\left(0.027777777777777776, \color{blue}{\frac{\sqrt{\log u1 \cdot -2}}{0.16666666666666666}}, 0.5\right) \]
9. Add Preprocessing

Alternative 6: 98.0% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[u1_, u2_] := N[(0.16666666666666666 * 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. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in u2 around 0

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

6. Applied rewrites 98.0%

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

Reproduce

herbie shell --seed 2025171 
(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))