normal distribution

Percentage Accurate: 99.4% → 99.7%

Time: 8.8s

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, 0.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(u1, u2)
	tmp = (sqrt(log((u1 ^ -0.05555555555555555))) * cos(((2.0 * pi) * 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 * Pi), $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. Step-by-step derivation

   1. add-sqr-sqrt 99.0%

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

   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. Taylor expanded in u1 around 0 99.6%

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

   1. log-pow 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[u1_, u2_] := N[(0.5 + N[(N[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * u2), $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. Step-by-step derivation

   1. add-sqr-sqrt 99.0%

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

   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%

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

Alternative 3: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[u1_, u2_] := N[(N[(0.16666666666666666 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-N[Log[u1], $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. *-commutative 99.3%

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

   2. associate-*l* 99.3%

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

   3. fma-def 99.3%

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

   4. unpow1/2 99.3%

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

   5. metadata-eval 99.3%

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

   6. associate-*l* 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in u2 around 0 97.6%

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

   1. pow1/2 97.6%

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

   2. add-cbrt-cube 97.3%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt[3]{\left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}}, 0.16666666666666666, 0.5\right) \]

   3. pow1/3 96.4%

      \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)}^{0.3333333333333333}}, 0.16666666666666666, 0.5\right) \]

   4. pow-prod-up 96.5%

      \[\leadsto \mathsf{fma}\left({\left(\color{blue}{{\left(-2 \cdot \log u1\right)}^{\left(0.5 + 0.5\right)}} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)}^{0.3333333333333333}, 0.16666666666666666, 0.5\right) \]

   5. metadata-eval 96.5%

      \[\leadsto \mathsf{fma}\left({\left({\left(-2 \cdot \log u1\right)}^{\color{blue}{1}} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)}^{0.3333333333333333}, 0.16666666666666666, 0.5\right) \]

   6. pow-prod-up 96.5%

      \[\leadsto \mathsf{fma}\left({\color{blue}{\left({\left(-2 \cdot \log u1\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}, 0.16666666666666666, 0.5\right) \]

   7. metadata-eval 96.5%

      \[\leadsto \mathsf{fma}\left({\left({\left(-2 \cdot \log u1\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}, 0.16666666666666666, 0.5\right) \]

6. Applied egg-rr 96.5%

   \[\leadsto \mathsf{fma}\left(\color{blue}{{\left({\left(-2 \cdot \log u1\right)}^{1.5}\right)}^{0.3333333333333333}}, 0.16666666666666666, 0.5\right) \]

7. Taylor expanded in u1 around inf 97.7%

   \[\leadsto \color{blue}{0.5 + 0.16666666666666666 \cdot \left(\sqrt{2} \cdot \sqrt{\log \left(\frac{1}{u1}\right)}\right)} \]
8. Step-by-step derivation

   1. +-commutative 97.7%

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

   2. associate-*r* 97.9%

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

   3. fma-def 97.9%

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

   4. log-rec 97.9%

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

9. Simplified 97.9%

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

10. Final simplification 97.9%

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

Alternative 4: 98.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ 0.5 + \sqrt{-\log u1} \cdot \left(0.16666666666666666 \cdot {4}^{0.25}\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+ 0.5 (* (sqrt (- (log u1))) (* 0.16666666666666666 (pow 4.0 0.25)))))

C

double code(double u1, double u2) {
	return 0.5 + (sqrt(-log(u1)) * (0.16666666666666666 * pow(4.0, 0.25)));
}

Fortran

real(8) function code(u1, u2)
    real(8), intent (in) :: u1
    real(8), intent (in) :: u2
    code = 0.5d0 + (sqrt(-log(u1)) * (0.16666666666666666d0 * (4.0d0 ** 0.25d0)))
end function

Java

public static double code(double u1, double u2) {
	return 0.5 + (Math.sqrt(-Math.log(u1)) * (0.16666666666666666 * Math.pow(4.0, 0.25)));
}

Python

def code(u1, u2):
	return 0.5 + (math.sqrt(-math.log(u1)) * (0.16666666666666666 * math.pow(4.0, 0.25)))

Julia

function code(u1, u2)
	return Float64(0.5 + Float64(sqrt(Float64(-log(u1))) * Float64(0.16666666666666666 * (4.0 ^ 0.25))))
end

MATLAB

function tmp = code(u1, u2)
	tmp = 0.5 + (sqrt(-log(u1)) * (0.16666666666666666 * (4.0 ^ 0.25)));
end

Wolfram

code[u1_, u2_] := N[(0.5 + N[(N[Sqrt[(-N[Log[u1], $MachinePrecision])], $MachinePrecision] * N[(0.16666666666666666 * N[Power[4.0, 0.25], $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. *-commutative 99.3%

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

   2. associate-*l* 99.3%

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

   3. fma-def 99.3%

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

   4. unpow1/2 99.3%

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

   5. metadata-eval 99.3%

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

   6. associate-*l* 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in u2 around 0 97.6%

