normal distribution

Percentage Accurate: 99.4% → 99.7%

Time: 12.2s

Alternatives: 7

Speedup: 1.0×

Specification

\[\left(0 \leq u1 \land u1 \leq 1\right) \land \left(0 \leq u2 \land u2 \leq 1\right)\]

Math

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

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 PI) u2)))
  0.5))

C

double code(double u1, double u2) {
	return (((1.0 / 6.0) * pow((-2.0 * log(u1)), 0.5)) * cos(((2.0 * ((double) M_PI)) * u2))) + 0.5;
}

Java

public static double code(double u1, double u2) {
	return (((1.0 / 6.0) * Math.pow((-2.0 * Math.log(u1)), 0.5)) * Math.cos(((2.0 * Math.PI) * u2))) + 0.5;
}

Python

def code(u1, u2):
	return (((1.0 / 6.0) * math.pow((-2.0 * math.log(u1)), 0.5)) * math.cos(((2.0 * math.pi) * u2))) + 0.5

Julia

function code(u1, u2)
	return Float64(Float64(Float64(Float64(1.0 / 6.0) * (Float64(-2.0 * log(u1)) ^ 0.5)) * cos(Float64(Float64(2.0 * pi) * u2))) + 0.5)
end

MATLAB

function tmp = code(u1, u2)
	tmp = (((1.0 / 6.0) * ((-2.0 * log(u1)) ^ 0.5)) * cos(((2.0 * pi) * u2))) + 0.5;
end

Wolfram

code[u1_, u2_] := N[(N[(N[(N[(1.0 / 6.0), $MachinePrecision] * N[Power[N[(-2.0 * N[Log[u1], $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * u2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 PI) u2)))
  0.5))

C

double code(double u1, double u2) {
	return (((1.0 / 6.0) * pow((-2.0 * log(u1)), 0.5)) * cos(((2.0 * ((double) M_PI)) * u2))) + 0.5;
}

Java

public static double code(double u1, double u2) {
	return (((1.0 / 6.0) * Math.pow((-2.0 * Math.log(u1)), 0.5)) * Math.cos(((2.0 * Math.PI) * u2))) + 0.5;
}

Python

def code(u1, u2):
	return (((1.0 / 6.0) * math.pow((-2.0 * math.log(u1)), 0.5)) * math.cos(((2.0 * math.pi) * u2))) + 0.5

Julia

function code(u1, u2)
	return Float64(Float64(Float64(Float64(1.0 / 6.0) * (Float64(-2.0 * log(u1)) ^ 0.5)) * cos(Float64(Float64(2.0 * pi) * u2))) + 0.5)
end

MATLAB

function tmp = code(u1, u2)
	tmp = (((1.0 / 6.0) * ((-2.0 * log(u1)) ^ 0.5)) * cos(((2.0 * pi) * u2))) + 0.5;
end

Wolfram

code[u1_, u2_] := N[(N[(N[(N[(1.0 / 6.0), $MachinePrecision] * N[Power[N[(-2.0 * N[Log[u1], $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * u2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]

Alternative 1: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{\log \left({u1}^{-0.05555555555555555}\right)} \cdot \cos \left(\left(2 \cdot \sqrt[3]{{\pi}^{3}}\right) \cdot u2\right) + 0.5 \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  (*
   (sqrt (log (pow u1 -0.05555555555555555)))
   (cos (* (* 2.0 (cbrt (pow PI 3.0))) u2)))
  0.5))

C

double code(double u1, double u2) {
	return (sqrt(log(pow(u1, -0.05555555555555555))) * cos(((2.0 * cbrt(pow(((double) M_PI), 3.0))) * u2))) + 0.5;
}

Java

public static double code(double u1, double u2) {
	return (Math.sqrt(Math.log(Math.pow(u1, -0.05555555555555555))) * Math.cos(((2.0 * Math.cbrt(Math.pow(Math.PI, 3.0))) * u2))) + 0.5;
}

