Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%
Time: 830.0ms
Alternatives: 3
Speedup: 1.1×

Specification

?
\[\frac{\left|x - y\right|}{\left|y\right|} \]
(FPCore (x y)
  :precision binary64
  :pre TRUE
  (/ (fabs (- x y)) (fabs y)))
double code(double x, double y) {
	return fabs((x - y)) / fabs(y);
}

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = abs((x - y)) / abs(y)
end function

public static double code(double x, double y) {
	return Math.abs((x - y)) / Math.abs(y);
}

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

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

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

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

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	(abs((x - y))) / (abs(y))
END code
\frac{\left|x - y\right|}{\left|y\right|}

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 3 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 100.0% accurate, 1.0× speedup?

\[\frac{\left|x - y\right|}{\left|y\right|} \]
(FPCore (x y)
  :precision binary64
  :pre TRUE
  (/ (fabs (- x y)) (fabs y)))
double code(double x, double y) {
	return fabs((x - y)) / fabs(y);
}

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = abs((x - y)) / abs(y)
end function

public static double code(double x, double y) {
	return Math.abs((x - y)) / Math.abs(y);
}

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

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

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

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

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	(abs((x - y))) / (abs(y))
END code
\frac{\left|x - y\right|}{\left|y\right|}

Alternative 1: 100.0% accurate, 1.1× speedup?

\[\left|-1 + \frac{x}{y}\right| \]
(FPCore (x y)
  :precision binary64
  :pre TRUE
  (fabs (+ -1.0 (/ x y))))
double code(double x, double y) {
	return fabs((-1.0 + (x / y)));
}

real(8) function code(x, y)
use fmin_fmax_functions
    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((-1.0 + (x / y)));
}

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

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

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

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

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	abs(((-1) + (x / y)))
END code
\left|-1 + \frac{x}{y}\right|
Derivation

  1. Initial program 100.0%

    \[\frac{\left|x - y\right|}{\left|y\right|} \]
  2. Step-by-step derivation

    1. Applied rewrites100.0%

      \[\leadsto \left|\frac{x - y}{y}\right| \]
    2. Step-by-step derivation

      1. Applied rewrites100.0%

        \[\leadsto \left|-1 + \frac{x}{y}\right| \]
      2. Add Preprocessing

      Alternative 2: 100.0% accurate, 1.1× speedup?

      \[\left|\frac{x - y}{y}\right| \]
      (FPCore (x y)
  :precision binary64
  :pre TRUE
  (fabs (/ (- x y) y)))
      double code(double x, double y) {
	return fabs(((x - y) / y));
}

      real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = abs(((x - y) / y))
end function

      public static double code(double x, double y) {
	return Math.abs(((x - y) / y));
}

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

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

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

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

      f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	abs(((x - y) / y))
END code
      \left|\frac{x - y}{y}\right|
      Derivation

      1. Initial program 100.0%

        \[\frac{\left|x - y\right|}{\left|y\right|} \]
      2. Step-by-step derivation

        1. Applied rewrites100.0%

          \[\leadsto \left|\frac{x - y}{y}\right| \]
        2. Add Preprocessing

        Alternative 3: 52.6% accurate, 9.4× speedup?

        \[1 \]
        (FPCore (x y)
  :precision binary64
  :pre TRUE
  1.0)
        double code(double x, double y) {
	return 1.0;
}

        real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

        public static double code(double x, double y) {
	return 1.0;
}

        def code(x, y):
	return 1.0

        function code(x, y)
	return 1.0
end

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

        code[x_, y_] := 1.0

        f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	1
END code
        1
        Derivation

        1. Initial program 100.0%

          \[\frac{\left|x - y\right|}{\left|y\right|} \]
        2. Step-by-step derivation

          1. Applied rewrites100.0%

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

          2. Taylor expanded in x around 0

            \[\leadsto 1 \]
          3. Step-by-step derivation

            1. Applied rewrites52.6%

              \[\leadsto 1 \]
            2. Add Preprocessing

            Reproduce

            ?
            herbie shell --seed 2026092 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))