Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5

Percentage Accurate: 100.0% → 100.0%

Time: 2.4s

Alternatives: 2

Speedup: 2.0×

Specification

Math

\[\begin{array}{l} \\ \frac{\left|x - y\right|}{\left|y\right|} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left|x - y\right|}{\left|y\right|} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left|1 - \frac{x}{y}\right| \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fabs (- 1.0 (/ x y))))

C

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((1.0d0 - (x / y)))
end function

Java

public static double code(double x, double y) {
	return Math.abs((1.0 - (x / y)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. neg-fabs N/A

      \[\leadsto \frac{\left|\mathsf{neg}\left(\left(x - y\right)\right)\right|}{\left|\color{blue}{y}\right|} \]

   2. div-fabs N/A

      \[\leadsto \left|\frac{\mathsf{neg}\left(\left(x - y\right)\right)}{y}\right| \]

   3. fabs-lowering-fabs.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\mathsf{neg}\left(\left(x - y\right)\right)}{y}\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\mathsf{neg}\left(\left(x + \left(\mathsf{neg}\left(y\right)\right)\right)\right)}{y}\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(y\right)\right) + x\right)\right)}{y}\right)\right) \]

   6. distribute-neg-in N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) + \left(\mathsf{neg}\left(x\right)\right)}{y}\right)\right) \]

   7. remove-double-neg N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{y + \left(\mathsf{neg}\left(x\right)\right)}{y}\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{y - x}{y}\right)\right) \]

   9. div-sub N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{y}{y} - \frac{x}{y}\right)\right) \]

   10. *-inverses N/A

       \[\leadsto \mathsf{fabs.f64}\left(\left(1 - \frac{x}{y}\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{x}{y}\right)\right)\right) \]

   12. /-lowering-/.f64 100.0%

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(x, y\right)\right)\right) \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left|1 - \frac{x}{y}\right|} \]
5. Add Preprocessing

Alternative 2: 51.2% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

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

   1. neg-fabs N/A

      \[\leadsto \frac{\left|\mathsf{neg}\left(\left(x - y\right)\right)\right|}{\left|\color{blue}{y}\right|} \]

   2. div-fabs N/A

      \[\leadsto \left|\frac{\mathsf{neg}\left(\left(x - y\right)\right)}{y}\right| \]

   3. fabs-lowering-fabs.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\mathsf{neg}\left(\left(x - y\right)\right)}{y}\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\mathsf{neg}\left(\left(x + \left(\mathsf{neg}\left(y\right)\right)\right)\right)}{y}\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(y\right)\right) + x\right)\right)}{y}\right)\right) \]

   6. distribute-neg-in N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) + \left(\mathsf{neg}\left(x\right)\right)}{y}\right)\right) \]

   7. remove-double-neg N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{y + \left(\mathsf{neg}\left(x\right)\right)}{y}\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{y - x}{y}\right)\right) \]

   9. div-sub N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{y}{y} - \frac{x}{y}\right)\right) \]

   10. *-inverses N/A

       \[\leadsto \mathsf{fabs.f64}\left(\left(1 - \frac{x}{y}\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{x}{y}\right)\right)\right) \]

   12. /-lowering-/.f64 100.0%

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(x, y\right)\right)\right) \]

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{1}\right) \]
6. Step-by-step derivation

7. Simplified 47.1%

   \[\leadsto \left|\color{blue}{1}\right| \]
8. Step-by-step derivation

   1. metadata-eval 47.1%

      \[\leadsto 1 \]

9. Applied egg-rr 47.1%

   \[\leadsto \color{blue}{1} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(FPCore (x y)
  :name "Numeric.LinearAlgebra.Util:formatSparse from hmatrix-0.16.1.5"
  :precision binary64
  (/ (fabs (- x y)) (fabs y)))