Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Average Error: 0.0 → 0.0

Time: 1.3s

Precision: binary64

Cost: 6720

Math

\[\frac{\left|x - y\right|}{\left|y\right|} \]

↓

\[\left|\frac{x}{y} - 1\right| \]

FPCore

(FPCore (x y) :precision binary64 (/ (fabs (- x y)) (fabs y)))

↓

(FPCore (x y) :precision binary64 (fabs (- (/ x y) 1.0)))

C

double code(double x, double y) {
	return fabs((x - y)) / fabs(y);
}

↓

double code(double x, double y) {
	return fabs(((x / y) - 1.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = abs((x - y)) / abs(y)
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = abs(((x / y) - 1.0d0))
end function

Java

public static double code(double x, double y) {
	return Math.abs((x - y)) / Math.abs(y);
}

↓

public static double code(double x, double y) {
	return Math.abs(((x / y) - 1.0));
}

Python

def code(x, y):
	return math.fabs((x - y)) / math.fabs(y)

↓

def code(x, y):
	return math.fabs(((x / y) - 1.0))

Julia

function code(x, y)
	return Float64(abs(Float64(x - y)) / abs(y))
end

↓

function code(x, y)
	return abs(Float64(Float64(x / y) - 1.0))
end

MATLAB

function tmp = code(x, y)
	tmp = abs((x - y)) / abs(y);
end

↓

function tmp = code(x, y)
	tmp = abs(((x / y) - 1.0));
end

Wolfram

code[x_, y_] := N[(N[Abs[N[(x - y), $MachinePrecision]], $MachinePrecision] / N[Abs[y], $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := N[Abs[N[(N[(x / y), $MachinePrecision] - 1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 0.0

   \[\frac{\left|x - y\right|}{\left|y\right|} \]

2. Simplified 0.0

   \[\leadsto \color{blue}{\left|\frac{x - y}{y}\right|} \]
   Proof

   [Start] 0.0	\[ \frac{\left|x - y\right|}{\left|y\right|} \]
   rational.json-simplify-40 [=>] 0.0	\[ \color{blue}{\left|\frac{x - y}{y}\right|} \]

3. Taylor expanded in x around 0 0.0

   \[\leadsto \left|\color{blue}{\frac{x}{y} - 1}\right| \]

4. Final simplification 0.0

   \[\leadsto \left|\frac{x}{y} - 1\right| \]

Alternatives

Alternative 1
Error	17.9
Cost	6856

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{+76}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{+29}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	26.6
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023074 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))