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

   1. pow1/2 97.6%

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

   2. metadata-eval 97.6%

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

   3. metadata-eval 97.6%

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

   4. pow-sqr 97.5%

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

   5. pow-prod-down 97.6%

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

   6. swap-sqr 97.6%

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

   7. metadata-eval 97.6%

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

   8. pow2 97.6%

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

   9. metadata-eval 97.6%

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

6. Applied egg-rr 97.6%

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

7. Taylor expanded in u1 around inf 97.7%

   \[\leadsto \color{blue}{0.5 + 0.16666666666666666 \cdot \left({4}^{0.25} \cdot \sqrt{\log \left(\frac{1}{u1}\right)}\right)} \]
8. Step-by-step derivation

   1. associate-*r* 97.9%

      \[\leadsto 0.5 + \color{blue}{\left(0.16666666666666666 \cdot {4}^{0.25}\right) \cdot \sqrt{\log \left(\frac{1}{u1}\right)}} \]

   2. log-rec 97.9%

      \[\leadsto 0.5 + \left(0.16666666666666666 \cdot {4}^{0.25}\right) \cdot \sqrt{\color{blue}{-\log u1}} \]

9. Simplified 97.9%

   \[\leadsto \color{blue}{0.5 + \left(0.16666666666666666 \cdot {4}^{0.25}\right) \cdot \sqrt{-\log u1}} \]

10. Final simplification 97.9%

    \[\leadsto 0.5 + \sqrt{-\log u1} \cdot \left(0.16666666666666666 \cdot {4}^{0.25}\right) \]

Alternative 5: 98.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[u1_, u2_] := N[(0.5 + N[(0.16666666666666666 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[(-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. Step-by-step derivation

   1. *-commutative 99.3%

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

   2. associate-*l* 99.3%

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

   3. fma-def 99.3%

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

   4. unpow1/2 99.3%

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

   5. metadata-eval 99.3%

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

   6. associate-*l* 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in u2 around 0 97.6%

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

   1. pow1/2 97.6%

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

   2. add-cbrt-cube 97.3%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt[3]{\left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}}, 0.16666666666666666, 0.5\right) \]

   3. pow1/3 96.4%

      \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)}^{0.3333333333333333}}, 0.16666666666666666, 0.5\right) \]

   4. pow-prod-up 96.5%

      \[\leadsto \mathsf{fma}\left({\left(\color{blue}{{\left(-2 \cdot \log u1\right)}^{\left(0.5 + 0.5\right)}} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)}^{0.3333333333333333}, 0.16666666666666666, 0.5\right) \]

   5. metadata-eval 96.5%

      \[\leadsto \mathsf{fma}\left({\left({\left(-2 \cdot \log u1\right)}^{\color{blue}{1}} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)}^{0.3333333333333333}, 0.16666666666666666, 0.5\right) \]

   6. pow-prod-up 96.5%

      \[\leadsto \mathsf{fma}\left({\color{blue}{\left({\left(-2 \cdot \log u1\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}, 0.16666666666666666, 0.5\right) \]

   7. metadata-eval 96.5%

      \[\leadsto \mathsf{fma}\left({\left({\left(-2 \cdot \log u1\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}, 0.16666666666666666, 0.5\right) \]

6. Applied egg-rr 96.5%

   \[\leadsto \mathsf{fma}\left(\color{blue}{{\left({\left(-2 \cdot \log u1\right)}^{1.5}\right)}^{0.3333333333333333}}, 0.16666666666666666, 0.5\right) \]

7. Taylor expanded in u1 around inf 97.7%

   \[\leadsto \color{blue}{0.5 + 0.16666666666666666 \cdot \left(\sqrt{2} \cdot \sqrt{\log \left(\frac{1}{u1}\right)}\right)} \]
8. Step-by-step derivation

   1. *-commutative 97.7%

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

   2. log-rec 97.7%

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

9. Simplified 97.7%

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

10. Final simplification 97.7%

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

Alternative 6: 98.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[u1_, u2_] := N[(N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] * 0.16666666666666666 + 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. *-commutative 99.3%

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

   2. associate-*l* 99.3%

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

   3. fma-def 99.3%

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

   4. unpow1/2 99.3%

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

   5. metadata-eval 99.3%

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

   6. associate-*l* 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in u2 around 0 97.6%

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

5. Final simplification 97.6%

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

Alternative 7: 0.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 + 0.16666666666666666 \cdot \sqrt{2 \cdot \log u1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[u1_, u2_] := N[(0.5 + N[(0.16666666666666666 * N[Sqrt[N[(2.0 * 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. Step-by-step derivation

   1. *-commutative 99.3%

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

   2. associate-*l* 99.3%

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

   3. fma-def 99.3%

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

   4. unpow1/2 99.3%

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

   5. metadata-eval 99.3%

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

   6. associate-*l* 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in u2 around 0 97.6%

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

   1. fma-udef 97.6%

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

   2. metadata-eval 97.6%

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

   3. associate-*r* 97.6%

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

   4. neg-mul-1 97.6%

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

   5. sqrt-unprod 97.7%

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

   6. *-commutative 97.7%

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

   7. sqrt-unprod 97.6%

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

   8. add-sqr-sqrt 97.6%

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

   9. sqrt-unprod 97.6%

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

   10. sqr-neg 97.6%

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

   11. sqrt-unprod 0.0%

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

   12. add-sqr-sqrt 0.0%

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

6. Applied egg-rr 0.0%

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

7. Final simplification 0.0%

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

Reproduce

herbie shell --seed 2023228 
(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))