normal distribution

Percentage Accurate: 99.4% → 99.6%

Time: 9.5s

Alternatives: 4

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: 99.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. add-sqr-sqrt 98.9%

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

   2. sqrt-unprod 99.3%

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

   3. pow1/2 99.3%

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

   4. *-commutative 99.3%

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

   5. pow1/2 99.3%

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

   6. *-commutative 99.3%

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

   7. swap-sqr 99.4%

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

   8. add-sqr-sqrt 99.6%

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

   9. metadata-eval 99.6%

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

   10. metadata-eval 99.6%

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

   11. metadata-eval 99.6%

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

3. Applied egg-rr 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-*l* 99.6%

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

   3. metadata-eval 99.6%

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

5. Simplified 99.6%

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

   1. associate-*r* 99.6%

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

   2. add-sqr-sqrt 99.6%

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

   3. pow2 99.6%

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

7. Applied egg-rr 99.6%

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

8. Final simplification 99.6%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(u1, u2)
	tmp = 0.5 + (sqrt((log(u1) * -0.05555555555555555)) * cos((u2 * (2.0 * pi))));
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[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. add-sqr-sqrt 98.9%

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

   2. sqrt-unprod 99.3%

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

   3. pow1/2 99.3%

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

   4. *-commutative 99.3%

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

   5. pow1/2 99.3%

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

   6. *-commutative 99.3%

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

   7. swap-sqr 99.4%

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

   8. add-sqr-sqrt 99.6%

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

   9. metadata-eval 99.6%

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

   10. metadata-eval 99.6%

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

   11. metadata-eval 99.6%

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

3. Applied egg-rr 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-*l* 99.6%

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

   3. metadata-eval 99.6%

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 3: 98.4% accurate, 1.4× 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 99.3%

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

2. Taylor expanded in u2 around 0 97.3%

   \[\leadsto \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \color{blue}{1} + 0.5 \]
3. Step-by-step derivation

   1. *-rgt-identity 97.3%

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

   2. metadata-eval 97.3%

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

   3. pow1/2 97.3%

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

   4. expm1-log1p-u 97.1%

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

   5. expm1-udef 97.1%

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

   6. associate-+l- 97.1%

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

4. Applied egg-rr 97.5%

   \[\leadsto \color{blue}{\left(1 + \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}\right) - 0.5} \]
5. Step-by-step derivation

   1. associate--l+ 97.6%

      \[\leadsto \color{blue}{1 + \left(\sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776} - 0.5\right)} \]

   2. associate-*l* 97.6%

      \[\leadsto 1 + \left(\sqrt{\color{blue}{\log u1 \cdot \left(-2 \cdot 0.027777777777777776\right)}} - 0.5\right) \]

   3. metadata-eval 97.6%

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

   4. *-commutative 97.6%

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

   5. log-pow 97.6%

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

6. Simplified 97.6%

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

7. Taylor expanded in u1 around 0 97.6%

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

8. Final simplification 97.6%

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

Alternative 4: 98.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in u2 around 0 97.3%

   \[\leadsto \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \color{blue}{1} + 0.5 \]
3. Step-by-step derivation

   1. *-rgt-identity 97.3%

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

   2. metadata-eval 97.3%

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

   3. pow1/2 97.3%

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

   4. expm1-log1p-u 97.1%

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

   5. expm1-udef 97.1%

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

   6. associate-+l- 97.1%

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

4. Applied egg-rr 97.5%

   \[\leadsto \color{blue}{\left(1 + \sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}\right) - 0.5} \]
5. Step-by-step derivation

   1. +-commutative 97.5%

      \[\leadsto \color{blue}{\left(\sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776} + 1\right)} - 0.5 \]

   2. associate--l+ 97.6%

      \[\leadsto \color{blue}{\sqrt{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776} + \left(1 - 0.5\right)} \]

   3. metadata-eval 97.6%

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

   4. +-commutative 97.6%

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

   5. associate-*l* 97.6%

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

   6. metadata-eval 97.6%

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

   7. *-commutative 97.6%

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

6. Simplified 97.6%

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

7. Final simplification 97.6%

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

Reproduce

herbie shell --seed 2023196 
(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))