Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Average Error: 0.0 → 0.0

Time: 5.1s

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. Applied egg-rr 0.0

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

3. Taylor expanded in y around 0 0.0

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

4. Simplified 0.0

   \[\leadsto \left|\color{blue}{1 - \frac{x}{y}}\right| \]
   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 simplification 0.0

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

Alternatives

Alternative 1
Error	17.7
Cost	7516

\[\begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;x \leq -1.1329364809276365 \cdot 10^{+34}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.5555939960240409 \cdot 10^{-25}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -1.615185901717486 \cdot 10^{-41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.427075381726709 \cdot 10^{-154}:\\ \;\;\;\;\left(1 + \frac{y}{x + y}\right) + -1\\ \mathbf{elif}\;x \leq 9.712446762991152 \cdot 10^{-68}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.628969437061893 \cdot 10^{-47}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.0303605066907297 \cdot 10^{+44}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	25.4
Cost	976

\[\begin{array}{l} t_0 := y \cdot \frac{1}{y - x}\\ t_1 := \frac{x}{y} + -1\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{-233}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.716480590929634 \cdot 10^{-31}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.8135517851873296 \cdot 10^{-16}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	24.9
Cost	848

\[\begin{array}{l} t_0 := \frac{y}{x + y}\\ t_1 := \frac{x}{y} + -1\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{-233}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.716480590929634 \cdot 10^{-31}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.8135517851873296 \cdot 10^{-16}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	24.7
Cost	848

\[\begin{array}{l} t_0 := \frac{x}{y} + -1\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{-233}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-158}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.716480590929634 \cdot 10^{-31}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.8135517851873296 \cdot 10^{-16}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	49.2
Cost	192

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

Alternative 6
Error	63.1
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022317 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))