normal distribution

?

Percentage Accurate: 99.4% → 99.6%
Time: 13.9s
Precision: binary64
Cost: 32832

?

\[\left(0 \leq u1 \land u1 \leq 1\right) \land \left(0 \leq u2 \land u2 \leq 1\right)\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
↓
\[\left(\sqrt{-\log u1} \cdot \left(0.16666666666666666 \cdot \sqrt{2}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
(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 u1))) (* 0.16666666666666666 (sqrt 2.0)))
   (cos (* (* 2.0 PI) u2)))
  0.5))
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(u1)) * (0.16666666666666666 * sqrt(2.0))) * cos(((2.0 * ((double) M_PI)) * u2))) + 0.5;
}

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

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

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(sqrt(Float64(-log(u1))) * Float64(0.16666666666666666 * sqrt(2.0))) * cos(Float64(Float64(2.0 * pi) * u2))) + 0.5)
end

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)) * (0.16666666666666666 * sqrt(2.0))) * cos(((2.0 * pi) * u2))) + 0.5;
end

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[Sqrt[(-N[Log[u1], $MachinePrecision])], $MachinePrecision] * N[(0.16666666666666666 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * u2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]

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

Local Percentage Accuracy?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 99.3%

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

  2. Taylor expanded in u1 around inf 99.5%

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

  3. Simplified99.6%

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

    [Start]99.5

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

    associate-*r* [=>]99.6

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

    *-commutative [=>]99.6

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

    log-rec [=>]99.6

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

  4. Final simplification99.6%

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

Alternatives

Alternative 1
Accuracy99.6%
Cost26240
\[0.5 + \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\log u1 \cdot -0.05555555555555555} \]
Alternative 2
Accuracy98.5%
Cost19456
\[0.5 + \sqrt{\log \left({u1}^{-0.05555555555555555}\right)} \]
Alternative 3
Accuracy98.5%
Cost13120
\[0.5 + \sqrt{\log u1 \cdot -0.05555555555555555} \]
Alternative 4
Accuracy16.3%
Cost64
\[0.5 \]

Error

Reproduce?

herbie shell --seed 2023159 
(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))