Julia

function code(u1, u2)
	return Float64(Float64(sqrt(log((u1 ^ -0.05555555555555555))) * cos(Float64(Float64(2.0 * cbrt((pi ^ 3.0))) * u2))) + 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(N[Sqrt[N[Log[N[Power[u1, -0.05555555555555555], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(2.0 * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * u2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. pow1/2 99.3%

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

   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 99.3%

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

   4. *-commutative 99.3%

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

   5. *-commutative 99.3%

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

   6. swap-sqr 99.3%

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

   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. Step-by-step derivation

   1. add-log-exp 99.6%

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

   2. exp-to-pow 99.7%

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

8. Applied egg-rr 99.7%

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

10. Applied egg-rr 99.7%

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

Alternative 2: 99.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(u1, u2)
	return Float64(0.5 + Float64(sqrt(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[Log[N[Power[u1, -0.05555555555555555], $MachinePrecision]], $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. Add Preprocessing
3. Step-by-step derivation

   1. pow1/2 99.3%

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

   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 99.3%

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

   4. *-commutative 99.3%

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

   5. *-commutative 99.3%

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

   6. swap-sqr 99.3%

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

   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. Step-by-step derivation

   1. add-log-exp 99.6%

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

   2. exp-to-pow 99.7%

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[u1_, u2_] := N[(0.5 + N[(N[Cos[N[(u2 * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[(-0.05555555555555555 * N[Log[u1], $MachinePrecision]), $MachinePrecision], 0.5], $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. Add Preprocessing
3. Step-by-step derivation

   1. pow1/2 99.3%

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

   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 99.3%

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

   4. *-commutative 99.3%

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

   5. *-commutative 99.3%

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

   6. swap-sqr 99.3%

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

   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. Step-by-step derivation

   1. pow1/2 99.6%

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

8. Applied egg-rr 99.6%

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

9. Final simplification 99.6%

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

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. pow1/2 99.3%

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

   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 99.3%

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

   4. *-commutative 99.3%

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

   5. *-commutative 99.3%

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

   6. swap-sqr 99.3%

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

   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%

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

Alternative 5: 51.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+ 0.5 (* 0.16666666666666666 (sqrt (log (pow u1 -2.0))))))

C

double code(double u1, double u2) {
	return 0.5 + (0.16666666666666666 * sqrt(log(pow(u1, -2.0))));
}

Fortran

real(8) function code(u1, u2)
    real(8), intent (in) :: u1
    real(8), intent (in) :: u2
    code = 0.5d0 + (0.16666666666666666d0 * sqrt(log((u1 ** (-2.0d0)))))
end function

Java

public static double code(double u1, double u2) {
	return 0.5 + (0.16666666666666666 * Math.sqrt(Math.log(Math.pow(u1, -2.0))));
}

Python

def code(u1, u2):
	return 0.5 + (0.16666666666666666 * math.sqrt(math.log(math.pow(u1, -2.0))))

Julia

function code(u1, u2)
	return Float64(0.5 + Float64(0.16666666666666666 * sqrt(log((u1 ^ -2.0)))))
end

MATLAB

function tmp = code(u1, u2)
	tmp = 0.5 + (0.16666666666666666 * sqrt(log((u1 ^ -2.0))));
end

Wolfram

code[u1_, u2_] := N[(0.5 + N[(0.16666666666666666 * N[Sqrt[N[Log[N[Power[u1, -2.0], $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. associate-*l* 99.3%

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

   2. fma-define 99.3%

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

   3. metadata-eval 99.3%

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

   4. unpow1/2 99.3%

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

   5. *-commutative 99.3%

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

   6. associate-*l* 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in u2 around 0 97.7%

   \[\leadsto \mathsf{fma}\left(0.16666666666666666, \sqrt{-2 \cdot \log u1} \cdot \color{blue}{1}, 0.5\right) \]
6. Step-by-step derivation

