normal distribution

Percentage Accurate: 99.4% → 99.4%

Time: 13.5s

Alternatives: 6

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: 99.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. add-sqr-sqrt 99.0%

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

   2. sqrt-unprod 99.4%

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

   3. *-commutative 99.4%

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

   4. pow1/2 99.4%

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

   5. *-commutative 99.4%

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

   6. pow1/2 99.4%

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

   7. swap-sqr 99.4%

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

4. Applied egg-rr 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-*r* 99.6%

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

   3. *-commutative 99.6%

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

   4. associate-*l* 99.6%

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

   5. metadata-eval 99.6%

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

6. Simplified 99.6%

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

   1. add-log-exp 99.6%

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

   2. exp-to-pow 99.7%

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

8. Applied egg-rr 99.7%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 + \sqrt{\frac{1 + \cos \left(\pi \cdot \left(u2 \cdot 4\right)\right)}{2} \cdot \left(-0.05555555555555555 \cdot \log u1\right)} \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  0.5
  (sqrt
   (*
    (/ (+ 1.0 (cos (* PI (* u2 4.0)))) 2.0)
    (* -0.05555555555555555 (log u1))))))

C

double code(double u1, double u2) {
	return 0.5 + sqrt((((1.0 + cos((((double) M_PI) * (u2 * 4.0)))) / 2.0) * (-0.05555555555555555 * log(u1))));
}

Java

public static double code(double u1, double u2) {
	return 0.5 + Math.sqrt((((1.0 + Math.cos((Math.PI * (u2 * 4.0)))) / 2.0) * (-0.05555555555555555 * Math.log(u1))));
}

Python

def code(u1, u2):
	return 0.5 + math.sqrt((((1.0 + math.cos((math.pi * (u2 * 4.0)))) / 2.0) * (-0.05555555555555555 * math.log(u1))))

Julia

function code(u1, u2)
	return Float64(0.5 + sqrt(Float64(Float64(Float64(1.0 + cos(Float64(pi * Float64(u2 * 4.0)))) / 2.0) * Float64(-0.05555555555555555 * log(u1)))))
end

MATLAB

function tmp = code(u1, u2)
	tmp = 0.5 + sqrt((((1.0 + cos((pi * (u2 * 4.0)))) / 2.0) * (-0.05555555555555555 * log(u1))));
end

Wolfram

code[u1_, u2_] := N[(0.5 + N[Sqrt[N[(N[(N[(1.0 + N[Cos[N[(Pi * N[(u2 * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * N[(-0.05555555555555555 * N[Log[u1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. add-sqr-sqrt 99.0%

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

   2. sqrt-unprod 99.4%

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

   3. *-commutative 99.4%

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

   4. pow1/2 99.4%

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

   5. *-commutative 99.4%

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

   6. pow1/2 99.4%

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

   7. swap-sqr 99.4%

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

4. Applied egg-rr 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-*r* 99.6%

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

   3. *-commutative 99.6%

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

   4. associate-*l* 99.6%

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

   5. metadata-eval 99.6%

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

6. Simplified 99.6%

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

   1. unpow2 99.6%

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

   2. cos-mult 99.6%

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

8. Applied egg-rr 99.6%

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

   1. +-commutative 99.6%

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

   2. +-inverses 99.6%

      \[\leadsto \sqrt{\frac{\cos \color{blue}{0} + \cos \left(u2 \cdot \left(2 \cdot \pi\right) + u2 \cdot \left(2 \cdot \pi\right)\right)}{2} \cdot \left(\log u1 \cdot -0.05555555555555555\right)} + 0.5 \]

   3. cos-0 99.6%

      \[\leadsto \sqrt{\frac{\color{blue}{1} + \cos \left(u2 \cdot \left(2 \cdot \pi\right) + u2 \cdot \left(2 \cdot \pi\right)\right)}{2} \cdot \left(\log u1 \cdot -0.05555555555555555\right)} + 0.5 \]

