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. Taylor expanded in x around -inf 0.0

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

3. Simplified 0.0

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

   [Start] 0.0	\[ \frac{\left|-\left(y + -1 \cdot x\right)\right|}{\left|y\right|} \]
   mul-1-neg [=>] 0.0	\[ \frac{\left|-\left(y + \color{blue}{\left(-x\right)}\right)\right|}{\left|y\right|} \]
   sub-neg [<=] 0.0	\[ \frac{\left|-\color{blue}{\left(y - x\right)}\right|}{\left|y\right|} \]
   fabs-neg [=>] 0.0	\[ \frac{\color{blue}{\left|y - x\right|}}{\left|y\right|} \]
   fabs-div [<=] 0.0	\[ \color{blue}{\left|\frac{y - x}{y}\right|} \]
   div-sub [=>] 0.0	\[ \left|\color{blue}{\frac{y}{y} - \frac{x}{y}}\right| \]
   *-inverses [=>] 0.0	\[ \left|\color{blue}{1} - \frac{x}{y}\right| \]

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	49.4
Cost	320

\[\frac{x}{y} + -1 \]

Reproduce

herbie shell --seed 2023031 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))