normal distribution

Average Error: 0.4 → 0.4

Time: 9.7s

Precision: binary64

Cost: 32896

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

Math

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

↓

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

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) (cos (* 2.0 (* u2 PI)))) (sqrt (log (/ 1.0 u1)))))
  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) * cos((2.0 * (u2 * ((double) M_PI))))) * sqrt(log((1.0 / u1))))) + 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.cos((2.0 * (u2 * Math.PI)))) * Math.sqrt(Math.log((1.0 / u1))))) + 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.cos((2.0 * (u2 * math.pi)))) * math.sqrt(math.log((1.0 / u1))))) + 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(0.16666666666666666 * Float64(Float64(sqrt(2.0) * cos(Float64(2.0 * Float64(u2 * pi)))) * sqrt(log(Float64(1.0 / u1))))) + 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) * cos((2.0 * (u2 * pi)))) * sqrt(log((1.0 / u1))))) + 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[(0.16666666666666666 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Cos[N[(2.0 * N[(u2 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Log[N[(1.0 / u1), $MachinePrecision]], $MachinePrecision]], $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. Simplified 0.4

   \[\leadsto \color{blue}{0.5 + {\left(-2 \cdot \log u1\right)}^{0.5} \cdot \left(\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot 0.16666666666666666\right)} \]
   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_best_oopsla_all_46_json_45_simplify-35 [=>] 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_best_oopsla_all_46_json_45_simplify-74 [=>] 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_best_oopsla_all_46_json_45_simplify-74 [=>] 0.4	\[ 0.5 + \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \color{blue}{\left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \frac{1}{6}\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-7 [=>] 0.4	\[ 0.5 + \color{blue}{{\left(-2 \cdot \log u1\right)}^{0.5} \cdot \left(\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \frac{1}{6}\right)} \]
   metadata-eval [=>] 0.4	\[ 0.5 + {\left(-2 \cdot \log u1\right)}^{0.5} \cdot \left(\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \color{blue}{0.16666666666666666}\right) \]

3. Taylor expanded in u1 around inf 0.4

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

4. Simplified 0.4

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

   [Start] 0.4	\[ 0.5 + 0.16666666666666666 \cdot \left(\left(\sqrt{2} \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right) \cdot \sqrt{\log \left(\frac{1}{u1}\right)}\right) \]
   rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.4	\[ \color{blue}{0.16666666666666666 \cdot \left(\left(\sqrt{2} \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right) \cdot \sqrt{\log \left(\frac{1}{u1}\right)}\right) + 0.5} \]

5. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.4
Cost	26432

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

Alternative 2
Error	0.4
Cost	26432

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

Alternative 3
Error	1.1
Cost	19776

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

Alternative 4
Error	1.1
Cost	19776

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

Alternative 5
Error	64.0
Cost	19648

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

Reproduce

herbie shell --seed 2023090 
(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))