Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Average Error: 0.0 → 0.0

Time: 3.7s

Precision: binary64

Cost: 6720

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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((1.0d0 - (x / y)))
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((1.0 - (x / y)));
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

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

4. Taylor expanded in y around 0 0.0

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

5. Simplified 0.0

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

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

6. Final simplification 0.0

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

Alternatives

Alternative 1
Error	18.8
Cost	7121

\[\begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+46}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -12 \lor \neg \left(y \leq -2.8 \cdot 10^{-73}\right) \land y \leq 1450:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2
Error	25.2
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{-154}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-65}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{x + y}{x}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3
Error	48.9
Cost	576

\[x \cdot \frac{\frac{x}{x + y}}{y} \]

Alternative 4
Error	48.8
Cost	576

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

Alternative 5
Error	49.1
Cost	192

\[\frac{x}{y} \]

Alternative 6
Error	63.1
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023016 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))