normal distribution

Average Error: 0.4 → 0.3

Time: 11.8s

Precision: binary64

Cost: 32832

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

Math

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

↓

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

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

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

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

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

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

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

3. Taylor expanded in u1 around inf 64.0

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

4. Simplified 0.3

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

   [Start] 64.0	\[ 0.5 + 0.16666666666666666 \cdot \left(\left(\sqrt{-1} \cdot \left(\sqrt{-2} \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{\log \left(\frac{1}{u1}\right)}\right) \]
   rational.json-simplify-43 [=>] 64.0	\[ 0.5 + \color{blue}{\left(\sqrt{-1} \cdot \left(\sqrt{-2} \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)\right) \cdot \left(\sqrt{\log \left(\frac{1}{u1}\right)} \cdot 0.16666666666666666\right)} \]
   rational.json-simplify-43 [=>] 64.0	\[ 0.5 + \color{blue}{\sqrt{\log \left(\frac{1}{u1}\right)} \cdot \left(0.16666666666666666 \cdot \left(\sqrt{-1} \cdot \left(\sqrt{-2} \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)\right)\right)} \]
   rational.json-simplify-2 [=>] 64.0	\[ 0.5 + \sqrt{\log \left(\frac{1}{u1}\right)} \cdot \left(0.16666666666666666 \cdot \left(\sqrt{-1} \cdot \color{blue}{\left(\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \sqrt{-2}\right)}\right)\right) \]
   rational.json-simplify-2 [=>] 64.0	\[ 0.5 + \sqrt{\log \left(\frac{1}{u1}\right)} \cdot \left(0.16666666666666666 \cdot \left(\sqrt{-1} \cdot \left(\cos \left(2 \cdot \color{blue}{\left(\pi \cdot u2\right)}\right) \cdot \sqrt{-2}\right)\right)\right) \]
   rational.json-simplify-2 [=>] 64.0	\[ 0.5 + \sqrt{\log \left(\frac{1}{u1}\right)} \cdot \left(0.16666666666666666 \cdot \left(\sqrt{-1} \cdot \left(\cos \color{blue}{\left(\left(\pi \cdot u2\right) \cdot 2\right)} \cdot \sqrt{-2}\right)\right)\right) \]
   metadata-eval [<=] 64.0	\[ 0.5 + \sqrt{\log \left(\frac{1}{u1}\right)} \cdot \left(0.16666666666666666 \cdot \left(\sqrt{-1} \cdot \left(\cos \left(\left(\pi \cdot u2\right) \cdot \color{blue}{\frac{-2}{-1}}\right) \cdot \sqrt{-2}\right)\right)\right) \]
   rational.json-simplify-49 [<=] 64.0	\[ 0.5 + \sqrt{\log \left(\frac{1}{u1}\right)} \cdot \left(0.16666666666666666 \cdot \left(\sqrt{-1} \cdot \left(\cos \color{blue}{\left(\frac{-2 \cdot \left(\pi \cdot u2\right)}{-1}\right)} \cdot \sqrt{-2}\right)\right)\right) \]
   rational.json-simplify-10 [<=] 64.0	\[ 0.5 + \sqrt{\log \left(\frac{1}{u1}\right)} \cdot \left(0.16666666666666666 \cdot \left(\sqrt{-1} \cdot \left(\cos \color{blue}{\left(--2 \cdot \left(\pi \cdot u2\right)\right)} \cdot \sqrt{-2}\right)\right)\right) \]
   trig.json-simplify-24 [<=] 64.0	\[ 0.5 + \sqrt{\log \left(\frac{1}{u1}\right)} \cdot \left(0.16666666666666666 \cdot \left(\sqrt{-1} \cdot \left(\color{blue}{\cos \left(-2 \cdot \left(\pi \cdot u2\right)\right)} \cdot \sqrt{-2}\right)\right)\right) \]
   rational.json-simplify-43 [=>] 64.0	\[ 0.5 + \sqrt{\log \left(\frac{1}{u1}\right)} \cdot \left(0.16666666666666666 \cdot \color{blue}{\left(\cos \left(-2 \cdot \left(\pi \cdot u2\right)\right) \cdot \left(\sqrt{-2} \cdot \sqrt{-1}\right)\right)}\right) \]
   rational.json-simplify-2 [=>] 64.0	\[ 0.5 + \sqrt{\log \left(\frac{1}{u1}\right)} \cdot \left(0.16666666666666666 \cdot \color{blue}{\left(\left(\sqrt{-2} \cdot \sqrt{-1}\right) \cdot \cos \left(-2 \cdot \left(\pi \cdot u2\right)\right)\right)}\right) \]

5. Taylor expanded in u1 around 0 64.0

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

6. Simplified 0.3

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

   [Start] 64.0	\[ 0.5 + \left(\sqrt{-1} \cdot \sqrt{\log u1}\right) \cdot \left(0.16666666666666666 \cdot \left(\sqrt{2} \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\right)\right) \]
   exponential.json-simplify-20 [=>] 0.3	\[ 0.5 + \color{blue}{\sqrt{\log u1 \cdot -1}} \cdot \left(0.16666666666666666 \cdot \left(\sqrt{2} \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\right)\right) \]
   rational.json-simplify-9 [=>] 0.3	\[ 0.5 + \sqrt{\color{blue}{-\log u1}} \cdot \left(0.16666666666666666 \cdot \left(\sqrt{2} \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\right)\right) \]

7. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.4
Cost	26368

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

Alternative 2
Error	1.1
Cost	19712

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

Alternative 3
Error	1.2
Cost	13248

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

Reproduce

herbie shell --seed 2023068 
(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))