normal distribution

Average Error: 0.4 → 0.2

Time: 8.2s

Precision: binary64

Cost: 32704

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

Math

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

↓

\[\sqrt{\mathsf{log1p}\left({u1}^{-0.05555555555555555} + -1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

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 (log1p (+ (pow u1 -0.05555555555555555) -1.0)))
   (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(log1p((pow(u1, -0.05555555555555555) + -1.0))) * 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.log1p((Math.pow(u1, -0.05555555555555555) + -1.0))) * Math.cos(((2.0 * Math.PI) * u2))) + 0.5;
}

Python

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 (math.sqrt(math.log1p((math.pow(u1, -0.05555555555555555) + -1.0))) * 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(log1p(Float64((u1 ^ -0.05555555555555555) + -1.0))) * 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[Log[1 + N[(N[Power[u1, -0.05555555555555555], $MachinePrecision] + -1.0), $MachinePrecision]], $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 32.4

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

3. Simplified 0.2

   \[\leadsto \color{blue}{\sqrt{-0.05555555555555555 \cdot \log u1}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
   Proof
   (sqrt.f64 (*.f64 -1/18 (log.f64 u1))): 0 points increase in error, 0 points decrease in error
   (sqrt.f64 (*.f64 (Rewrite<= metadata-eval (*.f64 1/36 -2)) (log.f64 u1))): 0 points increase in error, 0 points decrease in error
   (sqrt.f64 (Rewrite<= associate-*r*_binary64 (*.f64 1/36 (*.f64 -2 (log.f64 u1))))): 0 points increase in error, 0 points decrease in error
   (sqrt.f64 (*.f64 1/36 (Rewrite<= log-pow_binary64 (log.f64 (pow.f64 u1 -2))))): 140 points increase in error, 0 points decrease in error
   (sqrt.f64 (Rewrite<= *-commutative_binary64 (*.f64 (log.f64 (pow.f64 u1 -2)) 1/36))): 0 points increase in error, 0 points decrease in error

4. Applied egg-rr 0.2

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

5. Final simplification 0.2

   \[\leadsto \sqrt{\mathsf{log1p}\left({u1}^{-0.05555555555555555} + -1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

Alternatives

Alternative 1
Error	0.2
Cost	32576

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

Alternative 2
Error	0.2
Cost	26240

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

Alternative 3
Error	64.0
Cost	19520

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

Reproduce

herbie shell --seed 2022329 
(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))