normal distribution

Percentage Accurate: 99.4% → 99.7%

Time: 8.3s

Precision: binary64

Cost: 32576

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

Math

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

↓

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

FPCore

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

↓

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

↓

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 (((1.0 / 6.0) * Math.pow((-2.0 * Math.log(u1)), 0.5)) * Math.cos(((2.0 * Math.PI) * u2))) + 0.5;
}

↓

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

↓

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

↓

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

↓

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[(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]

↓

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 99.4%

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

2. Applied egg-rr 99.7%

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

   [Start] 99.4%	\[ \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
   add-sqr-sqrt [=>] 99.1%	\[ \color{blue}{\left(\sqrt{\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}} \cdot \sqrt{\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
   sqrt-unprod [=>] 99.4%	\[ \color{blue}{\sqrt{\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
   pow1/2 [<=] 99.4%	\[ \sqrt{\left(\frac{1}{6} \cdot \color{blue}{\sqrt{-2 \cdot \log u1}}\right) \cdot \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
   *-commutative [=>] 99.4%	\[ \sqrt{\color{blue}{\left(\sqrt{-2 \cdot \log u1} \cdot \frac{1}{6}\right)} \cdot \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
   pow1/2 [<=] 99.4%	\[ \sqrt{\left(\sqrt{-2 \cdot \log u1} \cdot \frac{1}{6}\right) \cdot \left(\frac{1}{6} \cdot \color{blue}{\sqrt{-2 \cdot \log u1}}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
   *-commutative [=>] 99.4%	\[ \sqrt{\left(\sqrt{-2 \cdot \log u1} \cdot \frac{1}{6}\right) \cdot \color{blue}{\left(\sqrt{-2 \cdot \log u1} \cdot \frac{1}{6}\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
   swap-sqr [=>] 99.4%	\[ \sqrt{\color{blue}{\left(\sqrt{-2 \cdot \log u1} \cdot \sqrt{-2 \cdot \log u1}\right) \cdot \left(\frac{1}{6} \cdot \frac{1}{6}\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
   add-sqr-sqrt [<=] 99.7%	\[ \sqrt{\color{blue}{\left(-2 \cdot \log u1\right)} \cdot \left(\frac{1}{6} \cdot \frac{1}{6}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
   metadata-eval [=>] 99.7%	\[ \sqrt{\left(-2 \cdot \log u1\right) \cdot \left(\color{blue}{0.16666666666666666} \cdot \frac{1}{6}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
   metadata-eval [=>] 99.7%	\[ \sqrt{\left(-2 \cdot \log u1\right) \cdot \left(0.16666666666666666 \cdot \color{blue}{0.16666666666666666}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
   metadata-eval [=>] 99.7%	\[ \sqrt{\left(-2 \cdot \log u1\right) \cdot \color{blue}{0.027777777777777776}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

3. Simplified 99.7%

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

   [Start] 99.7%	\[ \sqrt{\left(-2 \cdot \log u1\right) \cdot 0.027777777777777776} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
   *-commutative [=>] 99.7%	\[ \sqrt{\color{blue}{\left(\log u1 \cdot -2\right)} \cdot 0.027777777777777776} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
   associate-*l* [=>] 99.7%	\[ \sqrt{\color{blue}{\log u1 \cdot \left(-2 \cdot 0.027777777777777776\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
   metadata-eval [=>] 99.7%	\[ \sqrt{\log u1 \cdot \color{blue}{-0.05555555555555555}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

4. Applied egg-rr 99.7%

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

   [Start] 99.7%	\[ \sqrt{\log u1 \cdot -0.05555555555555555} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
   add-log-exp [=>] 99.6%	\[ \sqrt{\color{blue}{\log \left(e^{\log u1 \cdot -0.05555555555555555}\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
   exp-to-pow [=>] 99.7%	\[ \sqrt{\log \color{blue}{\left({u1}^{-0.05555555555555555}\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

5. Final simplification 99.7%

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

Alternatives

Alternative 1
Accuracy	99.7%
Cost	32576

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

Alternative 2
Accuracy	99.6%
Cost	26240

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

Alternative 3
Accuracy	98.3%
Cost	19776

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

Alternative 4
Accuracy	98.1%
Cost	19520

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

Alternative 5
Accuracy	0.0%
Cost	13248

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

Reproduce

herbie shell --seed 2023178 
(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))