Examples.Basics.BasicTests:f2 from sbv-4.4

Percentage Accurate: 93.9% → 98.4%

Time: 3.5s

Alternatives: 4

Speedup: 0.4×

• 44.2% 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: 93.9% 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: 98.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.85 \cdot 10^{+253}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m, y \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m + y\right) \cdot \left(x\_m + y\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (if (<= x_m 1.85e+253) (fma x_m x_m (* y (- y))) (* (+ x_m y) (+ x_m y))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double tmp;
	if (x_m <= 1.85e+253) {
		tmp = fma(x_m, x_m, (y * -y));
	} else {
		tmp = (x_m + y) * (x_m + y);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y)
	tmp = 0.0
	if (x_m <= 1.85e+253)
		tmp = fma(x_m, x_m, Float64(y * Float64(-y)));
	else
		tmp = Float64(Float64(x_m + y) * 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, 1.85e+253], N[(x$95$m * x$95$m + N[(y * (-y)), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m + y), $MachinePrecision] * N[(x$95$m + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.85000000000000014e253

   1. Initial program 92.2%

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

      1. sqr-neg 92.2%

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

      2. cancel-sign-sub 92.2%

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

      3. fma-define 97.5%

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

   3. Simplified 97.5%

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

   if 1.85000000000000014e253 < x

   1. Initial program 66.7%

      \[x \cdot x - y \cdot y \]
   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 33.3%

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

      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 66.7%

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

      7. add-sqr-sqrt 100.0%

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{+253}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{+271}:\\ \;\;\;\;x\_m \cdot x\_m - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;-{y}^{2}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (if (<= (* y y) 1e+271) (- (* x_m x_m) (* y y)) (- (pow y 2.0))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double tmp;
	if ((y * y) <= 1e+271) {
		tmp = (x_m * x_m) - (y * y);
	} else {
		tmp = -pow(y, 2.0);
	}
	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+271) then
        tmp = (x_m * x_m) - (y * y)
    else
        tmp = -(y ** 2.0d0)
    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+271) {
		tmp = (x_m * x_m) - (y * y);
	} else {
		tmp = -Math.pow(y, 2.0);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y):
	tmp = 0
	if (y * y) <= 1e+271:
		tmp = (x_m * x_m) - (y * y)
	else:
		tmp = -math.pow(y, 2.0)
	return tmp

Julia

x_m = abs(x)
function code(x_m, y)
	tmp = 0.0
	if (Float64(y * y) <= 1e+271)
		tmp = Float64(Float64(x_m * x_m) - Float64(y * y));
	else
		tmp = Float64(-(y ^ 2.0));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y)
	tmp = 0.0;
	if ((y * y) <= 1e+271)
		tmp = (x_m * x_m) - (y * y);
	else
		tmp = -(y ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 1e+271], N[(N[(x$95$m * x$95$m), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision], (-N[Power[y, 2.0], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 9.99999999999999953e270

   1. Initial program 100.0%

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

   if 9.99999999999999953e270 < (*.f64 y y)

   1. Initial program 67.6%

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

   3. Taylor expanded in x around 0 85.9%

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

      1. mul-1-neg 85.9%

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

   5. Simplified 85.9%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{+271}:\\ \;\;\;\;x \cdot x - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;-{y}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := x\_m \cdot x\_m - y \cdot y\\ \mathbf{if}\;t\_0 \leq 10^{+257}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m + y\right) \cdot \left(x\_m + y\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (let* ((t_0 (- (* x_m x_m) (* y y))))
   (if (<= t_0 1e+257) t_0 (* (+ x_m y) (+ x_m y)))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double t_0 = (x_m * x_m) - (y * y);
	double tmp;
	if (t_0 <= 1e+257) {
		tmp = t_0;
	} else {
		tmp = (x_m + y) * (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) :: t_0
    real(8) :: tmp
    t_0 = (x_m * x_m) - (y * y)
    if (t_0 <= 1d+257) then
        tmp = t_0
    else
        tmp = (x_m + y) * (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 t_0 = (x_m * x_m) - (y * y);
	double tmp;
	if (t_0 <= 1e+257) {
		tmp = t_0;
	} else {
		tmp = (x_m + y) * (x_m + y);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y):
	t_0 = (x_m * x_m) - (y * y)
	tmp = 0
	if t_0 <= 1e+257:
		tmp = t_0
	else:
		tmp = (x_m + y) * (x_m + y)
	return tmp

Julia

x_m = abs(x)
function code(x_m, y)
	t_0 = Float64(Float64(x_m * x_m) - Float64(y * y))
	tmp = 0.0
	if (t_0 <= 1e+257)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_m + y) * Float64(x_m + y));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y)
	t_0 = (x_m * x_m) - (y * y);
	tmp = 0.0;
	if (t_0 <= 1e+257)
		tmp = t_0;
	else
		tmp = (x_m + y) * (x_m + y);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := Block[{t$95$0 = N[(N[(x$95$m * x$95$m), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e+257], t$95$0, N[(N[(x$95$m + y), $MachinePrecision] * N[(x$95$m + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x x) (*.f64 y y)) < 1.00000000000000003e257

   1. Initial program 100.0%

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

   if 1.00000000000000003e257 < (-.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 70.1%

      \[x \cdot x - y \cdot y \]
   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 48.1%

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

      4. sqrt-unprod 88.3%

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

      5. sqr-neg 88.3%

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

      6. sqrt-prod 44.2%

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

      7. add-sqr-sqrt 83.1%

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

   4. Applied egg-rr 83.1%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x - y \cdot y \leq 10^{+257}:\\ \;\;\;\;x \cdot x - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
double code(double x_m, double y) {
	return (x_m + y) * (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) * (x_m + y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.0%

   \[x \cdot x - y \cdot y \]
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 49.9%

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

   4. sqrt-unprod 77.5%

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

   5. sqr-neg 77.5%

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

   6. sqrt-prod 28.7%

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

   7. add-sqr-sqrt 54.4%

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

4. Applied egg-rr 54.4%

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

5. Final simplification 54.4%

   \[\leadsto \left(x + y\right) \cdot \left(x + y\right) \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024071 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f2 from sbv-4.4"
  :precision binary64
  (- (* x x) (* y y)))