normal distribution

Average Error: 0.4 → 0.3

Time: 12.3s

Precision: binary64

Cost: 32832

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

Math

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

↓

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

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
 (+
  (*
   (* (* 0.16666666666666666 (sqrt 2.0)) (sqrt (- (log u1))))
   (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 (((0.16666666666666666 * sqrt(2.0)) * sqrt(-log(u1))) * 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 (((0.16666666666666666 * Math.sqrt(2.0)) * Math.sqrt(-Math.log(u1))) * 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 (((0.16666666666666666 * math.sqrt(2.0)) * math.sqrt(-math.log(u1))) * 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(Float64(Float64(0.16666666666666666 * sqrt(2.0)) * sqrt(Float64(-log(u1)))) * 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 = (((0.16666666666666666 * sqrt(2.0)) * sqrt(-log(u1))) * 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[(N[(0.16666666666666666 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-N[Log[u1], $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * u2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 0.4

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

2. Taylor expanded in u1 around inf 0.3

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

3. Simplified 0.3

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

   [Start] 0.3	\[ \left(0.16666666666666666 \cdot \left(\sqrt{2} \cdot \sqrt{\log \left(\frac{1}{u1}\right)}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
   associate-*r* [=>] 0.3	\[ \color{blue}{\left(\left(0.16666666666666666 \cdot \sqrt{2}\right) \cdot \sqrt{\log \left(\frac{1}{u1}\right)}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
   log-rec [=>] 0.3	\[ \left(\left(0.16666666666666666 \cdot \sqrt{2}\right) \cdot \sqrt{\color{blue}{-\log u1}}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

4. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.2
Cost	26240

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

Alternative 2
Error	0.9
Cost	19456

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

Alternative 3
Error	0.9
Cost	13120

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

Reproduce

herbie shell --seed 2023011 
(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))