normal distribution

Average Error: 0.4 → 0.2

Time: 12.1s

Precision: binary64

Cost: 32512

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

Math

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

↓

\[\mathsf{fma}\left(\sqrt{\log u1 \cdot -0.05555555555555555}, \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), 0.5\right) \]

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
 (fma (sqrt (* (log u1) -0.05555555555555555)) (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 fma(sqrt((log(u1) * -0.05555555555555555)), cos((2.0 * (((double) M_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 fma(sqrt(Float64(log(u1) * -0.05555555555555555)), cos(Float64(2.0 * Float64(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[Sqrt[N[(N[Log[u1], $MachinePrecision] * -0.05555555555555555), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(2.0 * N[(Pi * 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. Simplified 0.4

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

   [Start] 0.4	\[ \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
   fma-def [=>] 0.4	\[ \color{blue}{\mathsf{fma}\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right)} \]
   metadata-eval [=>] 0.4	\[ \mathsf{fma}\left(\color{blue}{0.16666666666666666} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right) \]
   unpow1/2 [=>] 0.4	\[ \mathsf{fma}\left(0.16666666666666666 \cdot \color{blue}{\sqrt{-2 \cdot \log u1}}, \cos \left(\left(2 \cdot \pi\right) \cdot u2\right), 0.5\right) \]
   associate-*l* [=>] 0.4	\[ \mathsf{fma}\left(0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1}, \cos \color{blue}{\left(2 \cdot \left(\pi \cdot u2\right)\right)}, 0.5\right) \]

3. Applied egg-rr 0.2

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

4. Simplified 0.2

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

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

5. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.2
Cost	26240

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

Alternative 2
Error	1.0
Cost	19456

\[0.5 + \sqrt{\log \left({u1}^{-0.05555555555555555}\right)} \]

Alternative 3
Error	1.0
Cost	13120

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

Reproduce

herbie shell --seed 2023002 
(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))