   4. associate-*r* 99.6%

      \[\leadsto \sqrt{\frac{1 + \cos \left(\color{blue}{\left(u2 \cdot 2\right) \cdot \pi} + u2 \cdot \left(2 \cdot \pi\right)\right)}{2} \cdot \left(\log u1 \cdot -0.05555555555555555\right)} + 0.5 \]

   5. associate-*r* 99.6%

      \[\leadsto \sqrt{\frac{1 + \cos \left(\left(u2 \cdot 2\right) \cdot \pi + \color{blue}{\left(u2 \cdot 2\right) \cdot \pi}\right)}{2} \cdot \left(\log u1 \cdot -0.05555555555555555\right)} + 0.5 \]

   6. distribute-rgt-out 99.6%

      \[\leadsto \sqrt{\frac{1 + \cos \color{blue}{\left(\pi \cdot \left(u2 \cdot 2 + u2 \cdot 2\right)\right)}}{2} \cdot \left(\log u1 \cdot -0.05555555555555555\right)} + 0.5 \]

   7. distribute-lft-out 99.6%

      \[\leadsto \sqrt{\frac{1 + \cos \left(\pi \cdot \color{blue}{\left(u2 \cdot \left(2 + 2\right)\right)}\right)}{2} \cdot \left(\log u1 \cdot -0.05555555555555555\right)} + 0.5 \]

   8. metadata-eval 99.6%

      \[\leadsto \sqrt{\frac{1 + \cos \left(\pi \cdot \left(u2 \cdot \color{blue}{4}\right)\right)}{2} \cdot \left(\log u1 \cdot -0.05555555555555555\right)} + 0.5 \]

10. Simplified 99.6%

    \[\leadsto \sqrt{\color{blue}{\frac{1 + \cos \left(\pi \cdot \left(u2 \cdot 4\right)\right)}{2}} \cdot \left(\log u1 \cdot -0.05555555555555555\right)} + 0.5 \]

11. Final simplification 99.6%

    \[\leadsto 0.5 + \sqrt{\frac{1 + \cos \left(\pi \cdot \left(u2 \cdot 4\right)\right)}{2} \cdot \left(-0.05555555555555555 \cdot \log u1\right)} \]
12. Add Preprocessing

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. pow1/2 99.4%

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

   2. add-sqr-sqrt 99.0%

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

   3. sqrt-unprod 99.4%

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

   4. *-commutative 99.4%

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

   5. *-commutative 99.4%

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

   6. swap-sqr 99.4%

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

   7. add-sqr-sqrt 99.6%

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

   8. metadata-eval 99.6%

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

   9. metadata-eval 99.6%

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

   10. metadata-eval 99.6%

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

4. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\sqrt{\left(-2 \cdot \log u1\right) \cdot 0.027777777777777776}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
5. 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%

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

6. Simplified 99.6%

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

7. Final simplification 99.6%

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

Alternative 4: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(u1, u2)
    real(8), intent (in) :: u1
    real(8), intent (in) :: u2
    code = 0.5d0 + (((log(u1) ** 2.0d0) * 0.0030864197530864196d0) ** 0.25d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. add-sqr-sqrt 99.0%

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

   2. sqrt-unprod 99.4%

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

   3. *-commutative 99.4%

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

   4. pow1/2 99.4%

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

   5. *-commutative 99.4%

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

   6. pow1/2 99.4%

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

   7. swap-sqr 99.4%

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

4. Applied egg-rr 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-*r* 99.6%

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

   3. *-commutative 99.6%

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

   4. associate-*l* 99.6%

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

   5. metadata-eval 99.6%

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

6. Simplified 99.6%

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

7. Taylor expanded in u2 around 0 99.1%

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

   1. *-commutative 99.1%

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

9. Simplified 99.1%

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

    1. add-cbrt-cube 98.7%

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

    2. pow1/3 98.6%

       \[\leadsto \color{blue}{{\left(\left(\sqrt{\log u1 \cdot -0.05555555555555555} \cdot \sqrt{\log u1 \cdot -0.05555555555555555}\right) \cdot \sqrt{\log u1 \cdot -0.05555555555555555}\right)}^{0.3333333333333333}} + 0.5 \]

