Average Error: 0.4 → 0.3
Time: 4.5s
Precision: binary64
\[\left(0 \leq u1 \land u1 \leq 1\right) \land \left(0 \leq u2 \land u2 \leq 1\right)\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
↓
\[0.5 + \sqrt{\log u1 \cdot \left({\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)}^{2} \cdot -0.05555555555555555\right)} \]
(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) (* (pow (cos (* 2.0 (* u2 PI))) 2.0) -0.05555555555555555)))))
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) * (pow(cos((2.0 * (u2 * ((double) M_PI)))), 2.0) * -0.05555555555555555)));
}

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) * (Math.pow(Math.cos((2.0 * (u2 * Math.PI))), 2.0) * -0.05555555555555555)));
}

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) * (math.pow(math.cos((2.0 * (u2 * math.pi))), 2.0) * -0.05555555555555555)))

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 + sqrt(Float64(log(u1) * Float64((cos(Float64(2.0 * Float64(u2 * pi))) ^ 2.0) * -0.05555555555555555))))
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 = 0.5 + sqrt((log(u1) * ((cos((2.0 * (u2 * pi))) ^ 2.0) * -0.05555555555555555)));
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[(0.5 + N[Sqrt[N[(N[Log[u1], $MachinePrecision] * N[(N[Power[N[Cos[N[(2.0 * N[(u2 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * -0.05555555555555555), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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
↓
0.5 + \sqrt{\log u1 \cdot \left({\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)}^{2} \cdot -0.05555555555555555\right)}

Error

Bits error versus u1

Bits error versus u2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Simplified0.4

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

  3. Taylor expanded in u1 around inf 0.3

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

  4. Applied egg-rr0.3

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

  5. Applied egg-rr0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2022162 
(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))