normal distribution

Average Error: 0.4 → 0.3

Time: 18.6s

Precision: binary64

Cost: 32832

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

Math

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

↓

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

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
 (+
  0.5
  (*
   0.16666666666666666
   (* (* (sqrt 2.0) (sqrt (- (log u1)))) (cos (* (* 2.0 PI) u2))))))

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 0.5 + (0.16666666666666666 * ((sqrt(2.0) * sqrt(-log(u1))) * cos(((2.0 * ((double) M_PI)) * u2))));
}

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 0.5 + (0.16666666666666666 * ((Math.sqrt(2.0) * Math.sqrt(-Math.log(u1))) * Math.cos(((2.0 * Math.PI) * u2))));
}

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 0.5 + (0.16666666666666666 * ((math.sqrt(2.0) * math.sqrt(-math.log(u1))) * math.cos(((2.0 * math.pi) * u2))))

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(0.5 + Float64(0.16666666666666666 * Float64(Float64(sqrt(2.0) * sqrt(Float64(-log(u1)))) * cos(Float64(Float64(2.0 * pi) * u2)))))
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 = 0.5 + (0.16666666666666666 * ((sqrt(2.0) * sqrt(-log(u1))) * cos(((2.0 * pi) * u2))));
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[(0.5 + N[(0.16666666666666666 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[(-N[Log[u1], $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * u2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. Simplified 0.4

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

   [Start] 0.4	\[ \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
   rational.json-simplify-1 [=>] 0.4	\[ \color{blue}{0.5 + \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)} \]
   rational.json-simplify-2 [=>] 0.4	\[ 0.5 + \color{blue}{\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)} \]
   rational.json-simplify-43 [=>] 0.4	\[ 0.5 + \color{blue}{\frac{1}{6} \cdot \left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\right)} \]
   metadata-eval [=>] 0.4	\[ 0.5 + \color{blue}{0.16666666666666666} \cdot \left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\right) \]

3. Taylor expanded in u1 around inf 0.4

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

4. Taylor expanded in u1 around 0 0.3

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

5. Simplified 0.3

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

   [Start] 0.3	\[ 0.5 + 0.16666666666666666 \cdot \left(\left(\sqrt{2} \cdot \sqrt{-1 \cdot \log u1}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\right) \]
   rational.json-simplify-2 [=>] 0.3	\[ 0.5 + 0.16666666666666666 \cdot \left(\left(\sqrt{2} \cdot \sqrt{\color{blue}{\log u1 \cdot -1}}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\right) \]
   rational.json-simplify-9 [=>] 0.3	\[ 0.5 + 0.16666666666666666 \cdot \left(\left(\sqrt{2} \cdot \sqrt{\color{blue}{-\log u1}}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\right) \]

6. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.4
Cost	26368

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

Alternative 2
Error	1.2
Cost	19712

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

Alternative 3
Error	1.3
Cost	13248

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

Reproduce

herbie shell --seed 2023069 
(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))