   1. fma-undefine 97.7%

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

   2. *-rgt-identity 97.7%

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

   3. *-commutative 97.7%

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

   4. add-log-exp 49.5%

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

   5. exp-to-pow 49.5%

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

7. Applied egg-rr 49.5%

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

8. Final simplification 49.5%

   \[\leadsto 0.5 + 0.16666666666666666 \cdot \sqrt{\log \left({u1}^{-2}\right)} \]
9. Add Preprocessing

Alternative 6: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(u1, u2)
	return Float64(sqrt(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[Log[N[Power[u1, -0.05555555555555555], $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. Add Preprocessing
3. Step-by-step derivation

   1. pow1/2 99.3%

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

   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 99.3%

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

   4. *-commutative 99.3%

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

   5. *-commutative 99.3%

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

   6. swap-sqr 99.3%

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

   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. Step-by-step derivation

   1. add-log-exp 99.6%

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

   2. exp-to-pow 99.7%

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

8. Applied egg-rr 99.7%

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

9. Taylor expanded in u2 around 0 98.0%

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

10. Final simplification 98.0%

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

Alternative 7: 0.0% 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 99.3%

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

   1. associate-*l* 99.3%

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

   2. fma-define 99.3%

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

   3. metadata-eval 99.3%

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

   4. unpow1/2 99.3%

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

   5. *-commutative 99.3%

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

   6. associate-*l* 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in u2 around 0 97.7%

   \[\leadsto \mathsf{fma}\left(0.16666666666666666, \sqrt{-2 \cdot \log u1} \cdot \color{blue}{1}, 0.5\right) \]
6. Step-by-step derivation

   1. pow1/2 97.7%

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

   2. metadata-eval 97.7%

      \[\leadsto \mathsf{fma}\left(0.16666666666666666, {\left(-2 \cdot \log u1\right)}^{\color{blue}{\left(2 \cdot 0.25\right)}} \cdot 1, 0.5\right) \]

   3. metadata-eval 97.7%

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

   4. pow-sqr 97.6%

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

   5. pow-prod-down 97.8%

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

   6. *-commutative 97.8%

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

   7. *-commutative 97.8%

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

   8. swap-sqr 97.8%

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

   9. pow2 97.8%

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

   10. metadata-eval 97.8%

       \[\leadsto \mathsf{fma}\left(0.16666666666666666, {\left({\log u1}^{2} \cdot \color{blue}{4}\right)}^{\left(0.5 \cdot 0.5\right)} \cdot 1, 0.5\right) \]

   11. metadata-eval 97.8%

       \[\leadsto \mathsf{fma}\left(0.16666666666666666, {\left({\log u1}^{2} \cdot 4\right)}^{\color{blue}{0.25}} \cdot 1, 0.5\right) \]

7. Applied egg-rr 97.8%

   \[\leadsto \mathsf{fma}\left(0.16666666666666666, \color{blue}{{\left({\log u1}^{2} \cdot 4\right)}^{0.25}} \cdot 1, 0.5\right) \]

9. Applied egg-rr 0.0%

   \[\leadsto \color{blue}{0.16666666666666666 \cdot \sqrt{2 \cdot \log u1} + 0.5} \]
10. Step-by-step derivation

    1. add-sqr-sqrt 0.0%

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

    2. sqrt-unprod 0.0%

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

    3. *-commutative 0.0%

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

    4. *-commutative 0.0%

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

    5. swap-sqr 0.0%

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

    6. add-sqr-sqrt 0.0%

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

    7. metadata-eval 0.0%

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

11. Applied egg-rr 0.0%

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

    1. *-commutative 0.0%

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

    2. associate-*r* 0.0%

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

    3. metadata-eval 0.0%

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

13. Simplified 0.0%

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

14. Final simplification 0.0%

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

Reproduce

herbie shell --seed 2024180 
(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))