normal distribution

Average Error: 0.57% → 0.33%

Time: 13.2s

Precision: binary64

Cost: 38848

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

Math

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

↓

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

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 (pow u1 -0.05555555555555555))) (cos (* PI (* 2.0 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(pow(u1, -0.05555555555555555))), cos((((double) M_PI) * (2.0 * 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(log((u1 ^ -0.05555555555555555))), cos(Float64(pi * Float64(2.0 * 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[Log[N[Power[u1, -0.05555555555555555], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[N[(Pi * N[(2.0 * u2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 0.57

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

2. Applied egg-rr 0.36

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

3. Simplified 0.36

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

   [Start] 0.36	\[ \sqrt{\left(-2 \cdot \log u1\right) \cdot 0.027777777777777776} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
   *-commutative [=>] 0.36	\[ \sqrt{\color{blue}{\left(\log u1 \cdot -2\right)} \cdot 0.027777777777777776} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
   associate-*l* [=>] 0.36	\[ \sqrt{\color{blue}{\log u1 \cdot \left(-2 \cdot 0.027777777777777776\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
   metadata-eval [=>] 0.36	\[ \sqrt{\log u1 \cdot \color{blue}{-0.05555555555555555}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

4. Applied egg-rr 1.02

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\sqrt{\log u1 \cdot -0.05555555555555555}, \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), 0.5\right)\right)} - 1} \]

5. Simplified 0.33

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

   [Start] 1.02	\[ e^{\mathsf{log1p}\left(\mathsf{fma}\left(\sqrt{\log u1 \cdot -0.05555555555555555}, \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), 0.5\right)\right)} - 1 \]
   expm1-def [=>] 1.02	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\sqrt{\log u1 \cdot -0.05555555555555555}, \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), 0.5\right)\right)\right)} \]
   expm1-log1p [=>] 0.36	\[ \color{blue}{\mathsf{fma}\left(\sqrt{\log u1 \cdot -0.05555555555555555}, \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), 0.5\right)} \]
   *-commutative [=>] 0.36	\[ \mathsf{fma}\left(\sqrt{\color{blue}{-0.05555555555555555 \cdot \log u1}}, \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), 0.5\right) \]
   log-pow [<=] 0.33	\[ \mathsf{fma}\left(\sqrt{\color{blue}{\log \left({u1}^{-0.05555555555555555}\right)}}, \cos \left(2 \cdot \left(\pi \cdot u2\right)\right), 0.5\right) \]
   *-commutative [<=] 0.33	\[ \mathsf{fma}\left(\sqrt{\log \left({u1}^{-0.05555555555555555}\right)}, \cos \left(2 \cdot \color{blue}{\left(u2 \cdot \pi\right)}\right), 0.5\right) \]
   associate-*r* [=>] 0.33	\[ \mathsf{fma}\left(\sqrt{\log \left({u1}^{-0.05555555555555555}\right)}, \cos \color{blue}{\left(\left(2 \cdot u2\right) \cdot \pi\right)}, 0.5\right) \]
   *-commutative [=>] 0.33	\[ \mathsf{fma}\left(\sqrt{\log \left({u1}^{-0.05555555555555555}\right)}, \cos \color{blue}{\left(\pi \cdot \left(2 \cdot u2\right)\right)}, 0.5\right) \]

6. Final simplification 0.33

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

Alternatives

Alternative 1
Error	0.36%
Cost	26240

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

Alternative 2
Error	1.47%
Cost	19456

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

Alternative 3
Error	1.51%
Cost	13120

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

Reproduce

herbie shell --seed 2023115 
(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))