Examples.Basics.BasicTests:f2 from sbv-4.4

Percentage Accurate: 93.5% → 100.0%

Time: 3.1s

Alternatives: 3

Speedup: 1.2×

• 44.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot x - y \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (* x x) (* y y)))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * x) - (y * y)
end function

Java

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

Python

def code(x, y):
	return (x * x) - (y * y)

Julia

function code(x, y)
	return Float64(Float64(x * x) - Float64(y * y))
end

MATLAB

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

Wolfram

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

Initial Program: 93.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot x - y \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (* x x) (* y y)))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * x) - (y * y)
end function

Java

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

Python

def code(x, y):
	return (x * x) - (y * y)

Julia

function code(x, y)
	return Float64(Float64(x * x) - Float64(y * y))
end

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(x - y\right) \cdot \left(y + x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (- x y) (+ y x)))

C

double code(double x, double y) {
	return (x - y) * (y + x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x - y) * (y + x)
end function

Java

public static double code(double x, double y) {
	return (x - y) * (y + x);
}

Python

def code(x, y):
	return (x - y) * (y + x)

Julia

function code(x, y)
	return Float64(Float64(x - y) * Float64(y + x))
end

MATLAB

function tmp = code(x, y)
	tmp = (x - y) * (y + x);
end

Wolfram

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

Derivation

1. Initial program 93.8%

   \[x \cdot x - y \cdot y \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{x \cdot x - y \cdot y} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot x} - y \cdot y \]

   3. lift-*.f64 N/A

      \[\leadsto x \cdot x - \color{blue}{y \cdot y} \]

   4. difference-of-squares N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower--.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 100.0

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\left(x - y\right) \cdot \left(y + x\right)} \]
5. Add Preprocessing

Alternative 2: 96.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot x - y \cdot y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-310} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(-y\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (* x x) (* y y))))
   (if (or (<= t_0 -5e-310) (not (<= t_0 INFINITY))) (* (- y) y) (* x x))))

C

double code(double x, double y) {
	double t_0 = (x * x) - (y * y);
	double tmp;
	if ((t_0 <= -5e-310) || !(t_0 <= ((double) INFINITY))) {
		tmp = -y * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = (x * x) - (y * y);
	double tmp;
	if ((t_0 <= -5e-310) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = -y * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * x) - (y * y)
	tmp = 0
	if (t_0 <= -5e-310) or not (t_0 <= math.inf):
		tmp = -y * y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * x) - Float64(y * y))
	tmp = 0.0
	if ((t_0 <= -5e-310) || !(t_0 <= Inf))
		tmp = Float64(Float64(-y) * y);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * x) - (y * y);
	tmp = 0.0;
	if ((t_0 <= -5e-310) || ~((t_0 <= Inf)))
		tmp = -y * y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e-310], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[((-y) * y), $MachinePrecision], N[(x * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x x) (*.f64 y y)) < -4.999999999999985e-310 or +inf.0 < (-.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 88.3%

      \[x \cdot x - y \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot {y}^{2}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({y}^{2}\right)} \]

      2. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{y \cdot y}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot y} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot y} \]

      5. lower-neg.f64 96.3

         \[\leadsto \color{blue}{\left(-y\right)} \cdot y \]

   5. Applied rewrites 96.3%

      \[\leadsto \color{blue}{\left(-y\right) \cdot y} \]

   if -4.999999999999985e-310 < (-.f64 (*.f64 x x) (*.f64 y y)) < +inf.0

   1. Initial program 100.0%

      \[x \cdot x - y \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot {y}^{2}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({y}^{2}\right)} \]

      2. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{y \cdot y}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot y} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot y} \]

      5. lower-neg.f64 10.1

         \[\leadsto \color{blue}{\left(-y\right)} \cdot y \]

   5. Applied rewrites 10.1%

      \[\leadsto \color{blue}{\left(-y\right) \cdot y} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \color{blue}{x \cdot x} \]

      2. lower-*.f64 99.8

         \[\leadsto \color{blue}{x \cdot x} \]

   8. Applied rewrites 99.8%

      \[\leadsto \color{blue}{x \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x - y \cdot y \leq -5 \cdot 10^{-310} \lor \neg \left(x \cdot x - y \cdot y \leq \infty\right):\\ \;\;\;\;\left(-y\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ x \cdot x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x x))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x * x

Julia

function code(x, y)
	return Float64(x * x)
end

MATLAB

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

Wolfram

code[x_, y_] := N[(x * x), $MachinePrecision]

Derivation

1. Initial program 93.8%

   \[x \cdot x - y \cdot y \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{-1 \cdot {y}^{2}} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left({y}^{2}\right)} \]

   2. unpow2 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{y \cdot y}\right) \]

   3. distribute-lft-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot y} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot y} \]

   5. lower-neg.f64 56.2

      \[\leadsto \color{blue}{\left(-y\right)} \cdot y \]

5. Applied rewrites 56.2%

   \[\leadsto \color{blue}{\left(-y\right) \cdot y} \]

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{{x}^{2}} \]
7. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{x \cdot x} \]

   2. lower-*.f64 48.9

      \[\leadsto \color{blue}{x \cdot x} \]

8. Applied rewrites 48.9%

   \[\leadsto \color{blue}{x \cdot x} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024320 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f2 from sbv-4.4"
  :precision binary64
  (- (* x x) (* y y)))