Average Error: 0.4 → 0.2
Time: 5.7s
Precision: binary64
\[\left(0 \leq u1 \land u1 \leq 1\right) \land \left(0 \leq u2 \land u2 \leq 1\right)\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
↓
\[\mathsf{fma}\left(\sqrt{-\log u1}, 0.16666666666666666 \cdot \left(\sqrt{2} \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right), 0.5\right) \]
(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
 (fma
  (sqrt (- (log u1)))
  (* 0.16666666666666666 (* (sqrt 2.0) (cos (* 2.0 (* u2 PI)))))
  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 fma(sqrt(-log(u1)), (0.16666666666666666 * (sqrt(2.0) * cos((2.0 * (u2 * ((double) M_PI)))))), 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 fma(sqrt(Float64(-log(u1))), Float64(0.16666666666666666 * Float64(sqrt(2.0) * cos(Float64(2.0 * Float64(u2 * pi))))), 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[Sqrt[(-N[Log[u1], $MachinePrecision])], $MachinePrecision] * N[(0.16666666666666666 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Cos[N[(2.0 * N[(u2 * Pi), $MachinePrecision]), $MachinePrecision]], $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
↓
\mathsf{fma}\left(\sqrt{-\log u1}, 0.16666666666666666 \cdot \left(\sqrt{2} \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right), 0.5\right)

Error

Bits error versus u1

Bits error versus u2

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

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

  4. Simplified0.3

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

  5. Taylor expanded in u1 around 0 0.3

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

  6. Simplified0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2022152 
(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))