Average Error: 0.0 → 0.0
Time: 3.9s
Precision: binary64
Cost: 6720
\[\frac{\left|x - y\right|}{\left|y\right|} \]
↓
\[\left|1 - \frac{x}{y}\right| \]
(FPCore (x y) :precision binary64 (/ (fabs (- x y)) (fabs y)))
↓
(FPCore (x y) :precision binary64 (fabs (- 1.0 (/ x y))))
double code(double x, double y) {
	return fabs((x - y)) / fabs(y);
}
↓
double code(double x, double y) {
	return fabs((1.0 - (x / y)));
}

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

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)));
}

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

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

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

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]

\frac{\left|x - y\right|}{\left|y\right|}
↓
\left|1 - \frac{x}{y}\right|

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

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

  2. Applied egg-rr0.0

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

  3. Taylor expanded in y around 0 0.0

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

  4. Simplified0.0

    \[\leadsto \left|\color{blue}{1 - \frac{x}{y}}\right| \]
    Proof
    (-.f64 1 (/.f64 x y)): 0 points increase in error, 0 points decrease in error
    (Rewrite<= unsub-neg_binary64 (+.f64 1 (neg.f64 (/.f64 x y)))): 0 points increase in error, 0 points decrease in error
    (+.f64 1 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 x y)))): 0 points increase in error, 0 points decrease in error

  5. Final simplification0.0

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

Alternatives

Alternative 1
Error18.2
Cost6856
\[\begin{array}{l} \mathbf{if}\;y \leq -2.6648091360231168 \cdot 10^{-67}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 7.533623143703852 \cdot 10^{-28}:\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
Alternative 2
Error25.1
Cost840
\[\begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+126}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 6.0376406963748116 \cdot 10^{+50}:\\ \;\;\;\;\left(1 + \frac{y}{x + y}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
Alternative 3
Error25.4
Cost712
\[\begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+126}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 6.0376406963748116 \cdot 10^{+50}:\\ \;\;\;\;y \cdot \frac{1}{y - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
Alternative 4
Error25.2
Cost712
\[\begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+126}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 6.0376406963748116 \cdot 10^{+50}:\\ \;\;\;\;y \cdot \frac{1}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
Alternative 5
Error25.1
Cost584
\[\begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+126}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 6.0376406963748116 \cdot 10^{+50}:\\ \;\;\;\;\frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
Alternative 6
Error24.8
Cost456
\[\begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+140}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;x \leq 6.0376406963748116 \cdot 10^{+50}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
Alternative 7
Error49.3
Cost192
\[\frac{x}{y} \]
Alternative 8
Error63.1
Cost64
\[-1 \]

Error

Reproduce

herbie shell --seed 2022300 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))