Examples.Basics.BasicTests:f2 from sbv-4.4

Percentage Accurate: 94.0% → 97.7%

Time: 3.5s

Alternatives: 4

Speedup: 0.4×

• 44.9% 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: 94.0% 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: 97.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{+217}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 7.2e+217) (fma x x (* y (- y))) (* (+ x y) (+ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 7.2e+217) {
		tmp = fma(x, x, (y * -y));
	} else {
		tmp = (x + y) * (x + y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 7.2e+217)
		tmp = fma(x, x, Float64(y * Float64(-y)));
	else
		tmp = Float64(Float64(x + y) * Float64(x + y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 7.2e+217], N[(x * x + N[(y * (-y)), $MachinePrecision]), $MachinePrecision], N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.2000000000000004e217

   1. Initial program 94.4%

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

      1. sqr-neg 94.4%

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

      2. cancel-sign-sub 94.4%

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

      3. fma-define 98.7%

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

   3. Simplified 98.7%

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

   if 7.2000000000000004e217 < x

   1. Initial program 76.0%

      \[x \cdot x - y \cdot y \]
   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 56.0%

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

      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 44.0%

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

      7. add-sqr-sqrt 100.0%

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{+217}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{+292}:\\ \;\;\;\;x \cdot x - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;-{y}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 1e+292) (- (* x x) (* y y)) (- (pow y 2.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1e+292) {
		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+292) 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+292) {
		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+292:
		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+292)
		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+292)
		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+292], 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) < 1e292

   1. Initial program 100.0%

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

   if 1e292 < (*.f64 y y)

   1. Initial program 69.8%

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

   3. Taylor expanded in x around 0 85.7%

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

      1. mul-1-neg 85.7%

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

   5. Simplified 85.7%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{+292}:\\ \;\;\;\;x \cdot x - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;-{y}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot x - y \cdot y\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{+298}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (* x x) (* y y))))
   (if (<= t_0 4e+298) 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 <= 4e+298) {
		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 <= 4d+298) 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 <= 4e+298) {
		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 <= 4e+298:
		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 <= 4e+298)
		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 <= 4e+298)
		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, 4e+298], 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)) < 3.9999999999999998e298

   1. Initial program 100.0%

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

   if 3.9999999999999998e298 < (-.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 78.2%

      \[x \cdot x - y \cdot y \]
   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 49.4%

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

      4. sqrt-unprod 93.1%

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

      5. sqr-neg 93.1%

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

      6. sqrt-prod 44.8%

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

      7. add-sqr-sqrt 88.5%

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

   4. Applied egg-rr 88.5%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x - y \cdot y \leq 4 \cdot 10^{+298}:\\ \;\;\;\;x \cdot x - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot \left(x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.5% 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 92.6%

   \[x \cdot x - y \cdot y \]
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 78.4%

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

   5. sqr-neg 78.4%

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

   6. sqrt-prod 28.8%

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

   7. add-sqr-sqrt 60.7%

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

4. Applied egg-rr 60.7%

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

5. Final simplification 60.7%

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

Reproduce

herbie shell --seed 2024079 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f2 from sbv-4.4"
  :precision binary64
  (- (* x x) (* y y)))