Examples.Basics.BasicTests:f2 from sbv-4.4

Percentage Accurate: 94.2% → 100.0%

Time: 6.2s

Alternatives: 3

Speedup: 1.0×

• 43.1% 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: 94.2% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.9%

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

   1. add-sqr-sqrt 49.1%

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

   2. associate-*l* 49.1%

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

   3. prod-diff 40.1%

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

4. Applied egg-rr 40.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, -\left(\sqrt{y} \cdot y\right) \cdot \sqrt{y}\right) + \mathsf{fma}\left(-\sqrt{y} \cdot y, \sqrt{y}, \left(\sqrt{y} \cdot y\right) \cdot \sqrt{y}\right)} \]
5. Step-by-step derivation

   1. prod-diff 49.1%

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

   2. associate-*r* 49.1%

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

   3. add-sqr-sqrt 94.9%

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

   4. difference-of-squares 100.0%

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

   5. +-commutative 100.0%

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

6. Applied egg-rr 100.0%

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

Alternative 2: 76.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-161}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* x x) 5e-161) (* y (- y)) (* x x)))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 5e-161) {
		tmp = y * -y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x * x) <= 5d-161) then
        tmp = y * -y
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x * x) <= 5e-161) {
		tmp = y * -y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 5e-161:
		tmp = y * -y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 5e-161)
		tmp = Float64(y * Float64(-y));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * x) <= 5e-161)
		tmp = y * -y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-161], N[(y * (-y)), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.9999999999999999e-161

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 93.0%

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

      1. neg-mul-1 93.0%

         \[\leadsto \color{blue}{-{y}^{2}} \]

   5. Simplified 93.0%

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

      1. unpow2 93.0%

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

      2. distribute-lft-neg-in 93.0%

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

   7. Applied egg-rr 93.0%

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

   if 4.9999999999999999e-161 < (*.f64 x x)

   1. Initial program 92.1%

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

      1. difference-of-squares 100.0%

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

      2. sub-neg 100.0%

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

      3. add-sqr-sqrt 45.4%

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

      4. sqrt-unprod 83.1%

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

      5. sqr-neg 83.1%

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

      6. sqrt-prod 40.0%

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

      7. add-sqr-sqrt 77.1%

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

   4. Applied egg-rr 77.1%

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

   5. Taylor expanded in x around inf 80.5%

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

   6. Taylor expanded in x around inf 77.6%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-161}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.6% 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 94.9%

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

   1. difference-of-squares 100.0%

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

   2. sub-neg 100.0%

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

   3. add-sqr-sqrt 47.6%

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

   4. sqrt-unprod 74.5%

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

   5. sqr-neg 74.5%

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

   6. sqrt-prod 28.4%

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

   7. add-sqr-sqrt 54.5%

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

4. Applied egg-rr 54.5%

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

5. Taylor expanded in x around inf 57.3%

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

6. Taylor expanded in x around inf 55.3%

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

Reproduce

herbie shell --seed 2024170 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f2 from sbv-4.4"
  :precision binary64
  (- (* x x) (* y y)))