Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Average Accuracy: 100.0% → 100.0%

Time: 5.1s

Precision: binary64

Cost: 6720

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

function tmp = code(x, y)
	tmp = abs(((y - 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[(N[(y - x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around -inf 100.0%

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

3. Simplified 100.0%

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

   [Start] 100.0	\[ \frac{\left|-\left(y + -1 \cdot x\right)\right|}{\left|y\right|} \]
   mul-1-neg [=>] 100.0	\[ \frac{\left|-\left(y + \color{blue}{\left(-x\right)}\right)\right|}{\left|y\right|} \]
   sub-neg [<=] 100.0	\[ \frac{\left|-\color{blue}{\left(y - x\right)}\right|}{\left|y\right|} \]
   fabs-neg [=>] 100.0	\[ \frac{\color{blue}{\left|y - x\right|}}{\left|y\right|} \]
   fabs-div [<=] 100.0	\[ \color{blue}{\left|\frac{y - x}{y}\right|} \]

4. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	69.7%
Cost	7387

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-51} \lor \neg \left(x \leq -1.7 \cdot 10^{-122} \lor \neg \left(x \leq -1.4 \cdot 10^{-149}\right) \land \left(x \leq 1.8 \cdot 10^{-13} \lor \neg \left(x \leq 1.65 \cdot 10^{+52}\right) \land x \leq 6 \cdot 10^{+112}\right)\right):\\ \;\;\;\;\left|\frac{x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2
Accuracy	100.0%
Cost	6720

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

Alternative 3
Accuracy	59.8%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-182} \lor \neg \left(y \leq 14\right):\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -1\\ \end{array} \]

Alternative 4
Accuracy	59.8%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{-181} \lor \neg \left(y \leq 98\right):\\ \;\;\;\;\frac{y}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y}\\ \end{array} \]

Alternative 5
Accuracy	60.1%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-181}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 14:\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Accuracy	22.8%
Cost	320

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

Alternative 7
Accuracy	23.4%
Cost	192

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

Alternative 8
Accuracy	1.4%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023140 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))