normal distribution

Average Error: 0.4 → 0.2

Time: 9.8s

Precision: binary64

Cost: 32640

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

Math

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

↓

\[\sqrt{\log \left(\sqrt[3]{u1}\right) \cdot -0.16666666666666666} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

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
 (+
  (* (sqrt (* (log (cbrt u1)) -0.16666666666666666)) (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 (sqrt((log(cbrt(u1)) * -0.16666666666666666)) * 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 (Math.sqrt((Math.log(Math.cbrt(u1)) * -0.16666666666666666)) * 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(sqrt(Float64(log(cbrt(u1)) * -0.16666666666666666)) * cos(Float64(Float64(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[Sqrt[N[(N[Log[N[Power[u1, 1/3], $MachinePrecision]], $MachinePrecision] * -0.16666666666666666), $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. Applied egg-rr 0.2

   \[\leadsto \color{blue}{\sqrt{\left(-2 \cdot \log u1\right) \cdot 0.027777777777777776}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

3. Simplified 0.2

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

   [Start] 0.2	\[ \sqrt{\left(-2 \cdot \log u1\right) \cdot 0.027777777777777776} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
   *-commutative [=>] 0.2	\[ \sqrt{\color{blue}{\left(\log u1 \cdot -2\right)} \cdot 0.027777777777777776} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
   associate-*l* [=>] 0.2	\[ \sqrt{\color{blue}{\log u1 \cdot \left(-2 \cdot 0.027777777777777776\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
   metadata-eval [=>] 0.2	\[ \sqrt{\log u1 \cdot \color{blue}{-0.05555555555555555}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

4. Applied egg-rr 0.2

   \[\leadsto \sqrt{\color{blue}{-0.05555555555555555 \cdot \log \left({\left(\sqrt[3]{u1}\right)}^{2}\right) + -0.05555555555555555 \cdot \log \left(\sqrt[3]{u1}\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

5. Simplified 0.2

   \[\leadsto \sqrt{\color{blue}{\log \left(\sqrt[3]{u1}\right) \cdot -0.16666666666666666}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
   Proof

   [Start] 0.2	\[ \sqrt{-0.05555555555555555 \cdot \log \left({\left(\sqrt[3]{u1}\right)}^{2}\right) + -0.05555555555555555 \cdot \log \left(\sqrt[3]{u1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
   log-pow [=>] 0.2	\[ \sqrt{-0.05555555555555555 \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt[3]{u1}\right)\right)} + -0.05555555555555555 \cdot \log \left(\sqrt[3]{u1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
   associate-*r* [=>] 0.2	\[ \sqrt{\color{blue}{\left(-0.05555555555555555 \cdot 2\right) \cdot \log \left(\sqrt[3]{u1}\right)} + -0.05555555555555555 \cdot \log \left(\sqrt[3]{u1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
   metadata-eval [=>] 0.2	\[ \sqrt{\color{blue}{-0.1111111111111111} \cdot \log \left(\sqrt[3]{u1}\right) + -0.05555555555555555 \cdot \log \left(\sqrt[3]{u1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
   metadata-eval [<=] 0.2	\[ \sqrt{\color{blue}{\left(-0.05555555555555555 + -0.05555555555555555\right)} \cdot \log \left(\sqrt[3]{u1}\right) + -0.05555555555555555 \cdot \log \left(\sqrt[3]{u1}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
   distribute-rgt-out [=>] 0.2	\[ \sqrt{\color{blue}{\log \left(\sqrt[3]{u1}\right) \cdot \left(\left(-0.05555555555555555 + -0.05555555555555555\right) + -0.05555555555555555\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
   metadata-eval [=>] 0.2	\[ \sqrt{\log \left(\sqrt[3]{u1}\right) \cdot \left(\color{blue}{-0.1111111111111111} + -0.05555555555555555\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
   metadata-eval [=>] 0.2	\[ \sqrt{\log \left(\sqrt[3]{u1}\right) \cdot \color{blue}{-0.16666666666666666}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

6. Final simplification 0.2

   \[\leadsto \sqrt{\log \left(\sqrt[3]{u1}\right) \cdot -0.16666666666666666} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

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	64.0
Cost	19520

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

Reproduce

herbie shell --seed 2023028 
(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))