normal distribution

Percentage Accurate: 99.4% → 99.5%

Time: 11.4s

Alternatives: 5

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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. add-sqr-sqrt 99.1%

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

   2. sqrt-unprod 99.6%

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

   3. *-commutative 99.6%

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

   4. pow1/2 99.6%

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

   5. *-commutative 99.6%

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

   6. pow1/2 99.6%

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

   7. swap-sqr 99.6%

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

4. Applied egg-rr 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-*r* 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-*l* 99.7%

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

   5. metadata-eval 99.7%

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

6. Simplified 99.7%

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

   1. unpow2 99.7%

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

   2. cos-mult 99.7%

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

8. Applied egg-rr 99.7%

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

   1. +-commutative 99.7%

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

   2. +-inverses 99.7%

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

   3. cos-0 99.7%

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

   4. distribute-lft-out 99.7%

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

   5. distribute-rgt-out 99.7%

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

   6. metadata-eval 99.7%

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

10. Simplified 99.7%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. pow1/2 99.6%

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

   2. add-sqr-sqrt 99.1%

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

   3. sqrt-unprod 99.6%

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

   4. *-commutative 99.6%

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

   5. *-commutative 99.6%

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

   6. swap-sqr 99.6%

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

   7. add-sqr-sqrt 99.7%

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

   8. metadata-eval 99.7%

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

   9. metadata-eval 99.7%

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

   10. metadata-eval 99.7%

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

4. Applied egg-rr 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-*l* 99.7%

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

   3. metadata-eval 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 3: 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 99.6%

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

   1. add-sqr-sqrt 99.1%

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

   2. sqrt-unprod 99.6%

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

   3. *-commutative 99.6%

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

   4. pow1/2 99.6%

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

   5. *-commutative 99.6%

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

   6. pow1/2 99.6%

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

   7. swap-sqr 99.6%

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

4. Applied egg-rr 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-*r* 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-*l* 99.7%

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

   5. metadata-eval 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in u2 around 0 99.0%

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

   1. *-commutative 99.0%

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

9. Simplified 99.0%

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

    1. add-log-exp 99.0%

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

    2. exp-to-pow 99.1%

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

11. Applied egg-rr 99.1%

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

12. Final simplification 99.1%

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

Alternative 4: 98.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ 0.5 + \sqrt{0.05555555555555555 \cdot \log \left(\frac{1}{u1}\right)} \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+ 0.5 (sqrt (* 0.05555555555555555 (log (/ 1.0 u1))))))

C

double code(double u1, double u2) {
	return 0.5 + sqrt((0.05555555555555555 * log((1.0 / 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((1.0d0 / u1))))
end function

Java

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

Python

def code(u1, u2):
	return 0.5 + math.sqrt((0.05555555555555555 * math.log((1.0 / u1))))

Julia

function code(u1, u2)
	return Float64(0.5 + sqrt(Float64(0.05555555555555555 * log(Float64(1.0 / u1)))))
end

MATLAB

function tmp = code(u1, u2)
	tmp = 0.5 + sqrt((0.05555555555555555 * log((1.0 / u1))));
end

Wolfram

code[u1_, u2_] := N[(0.5 + N[Sqrt[N[(0.05555555555555555 * N[Log[N[(1.0 / u1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. add-sqr-sqrt 99.1%

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

   2. sqrt-unprod 99.6%

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

   3. *-commutative 99.6%

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

   4. pow1/2 99.6%

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

   5. *-commutative 99.6%

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

   6. pow1/2 99.6%

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

   7. swap-sqr 99.6%

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

4. Applied egg-rr 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-*r* 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-*l* 99.7%

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

   5. metadata-eval 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in u2 around 0 99.0%

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

   1. *-commutative 99.0%

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

9. Simplified 99.0%

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

10. Taylor expanded in u1 around inf 99.0%

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

11. Final simplification 99.0%

    \[\leadsto 0.5 + \sqrt{0.05555555555555555 \cdot \log \left(\frac{1}{u1}\right)} \]
12. Add Preprocessing

Alternative 5: 98.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(u1, u2)
	return Float64(0.5 + sqrt(Float64(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[(N[Log[u1], $MachinePrecision] * -0.05555555555555555), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. add-sqr-sqrt 99.1%

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

   2. sqrt-unprod 99.6%

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

   3. *-commutative 99.6%

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

   4. pow1/2 99.6%

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

   5. *-commutative 99.6%

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

   6. pow1/2 99.6%

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

   7. swap-sqr 99.6%

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

4. Applied egg-rr 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-*r* 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-*l* 99.7%

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

   5. metadata-eval 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in u2 around 0 99.0%

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

   1. *-commutative 99.0%

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

9. Simplified 99.0%

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

10. Final simplification 99.0%

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

Reproduce

herbie shell --seed 2024137 
(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))