    3. pow3 98.6%

       \[\leadsto {\color{blue}{\left({\left(\sqrt{\log u1 \cdot -0.05555555555555555}\right)}^{3}\right)}}^{0.3333333333333333} + 0.5 \]

    4. sqrt-pow2 98.6%

       \[\leadsto {\color{blue}{\left({\left(\log u1 \cdot -0.05555555555555555\right)}^{\left(\frac{3}{2}\right)}\right)}}^{0.3333333333333333} + 0.5 \]

    5. *-commutative 98.6%

       \[\leadsto {\left({\color{blue}{\left(-0.05555555555555555 \cdot \log u1\right)}}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} + 0.5 \]

    6. metadata-eval 98.6%

       \[\leadsto {\left({\left(-0.05555555555555555 \cdot \log u1\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} + 0.5 \]

11. Applied egg-rr 98.6%

    \[\leadsto \color{blue}{{\left({\left(-0.05555555555555555 \cdot \log u1\right)}^{1.5}\right)}^{0.3333333333333333}} + 0.5 \]
12. Step-by-step derivation

    1. unpow1/3 98.7%

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

    2. *-commutative 98.7%

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

13. Simplified 98.7%

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

    1. pow1/3 98.6%

       \[\leadsto \color{blue}{{\left({\left(\log u1 \cdot -0.05555555555555555\right)}^{1.5}\right)}^{0.3333333333333333}} + 0.5 \]

    2. pow-pow 99.1%

       \[\leadsto \color{blue}{{\left(\log u1 \cdot -0.05555555555555555\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} + 0.5 \]

    3. metadata-eval 99.1%

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

    4. metadata-eval 99.1%

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

    5. pow-prod-up 98.7%

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

    6. pow-prod-down 99.2%

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

    7. swap-sqr 99.2%

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

    8. unpow2 99.2%

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

    9. metadata-eval 99.2%

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

15. Applied egg-rr 99.2%

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

16. Final simplification 99.2%

    \[\leadsto 0.5 + {\left({\log u1}^{2} \cdot 0.0030864197530864196\right)}^{0.25} \]
17. Add Preprocessing

Alternative 5: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double u1, double u2) {
	return 0.5 + sqrt(log(pow(u1, -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 ** (-0.05555555555555555d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. add-sqr-sqrt 99.0%

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

   2. sqrt-unprod 99.4%

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

   3. *-commutative 99.4%

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

   4. pow1/2 99.4%

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

   5. *-commutative 99.4%

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

   6. pow1/2 99.4%

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

   7. swap-sqr 99.4%

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

4. Applied egg-rr 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-*r* 99.6%

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

   3. *-commutative 99.6%

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

   4. associate-*l* 99.6%

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

   5. metadata-eval 99.6%

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

6. Simplified 99.6%

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

7. Taylor expanded in u2 around 0 99.1%

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

   1. *-commutative 99.1%

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

9. Simplified 99.1%

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

    1. add-log-exp 99.6%

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

    2. exp-to-pow 99.7%

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

11. Applied egg-rr 99.2%

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

12. Final simplification 99.2%

    \[\leadsto 0.5 + \sqrt{\log \left({u1}^{-0.05555555555555555}\right)} \]
13. Add Preprocessing

Alternative 6: 98.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. add-sqr-sqrt 99.0%

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

   2. sqrt-unprod 99.4%

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

   3. *-commutative 99.4%

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

   4. pow1/2 99.4%

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

   5. *-commutative 99.4%

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

   6. pow1/2 99.4%

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

   7. swap-sqr 99.4%

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

4. Applied egg-rr 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-*r* 99.6%

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

   3. *-commutative 99.6%

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

   4. associate-*l* 99.6%

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

   5. metadata-eval 99.6%

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

6. Simplified 99.6%

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

7. Taylor expanded in u2 around 0 99.1%

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

   1. *-commutative 99.1%

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

9. Simplified 99.1%

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

10. Final simplification 99.1%

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

Reproduce

herbie shell --seed 2024130 
(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))