Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Average Error: 0.02% → 0%

Time: 4.4s

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 0.02

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

2. Taylor expanded in x around -inf 0.02

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

3. Simplified 0.02

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

   [Start] 0.02	\[ \frac{\left|-\left(y + -1 \cdot x\right)\right|}{\left|y\right|} \]
   mul-1-neg [=>] 0.02	\[ \frac{\left|-\left(y + \color{blue}{\left(-x\right)}\right)\right|}{\left|y\right|} \]
   sub-neg [<=] 0.02	\[ \frac{\left|-\color{blue}{\left(y - x\right)}\right|}{\left|y\right|} \]
   fabs-neg [=>] 0.02	\[ \frac{\color{blue}{\left|y - x\right|}}{\left|y\right|} \]
   fabs-div [<=] 0.02	\[ \color{blue}{\left|\frac{y - x}{y}\right|} \]

4. Taylor expanded in y around 0 0

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

5. Simplified 0

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

   [Start] 0	\[ \left|1 + -1 \cdot \frac{x}{y}\right| \]
   mul-1-neg [=>] 0	\[ \left|1 + \color{blue}{\left(-\frac{x}{y}\right)}\right| \]
   sub-neg [<=] 0	\[ \left|\color{blue}{1 - \frac{x}{y}}\right| \]

6. Final simplification 0

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

Alternatives

Alternative 1
Error	30.71%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;x \leq -360000000000 \lor \neg \left(x \leq 3.6 \cdot 10^{+121}\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x + y}\\ \end{array} \]

Alternative 2
Error	42.75%
Cost	1364

\[\begin{array}{l} t_0 := \frac{y}{x + y}\\ \mathbf{if}\;y \leq -1.2 \cdot 10^{+68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{-62}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq -1.75 \cdot 10^{-169}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-202}:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{x + y}\\ \mathbf{elif}\;y \leq -7.5 \cdot 10^{-234}:\\ \;\;\;\;y \cdot \frac{\frac{1}{y}}{\frac{x + y}{y}}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-193}:\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	42.54%
Cost	1364

\[\begin{array}{l} t_0 := \frac{y}{x + y}\\ \mathbf{if}\;y \leq -1.2 \cdot 10^{+68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{-62}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-169}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -7 \cdot 10^{-202}:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{x + y}\\ \mathbf{elif}\;y \leq -7.5 \cdot 10^{-234}:\\ \;\;\;\;y \cdot \frac{\frac{1}{y}}{\frac{x + y}{y}}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-192}:\\ \;\;\;\;\frac{x}{y} + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Error	42.61%
Cost	1114

\[\begin{array}{l} t_0 := \frac{y}{x + y}\\ \mathbf{if}\;y \leq -1.2 \cdot 10^{+68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{-62}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{-170} \lor \neg \left(y \leq -8.5 \cdot 10^{-202}\right) \land \left(y \leq -7.2 \cdot 10^{-234} \lor \neg \left(y \leq 8.6 \cdot 10^{-193}\right)\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -1\\ \end{array} \]

Alternative 5
Error	42.84%
Cost	1113

\[\begin{array}{l} t_0 := \frac{y}{x + y}\\ \mathbf{if}\;y \leq -1.2 \cdot 10^{+68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{-62}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq -3 \cdot 10^{-169}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -8.1 \cdot 10^{-202}:\\ \;\;\;\;\frac{x \cdot \frac{x}{y}}{x + y}\\ \mathbf{elif}\;y \leq -6.2 \cdot 10^{-234} \lor \neg \left(y \leq 2.15 \cdot 10^{-194}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -1\\ \end{array} \]

Alternative 6
Error	77.1%
Cost	192

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

Alternative 7
Error	98.62%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023088 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))