normal distribution

Percentage Accurate: 99.4% → 99.7%

Time: 9.9s

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

\[\begin{array}{l} \\ {\left({\log u1}^{2} \cdot 0.0030864197530864196\right)}^{0.25} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  (*
   (pow (* (pow (log u1) 2.0) 0.0030864197530864196) 0.25)
   (cos (* (* 2.0 PI) u2)))
  0.5))

C

double code(double u1, double u2) {
	return (pow((pow(log(u1), 2.0) * 0.0030864197530864196), 0.25) * cos(((2.0 * ((double) M_PI)) * u2))) + 0.5;
}

Java

public static double code(double u1, double u2) {
	return (Math.pow((Math.pow(Math.log(u1), 2.0) * 0.0030864197530864196), 0.25) * Math.cos(((2.0 * Math.PI) * u2))) + 0.5;
}

Python

def code(u1, u2):
	return (math.pow((math.pow(math.log(u1), 2.0) * 0.0030864197530864196), 0.25) * math.cos(((2.0 * math.pi) * u2))) + 0.5

Julia

function code(u1, u2)
	return Float64(Float64((Float64((log(u1) ^ 2.0) * 0.0030864197530864196) ^ 0.25) * cos(Float64(Float64(2.0 * pi) * u2))) + 0.5)
end

MATLAB

function tmp = code(u1, u2)
	tmp = ((((log(u1) ^ 2.0) * 0.0030864197530864196) ^ 0.25) * cos(((2.0 * pi) * u2))) + 0.5;
end

Wolfram

code[u1_, u2_] := N[(N[(N[Power[N[(N[Power[N[Log[u1], $MachinePrecision], 2.0], $MachinePrecision] * 0.0030864197530864196), $MachinePrecision], 0.25], $MachinePrecision] * N[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * u2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. pow1/2 99.5%

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

   2. add-log-exp 99.5%

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

   3. *-commutative 99.5%

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

   4. exp-prod 99.5%

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

   5. metadata-eval 99.5%

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

3. Applied egg-rr 99.5%

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

   1. log-pow 99.5%

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

   2. rem-log-exp 99.5%

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

   3. *-commutative 99.5%

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

5. Simplified 99.5%

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

   1. metadata-eval 99.5%

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

   2. sqrt-prod 99.7%

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

   3. *-commutative 99.7%

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

   4. *-commutative 99.7%

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

   5. pow1/2 99.7%

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

   6. pow-to-exp 99.4%

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

   7. *-commutative 99.4%

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

   8. associate-*l* 99.4%

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

   9. metadata-eval 99.4%

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

7. Applied egg-rr 99.4%

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

   1. exp-to-pow 99.7%

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

   2. metadata-eval 99.7%

      \[\leadsto {\left(\log u1 \cdot -0.05555555555555555\right)}^{\color{blue}{\left(0.25 + 0.25\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   3. pow-prod-up 99.2%

      \[\leadsto \color{blue}{\left({\left(\log u1 \cdot -0.05555555555555555\right)}^{0.25} \cdot {\left(\log u1 \cdot -0.05555555555555555\right)}^{0.25}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   4. pow-prod-down 99.6%

      \[\leadsto \color{blue}{{\left(\left(\log u1 \cdot -0.05555555555555555\right) \cdot \left(\log u1 \cdot -0.05555555555555555\right)\right)}^{0.25}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   5. swap-sqr 99.7%

      \[\leadsto {\color{blue}{\left(\left(\log u1 \cdot \log u1\right) \cdot \left(-0.05555555555555555 \cdot -0.05555555555555555\right)\right)}}^{0.25} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   6. pow2 99.7%

      \[\leadsto {\left(\color{blue}{{\log u1}^{2}} \cdot \left(-0.05555555555555555 \cdot -0.05555555555555555\right)\right)}^{0.25} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   7. metadata-eval 99.7%

      \[\leadsto {\left({\log u1}^{2} \cdot \color{blue}{0.0030864197530864196}\right)}^{0.25} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

9. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{{\left({\log u1}^{2} \cdot 0.0030864197530864196\right)}^{0.25}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

10. Final simplification 99.7%

    \[\leadsto {\left({\log u1}^{2} \cdot 0.0030864197530864196\right)}^{0.25} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. expm1-log1p-u 99.3%

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

   2. expm1-udef 99.2%

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

3. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\left(-2 \cdot \log u1\right) \cdot 0.027777777777777776}\right)} - 1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
4. Step-by-step derivation

   1. expm1-def 99.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(-2 \cdot \log u1\right) \cdot 0.027777777777777776}\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   2. expm1-log1p 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-*l* 99.7%

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

   5. metadata-eval 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 3: 0.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. pow1/2 99.5%

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

   2. add-log-exp 99.5%

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

   3. *-commutative 99.5%

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

   4. exp-prod 99.5%

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

   5. metadata-eval 99.5%

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

3. Applied egg-rr 99.5%

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

   1. log-pow 99.5%

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

   2. rem-log-exp 99.5%

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

   3. *-commutative 99.5%

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

5. Simplified 99.5%

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

   1. metadata-eval 99.5%

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

   2. sqrt-prod 99.7%

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

   3. *-commutative 99.7%

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

   4. *-commutative 99.7%

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

   5. add-sqr-sqrt 99.2%

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

   6. unpow2 99.2%

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

   7. pow1/2 99.2%

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

   8. sqrt-pow1 99.2%

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

   9. *-commutative 99.2%

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

   10. associate-*l* 99.2%

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

   11. metadata-eval 99.2%

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

   12. metadata-eval 99.2%

       \[\leadsto {\left({\left(\log u1 \cdot -0.05555555555555555\right)}^{\color{blue}{0.25}}\right)}^{2} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

7. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{{\left({\left(\log u1 \cdot -0.05555555555555555\right)}^{0.25}\right)}^{2}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

8. Taylor expanded in u2 around 0 0.0%

   \[\leadsto \color{blue}{0.5 + \sqrt{\log u1} \cdot \sqrt{-0.05555555555555555}} \]
9. Step-by-step derivation

   1. +-commutative 0.0%

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

10. Simplified 0.0%

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

11. Final simplification 0.0%

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

Reproduce

herbie shell --seed 2023309 
(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))