Octave 3.8, oct_fill_randg

Average Error: 0.1 → 0.1

Time: 4.4s

Precision: binary64

Math

\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]

↓

\[\left(a - 0.3333333333333333\right) \cdot \left(1 + {\left(\left(a - 0.3333333333333333\right) \cdot 9\right)}^{-0.5} \cdot rand\right) \]

FPCore

(FPCore (a rand)
 :precision binary64
 (*
  (- a (/ 1.0 3.0))
  (+ 1.0 (* (/ 1.0 (sqrt (* 9.0 (- a (/ 1.0 3.0))))) rand))))

↓

(FPCore (a rand)
 :precision binary64
 (*
  (- a 0.3333333333333333)
  (+ 1.0 (* (pow (* (- a 0.3333333333333333) 9.0) -0.5) rand))))

C

double code(double a, double rand) {
	return (a - (1.0 / 3.0)) * (1.0 + ((1.0 / sqrt((9.0 * (a - (1.0 / 3.0))))) * rand));
}

↓

double code(double a, double rand) {
	return (a - 0.3333333333333333) * (1.0 + (pow(((a - 0.3333333333333333) * 9.0), -0.5) * rand));
}

Fortran

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    code = (a - (1.0d0 / 3.0d0)) * (1.0d0 + ((1.0d0 / sqrt((9.0d0 * (a - (1.0d0 / 3.0d0))))) * rand))
end function

↓

real(8) function code(a, rand)
    real(8), intent (in) :: a
    real(8), intent (in) :: rand
    code = (a - 0.3333333333333333d0) * (1.0d0 + ((((a - 0.3333333333333333d0) * 9.0d0) ** (-0.5d0)) * rand))
end function

Java

public static double code(double a, double rand) {
	return (a - (1.0 / 3.0)) * (1.0 + ((1.0 / Math.sqrt((9.0 * (a - (1.0 / 3.0))))) * rand));
}

↓

public static double code(double a, double rand) {
	return (a - 0.3333333333333333) * (1.0 + (Math.pow(((a - 0.3333333333333333) * 9.0), -0.5) * rand));
}

Python

def code(a, rand):
	return (a - (1.0 / 3.0)) * (1.0 + ((1.0 / math.sqrt((9.0 * (a - (1.0 / 3.0))))) * rand))

↓

def code(a, rand):
	return (a - 0.3333333333333333) * (1.0 + (math.pow(((a - 0.3333333333333333) * 9.0), -0.5) * rand))

Julia

function code(a, rand)
	return Float64(Float64(a - Float64(1.0 / 3.0)) * Float64(1.0 + Float64(Float64(1.0 / sqrt(Float64(9.0 * Float64(a - Float64(1.0 / 3.0))))) * rand)))
end

↓

function code(a, rand)
	return Float64(Float64(a - 0.3333333333333333) * Float64(1.0 + Float64((Float64(Float64(a - 0.3333333333333333) * 9.0) ^ -0.5) * rand)))
end

MATLAB

function tmp = code(a, rand)
	tmp = (a - (1.0 / 3.0)) * (1.0 + ((1.0 / sqrt((9.0 * (a - (1.0 / 3.0))))) * rand));
end

↓

function tmp = code(a, rand)
	tmp = (a - 0.3333333333333333) * (1.0 + ((((a - 0.3333333333333333) * 9.0) ^ -0.5) * rand));
end

Wolfram

code[a_, rand_] := N[(N[(a - N[(1.0 / 3.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(1.0 / N[Sqrt[N[(9.0 * N[(a - N[(1.0 / 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * rand), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[a_, rand_] := N[(N[(a - 0.3333333333333333), $MachinePrecision] * N[(1.0 + N[(N[Power[N[(N[(a - 0.3333333333333333), $MachinePrecision] * 9.0), $MachinePrecision], -0.5], $MachinePrecision] * rand), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Error

Bits error versus a

Bits error versus rand

Derivation

1. Initial program 0.1

   \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \]

2. Applied pow1/2_binary64 0.1

   \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\color{blue}{{\left(9 \cdot \left(a - \frac{1}{3}\right)\right)}^{0.5}}} \cdot rand\right) \]

3. Applied pow-flip_binary64 0.1

   \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + \color{blue}{{\left(9 \cdot \left(a - \frac{1}{3}\right)\right)}^{\left(-0.5\right)}} \cdot rand\right) \]

4. Simplified 0.1

   \[\leadsto \left(a - \frac{1}{3}\right) \cdot \left(1 + {\left(9 \cdot \left(a - \frac{1}{3}\right)\right)}^{\color{blue}{-0.5}} \cdot rand\right) \]

5. Final simplification 0.1

   \[\leadsto \left(a - 0.3333333333333333\right) \cdot \left(1 + {\left(\left(a - 0.3333333333333333\right) \cdot 9\right)}^{-0.5} \cdot rand\right) \]

Reproduce

herbie shell --seed 2022137 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  :precision binary64
  (* (- a (/ 1.0 3.0)) (+ 1.0 (* (/ 1.0 (sqrt (* 9.0 (- a (/ 1.0 3.0))))) rand))))