normal distribution

Average Error: 0.4 → 0.2

Time: 7.9s

Precision: binary64

Cost: 26240

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

Math

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

↓

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

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
 (+ (* (cos (* (* 2.0 PI) u2)) (sqrt (* -0.05555555555555555 (log u1)))) 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 (cos(((2.0 * ((double) M_PI)) * u2)) * sqrt((-0.05555555555555555 * log(u1)))) + 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.cos(((2.0 * Math.PI) * u2)) * Math.sqrt((-0.05555555555555555 * Math.log(u1)))) + 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.cos(((2.0 * math.pi) * u2)) * math.sqrt((-0.05555555555555555 * math.log(u1)))) + 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(cos(Float64(Float64(2.0 * pi) * u2)) * sqrt(Float64(-0.05555555555555555 * log(u1)))) + 0.5)
end

MATLAB

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 = (cos(((2.0 * pi) * u2)) * sqrt((-0.05555555555555555 * log(u1)))) + 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[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * u2), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(-0.05555555555555555 * N[Log[u1], $MachinePrecision]), $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 31.9

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

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))))): 124 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. Final simplification 0.2

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

Alternatives

Alternative 1
Error	1.1
Cost	13120

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

Alternative 2
Error	53.6
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022328 
(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))