Examples.Basics.BasicTests:f2 from sbv-4.4

Percentage Accurate: 93.2% → 96.6%

Time: 3.2s

Alternatives: 4

Speedup: 0.4×

• 44.7% 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: 93.2% 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: 96.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.4%

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

   1. sqr-neg 93.4%

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

   2. cancel-sign-sub 93.4%

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

   3. fma-define 97.7%

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

3. Simplified 97.7%

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

5. Final simplification 97.7%

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

Alternative 2: 95.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{+249}:\\ \;\;\;\;x \cdot x - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;-{y}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 1e+249) (- (* x x) (* y y)) (- (pow y 2.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1e+249) {
		tmp = (x * x) - (y * y);
	} else {
		tmp = -pow(y, 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * y) <= 1d+249) then
        tmp = (x * x) - (y * y)
    else
        tmp = -(y ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1e+249) {
		tmp = (x * x) - (y * y);
	} else {
		tmp = -Math.pow(y, 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 1e+249:
		tmp = (x * x) - (y * y)
	else:
		tmp = -math.pow(y, 2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 1e+249)
		tmp = Float64(Float64(x * x) - Float64(y * y));
	else
		tmp = Float64(-(y ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 1e+249)
		tmp = (x * x) - (y * y);
	else
		tmp = -(y ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 1e+249], N[(N[(x * x), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision], (-N[Power[y, 2.0], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 9.9999999999999992e248

   1. Initial program 100.0%

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

   if 9.9999999999999992e248 < (*.f64 y y)

   1. Initial program 79.0%

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

   3. Taylor expanded in x around 0 92.6%

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

      1. mul-1-neg 92.6%

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

   5. Simplified 92.6%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{+249}:\\ \;\;\;\;x \cdot x - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;-{y}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot x - y \cdot y\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+44}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (* x x) (* y y))))
   (if (<= t_0 2e+44) t_0 (* (+ x y) (+ x y)))))

C

double code(double x, double y) {
	double t_0 = (x * x) - (y * y);
	double tmp;
	if (t_0 <= 2e+44) {
		tmp = t_0;
	} else {
		tmp = (x + y) * (x + y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * x) - (y * y)
    if (t_0 <= 2d+44) then
        tmp = t_0
    else
        tmp = (x + y) * (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * x) - (y * y);
	double tmp;
	if (t_0 <= 2e+44) {
		tmp = t_0;
	} else {
		tmp = (x + y) * (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * x) - (y * y)
	tmp = 0
	if t_0 <= 2e+44:
		tmp = t_0
	else:
		tmp = (x + y) * (x + y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * x) - Float64(y * y))
	tmp = 0.0
	if (t_0 <= 2e+44)
		tmp = t_0;
	else
		tmp = Float64(Float64(x + y) * Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * x) - (y * y);
	tmp = 0.0;
	if (t_0 <= 2e+44)
		tmp = t_0;
	else
		tmp = (x + y) * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+44], t$95$0, N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x x) (*.f64 y y)) < 2.0000000000000002e44

   1. Initial program 100.0%

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

   if 2.0000000000000002e44 < (-.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 81.5%

      \[x \cdot x - y \cdot y \]
   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 43.5%

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

      4. sqrt-unprod 89.1%

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

      5. sqr-neg 89.1%

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

      6. sqrt-prod 50.0%

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

      7. add-sqr-sqrt 88.0%

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

   4. Applied egg-rr 88.0%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x - y \cdot y \leq 2 \cdot 10^{+44}:\\ \;\;\;\;x \cdot x - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $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 48.4%

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

   4. sqrt-unprod 74.2%

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

   5. sqr-neg 74.2%

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

   6. sqrt-prod 27.4%

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

   7. add-sqr-sqrt 51.7%

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

4. Applied egg-rr 51.7%

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

5. Final simplification 51.7%

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

Reproduce

herbie shell --seed 2024096 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f2 from sbv-4.4"
  :precision binary64
  (- (* x x) (* y y)))