Examples.Basics.BasicTests:f2 from sbv-4.4

Percentage Accurate: 93.7% → 98.9%

Time: 5.7s

Alternatives: 5

Speedup: 0.5×

• 43.5% 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: 93.7% 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: 98.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.35 \cdot 10^{+218}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m, y \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(x\_m + y\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (if (<= x_m 1.35e+218) (fma x_m x_m (* y (- y))) (* x_m (+ x_m y))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double tmp;
	if (x_m <= 1.35e+218) {
		tmp = fma(x_m, x_m, (y * -y));
	} else {
		tmp = x_m * (x_m + y);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y)
	tmp = 0.0
	if (x_m <= 1.35e+218)
		tmp = fma(x_m, x_m, Float64(y * Float64(-y)));
	else
		tmp = Float64(x_m * Float64(x_m + y));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := If[LessEqual[x$95$m, 1.35e+218], N[(x$95$m * x$95$m + N[(y * (-y)), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(x$95$m + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000006e218

   1. Initial program 94.3%

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

      1. sqr-neg 94.3%

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

      2. cancel-sign-sub 94.3%

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

      3. fma-define 98.4%

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

   3. Simplified 98.4%

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

   if 1.35000000000000006e218 < x

   1. Initial program 72.7%

      \[x \cdot x - y \cdot y \]
   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 36.4%

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

      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 63.6%

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

      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%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+218}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{+301}:\\ \;\;\;\;x\_m \cdot x\_m - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (if (<= (* y y) 2e+301) (- (* 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) <= 2e+301) {
		tmp = (x_m * x_m) - (y * y);
	} 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) <= 2d+301) then
        tmp = (x_m * x_m) - (y * y)
    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) <= 2e+301) {
		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) <= 2e+301:
		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) <= 2e+301)
		tmp = Float64(Float64(x_m * x_m) - Float64(y * y));
	else
		tmp = Float64(y * Float64(-y));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y)
	tmp = 0.0;
	if ((y * y) <= 2e+301)
		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], 2e+301], 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) < 2.00000000000000011e301

   1. Initial program 100.0%

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

   if 2.00000000000000011e301 < (*.f64 y y)

   1. Initial program 74.6%

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

   3. Taylor expanded in x around 0 89.6%

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

      1. neg-mul-1 89.6%

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

   5. Simplified 89.6%

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

      1. unpow2 89.6%

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

      2. distribute-lft-neg-in 89.6%

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

   7. Applied egg-rr 89.6%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{+301}:\\ \;\;\;\;x \cdot x - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{-23}:\\ \;\;\;\;x\_m \cdot \left(x\_m + y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (if (<= (* y y) 2e-23) (* x_m (+ x_m y)) (* y (- y))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double tmp;
	if ((y * y) <= 2e-23) {
		tmp = x_m * (x_m + y);
	} 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) <= 2d-23) then
        tmp = x_m * (x_m + y)
    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) <= 2e-23) {
		tmp = x_m * (x_m + y);
	} else {
		tmp = y * -y;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y):
	tmp = 0
	if (y * y) <= 2e-23:
		tmp = x_m * (x_m + y)
	else:
		tmp = y * -y
	return tmp

Julia

x_m = abs(x)
function code(x_m, y)
	tmp = 0.0
	if (Float64(y * y) <= 2e-23)
		tmp = Float64(x_m * Float64(x_m + y));
	else
		tmp = Float64(y * Float64(-y));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y)
	tmp = 0.0;
	if ((y * y) <= 2e-23)
		tmp = x_m * (x_m + 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], 2e-23], N[(x$95$m * N[(x$95$m + y), $MachinePrecision]), $MachinePrecision], N[(y * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1.99999999999999992e-23

   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 50.4%

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

      4. sqrt-unprod 92.7%

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

      5. sqr-neg 92.7%

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

      6. sqrt-prod 42.3%

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

      7. add-sqr-sqrt 84.7%

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

   4. Applied egg-rr 84.7%

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

   5. Taylor expanded in x around inf 85.1%

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

   if 1.99999999999999992e-23 < (*.f64 y y)

   1. Initial program 87.4%

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

   3. Taylor expanded in x around 0 80.4%

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

      1. neg-mul-1 80.4%

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

   5. Simplified 80.4%

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

      1. unpow2 80.4%

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

      2. distribute-lft-neg-in 80.4%

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

   7. Applied egg-rr 80.4%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{-23}:\\ \;\;\;\;x \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 7.8 \cdot 10^{+217}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (if (<= x_m 7.8e+217) (* y (- y)) (* x_m y)))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double tmp;
	if (x_m <= 7.8e+217) {
		tmp = y * -y;
	} else {
		tmp = x_m * 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 (x_m <= 7.8d+217) then
        tmp = y * -y
    else
        tmp = x_m * 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 (x_m <= 7.8e+217) {
		tmp = y * -y;
	} else {
		tmp = x_m * y;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y):
	tmp = 0
	if x_m <= 7.8e+217:
		tmp = y * -y
	else:
		tmp = x_m * y
	return tmp

Julia

x_m = abs(x)
function code(x_m, y)
	tmp = 0.0
	if (x_m <= 7.8e+217)
		tmp = Float64(y * Float64(-y));
	else
		tmp = Float64(x_m * y);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y)
	tmp = 0.0;
	if (x_m <= 7.8e+217)
		tmp = y * -y;
	else
		tmp = x_m * y;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := If[LessEqual[x$95$m, 7.8e+217], N[(y * (-y)), $MachinePrecision], N[(x$95$m * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.79999999999999986e217

   1. Initial program 94.3%

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

   3. Taylor expanded in x around 0 58.6%

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

      1. neg-mul-1 58.6%

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

   5. Simplified 58.6%

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

      1. unpow2 58.6%

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

      2. distribute-lft-neg-in 58.6%

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

   7. Applied egg-rr 58.6%

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

   if 7.79999999999999986e217 < x

   1. Initial program 72.7%

      \[x \cdot x - y \cdot y \]
   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 36.4%

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

      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 63.6%

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

      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 0 29.0%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.8 \cdot 10^{+217}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 15.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

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 * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.4%

   \[x \cdot x - y \cdot y \]
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 49.1%

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

   4. sqrt-unprod 73.6%

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

   5. sqr-neg 73.6%

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

   6. sqrt-prod 25.9%

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

   7. add-sqr-sqrt 50.3%

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

4. Applied egg-rr 50.3%

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

5. Taylor expanded in x around inf 54.4%

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

6. Taylor expanded in x around 0 14.2%

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

Reproduce

herbie shell --seed 2024111 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f2 from sbv-4.4"
  :precision binary64
  (- (* x x) (* y y)))