normal distribution

Percentage Accurate: 99.4% → 99.7%

Time: 9.3s

Alternatives: 3

Speedup: 1.0×

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: 99.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double u1, double u2) {
	return (Math.sqrt(Math.log(Math.pow(u1, -0.05555555555555555))) * Math.cos(((2.0 * Math.PI) * u2))) + 0.5;
}

Python

def code(u1, u2):
	return (math.sqrt(math.log(math.pow(u1, -0.05555555555555555))) * math.cos(((2.0 * math.pi) * u2))) + 0.5

Julia

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

MATLAB

function tmp = code(u1, u2)
	tmp = (sqrt(log((u1 ^ -0.05555555555555555))) * cos(((2.0 * pi) * u2))) + 0.5;
end

Wolfram

code[u1_, u2_] := N[(N[(N[Sqrt[N[Log[N[Power[u1, -0.05555555555555555], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * u2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. expm1-log1p-u 99.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   2. expm1-udef 99.1%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)} - 1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   3. metadata-eval 99.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{0.16666666666666666} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)} - 1\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   4. unpow1/2 99.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(0.16666666666666666 \cdot \color{blue}{\sqrt{-2 \cdot \log u1}}\right)} - 1\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

3. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1}\right)} - 1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
4. Step-by-step derivation

   1. expm1-def 99.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1}\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   2. expm1-log1p 99.3%

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

   3. *-commutative 99.3%

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

5. Simplified 99.3%

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

   1. *-commutative 99.3%

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

   2. rem-cbrt-cube 99.1%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1}\right)}^{3}}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   3. unpow1/3 98.8%

      \[\leadsto \color{blue}{{\left({\left(0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1}\right)}^{3}\right)}^{0.3333333333333333}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

7. Applied egg-rr 44.1%

   \[\leadsto \color{blue}{{\left({\left(\log \left({u1}^{-2}\right) \cdot 0.027777777777777776\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
8. Step-by-step derivation

   1. unpow1/3 44.1%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\log \left({u1}^{-2}\right) \cdot 0.027777777777777776\right)}^{1.5}}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   2. *-commutative 44.1%

      \[\leadsto \sqrt[3]{{\color{blue}{\left(0.027777777777777776 \cdot \log \left({u1}^{-2}\right)\right)}}^{1.5}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   3. log-pow 99.2%

      \[\leadsto \sqrt[3]{{\left(0.027777777777777776 \cdot \color{blue}{\left(-2 \cdot \log u1\right)}\right)}^{1.5}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   4. associate-*r* 99.2%

      \[\leadsto \sqrt[3]{{\color{blue}{\left(\left(0.027777777777777776 \cdot -2\right) \cdot \log u1\right)}}^{1.5}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   5. metadata-eval 99.2%

      \[\leadsto \sqrt[3]{{\left(\color{blue}{-0.05555555555555555} \cdot \log u1\right)}^{1.5}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

9. Simplified 99.2%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(-0.05555555555555555 \cdot \log u1\right)}^{1.5}}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
10. Step-by-step derivation

    1. add-sqr-sqrt 98.9%

       \[\leadsto \color{blue}{\left(\sqrt{\sqrt[3]{{\left(-0.05555555555555555 \cdot \log u1\right)}^{1.5}}} \cdot \sqrt{\sqrt[3]{{\left(-0.05555555555555555 \cdot \log u1\right)}^{1.5}}}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

    2. sqrt-prod 99.2%

       \[\leadsto \color{blue}{\sqrt{\sqrt[3]{{\left(-0.05555555555555555 \cdot \log u1\right)}^{1.5}} \cdot \sqrt[3]{{\left(-0.05555555555555555 \cdot \log u1\right)}^{1.5}}}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

    3. cbrt-unprod 99.5%

       \[\leadsto \sqrt{\color{blue}{\sqrt[3]{{\left(-0.05555555555555555 \cdot \log u1\right)}^{1.5} \cdot {\left(-0.05555555555555555 \cdot \log u1\right)}^{1.5}}}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

    4. pow-prod-up 99.5%

       \[\leadsto \sqrt{\sqrt[3]{\color{blue}{{\left(-0.05555555555555555 \cdot \log u1\right)}^{\left(1.5 + 1.5\right)}}}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

    5. metadata-eval 99.5%

       \[\leadsto \sqrt{\sqrt[3]{{\left(-0.05555555555555555 \cdot \log u1\right)}^{\color{blue}{3}}}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

    6. pow3 99.5%

       \[\leadsto \sqrt{\sqrt[3]{\color{blue}{\left(\left(-0.05555555555555555 \cdot \log u1\right) \cdot \left(-0.05555555555555555 \cdot \log u1\right)\right) \cdot \left(-0.05555555555555555 \cdot \log u1\right)}}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

    7. add-cbrt-cube 99.6%

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

    8. add-log-exp 99.6%

       \[\leadsto \sqrt{\color{blue}{\log \left(e^{-0.05555555555555555 \cdot \log u1}\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

    9. *-commutative 99.6%

       \[\leadsto \sqrt{\log \left(e^{\color{blue}{\log u1 \cdot -0.05555555555555555}}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

    10. exp-to-pow 99.6%

        \[\leadsto \sqrt{\log \color{blue}{\left({u1}^{-0.05555555555555555}\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

11. Applied egg-rr 99.6%

    \[\leadsto \color{blue}{\sqrt{\log \left({u1}^{-0.05555555555555555}\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

12. Final simplification 99.6%

    \[\leadsto \sqrt{\log \left({u1}^{-0.05555555555555555}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double u1, double u2) {
	return 0.5 + (Math.cos(((2.0 * Math.PI) * u2)) * Math.sqrt((-0.05555555555555555 * Math.log(u1))));
}

Python

def code(u1, u2):
	return 0.5 + (math.cos(((2.0 * math.pi) * u2)) * math.sqrt((-0.05555555555555555 * math.log(u1))))

Julia

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

MATLAB

function tmp = code(u1, u2)
	tmp = 0.5 + (cos(((2.0 * pi) * u2)) * sqrt((-0.05555555555555555 * log(u1))));
end

Wolfram

code[u1_, u2_] := N[(0.5 + N[(N[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * u2), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(-0.05555555555555555 * N[Log[u1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. add-sqr-sqrt 99.0%

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

   2. sqrt-unprod 99.3%

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

   3. *-commutative 99.3%

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

   4. *-commutative 99.3%

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

   5. swap-sqr 99.4%

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

   6. unpow1/2 99.4%

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

   7. unpow1/2 99.4%

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

   8. add-sqr-sqrt 99.6%

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

   9. metadata-eval 99.6%

      \[\leadsto \sqrt{\left(-2 \cdot \log u1\right) \cdot \left(\color{blue}{0.16666666666666666} \cdot \frac{1}{6}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   10. metadata-eval 99.6%

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

   11. metadata-eval 99.6%

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

3. Applied egg-rr 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-*l* 99.6%

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

   3. metadata-eval 99.6%

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

   \[\leadsto 0.5 + \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{-0.05555555555555555 \cdot \log u1} \]

Alternative 3: 98.0% accurate, 1.4× speedup

Math

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

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 99.3%

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

   1. *-commutative 99.3%

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

   2. associate-*r* 99.3%

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

   3. fma-def 99.3%

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

   4. associate-*l* 99.3%

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

   5. metadata-eval 99.3%

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

   6. unpow1/2 99.3%

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

3. Applied egg-rr 99.3%

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

4. Taylor expanded in u2 around 0 97.9%

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

5. Final simplification 97.9%

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

Reproduce

herbie shell --seed 2023174 
(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))