Examples.Basics.BasicTests:f2 from sbv-4.4

Percentage Accurate: 93.0% → 98.2%

Time: 5.6s

Alternatives: 4

Speedup: 0.5×

• 43.7% 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: 93.0% 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: 98.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2 \cdot 10^{+251}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m, -y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (if (<= x_m 2e+251) (fma x_m x_m (- (* y y))) (* x_m x_m)))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double tmp;
	if (x_m <= 2e+251) {
		tmp = fma(x_m, x_m, -(y * y));
	} else {
		tmp = x_m * x_m;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y)
	tmp = 0.0
	if (x_m <= 2e+251)
		tmp = fma(x_m, x_m, Float64(-Float64(y * y)));
	else
		tmp = Float64(x_m * x_m);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := If[LessEqual[x$95$m, 2e+251], N[(x$95$m * x$95$m + (-N[(y * y), $MachinePrecision])), $MachinePrecision], N[(x$95$m * x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.0000000000000001e251

   1. Initial program 92.7%

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

      1. sqr-neg 92.7%

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

      2. cancel-sign-sub 92.7%

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

      3. fma-define 98.0%

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

   3. Simplified 98.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, y \cdot \left(-y\right)\right)} \]
   4. Add Preprocessing

   if 2.0000000000000001e251 < x

   1. Initial program 80.0%

      \[x \cdot x - y \cdot y \]
   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 40.0%

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

      4. sqrt-unprod 100.0%

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

      5. sqr-neg 100.0%

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

      6. sqrt-prod 60.0%

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

      7. add-sqr-sqrt 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+251}:\\ \;\;\;\;\mathsf{fma}\left(x, x, -y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq \infty:\\ \;\;\;\;x\_m \cdot x\_m - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;-y \cdot y\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (if (<= (* y y) INFINITY) (- (* x_m x_m) (* y y)) (- (* y y))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double tmp;
	if ((y * y) <= ((double) INFINITY)) {
		tmp = (x_m * x_m) - (y * y);
	} else {
		tmp = -(y * y);
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m, double y) {
	double tmp;
	if ((y * y) <= Double.POSITIVE_INFINITY) {
		tmp = (x_m * x_m) - (y * y);
	} else {
		tmp = -(y * y);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y):
	tmp = 0
	if (y * y) <= math.inf:
		tmp = (x_m * x_m) - (y * y)
	else:
		tmp = -(y * y)
	return tmp

Julia

x_m = abs(x)
function code(x_m, y)
	tmp = 0.0
	if (Float64(y * y) <= Inf)
		tmp = Float64(Float64(x_m * x_m) - Float64(y * y));
	else
		tmp = Float64(-Float64(y * y));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y)
	tmp = 0.0;
	if ((y * y) <= Inf)
		tmp = (x_m * x_m) - (y * y);
	else
		tmp = -(y * y);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], Infinity], N[(N[(x$95$m * x$95$m), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision], (-N[(y * y), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < +inf.0

   1. Initial program 92.2%

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

   if +inf.0 < (*.f64 y y)

   1. Initial program 92.2%

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

   3. Taylor expanded in x around 0 54.9%

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

      1. neg-mul-1 54.9%

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

   5. Simplified 54.9%

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

      1. unpow2 54.9%

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

      2. distribute-lft-neg-in 54.9%

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

   7. Applied egg-rr 54.9%

      \[\leadsto \color{blue}{\left(-y\right) \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq \infty:\\ \;\;\;\;x \cdot x - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;-y \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{-44}:\\ \;\;\;\;x\_m \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;-y \cdot y\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (if (<= (* y y) 1e-44) (* x_m x_m) (- (* y y))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double tmp;
	if ((y * y) <= 1e-44) {
		tmp = x_m * x_m;
	} else {
		tmp = -(y * y);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * y) <= 1d-44) then
        tmp = x_m * x_m
    else
        tmp = -(y * y)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y) {
	double tmp;
	if ((y * y) <= 1e-44) {
		tmp = x_m * x_m;
	} else {
		tmp = -(y * y);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y):
	tmp = 0
	if (y * y) <= 1e-44:
		tmp = x_m * x_m
	else:
		tmp = -(y * y)
	return tmp

Julia

x_m = abs(x)
function code(x_m, y)
	tmp = 0.0
	if (Float64(y * y) <= 1e-44)
		tmp = Float64(x_m * x_m);
	else
		tmp = Float64(-Float64(y * y));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y)
	tmp = 0.0;
	if ((y * y) <= 1e-44)
		tmp = x_m * x_m;
	else
		tmp = -(y * y);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 1e-44], N[(x$95$m * x$95$m), $MachinePrecision], (-N[(y * y), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 9.99999999999999953e-45

   1. Initial program 100.0%

      \[x \cdot x - y \cdot y \]
   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 39.1%

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

      4. sqrt-unprod 85.9%

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

      5. sqr-neg 85.9%

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

      6. sqrt-prod 46.8%

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

      7. add-sqr-sqrt 84.0%

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

   4. Applied egg-rr 84.0%

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

   5. Taylor expanded in x around inf 84.3%

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

   6. Taylor expanded in x around inf 84.2%

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

   if 9.99999999999999953e-45 < (*.f64 y y)

   1. Initial program 84.4%

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

   3. Taylor expanded in x around 0 78.3%

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

      1. neg-mul-1 78.3%

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

   5. Simplified 78.3%

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

      1. unpow2 78.3%

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

      2. distribute-lft-neg-in 78.3%

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

   7. Applied egg-rr 78.3%

      \[\leadsto \color{blue}{\left(-y\right) \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{-44}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;-y \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x\_m \cdot x\_m \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y) :precision binary64 (* x_m x_m))

C

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

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m, double y) {
	return x_m * x_m;
}

Python

x_m = math.fabs(x)
def code(x_m, y):
	return x_m * x_m

Julia

x_m = abs(x)
function code(x_m, y)
	return Float64(x_m * x_m)
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := N[(x$95$m * x$95$m), $MachinePrecision]

Derivation

1. Initial program 92.2%

   \[x \cdot x - y \cdot y \]
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 43.7%

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

   4. sqrt-unprod 72.2%

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

   5. sqr-neg 72.2%

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

   6. sqrt-prod 30.0%

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

   7. add-sqr-sqrt 52.6%

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

4. Applied egg-rr 52.6%

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

5. Taylor expanded in x around inf 57.1%

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

6. Taylor expanded in x around inf 53.1%

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

Reproduce

herbie shell --seed 2024116 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f2 from sbv-4.4"
  :precision binary64
  (- (* x x) (* y y)))