Examples.Basics.BasicTests:f2 from sbv-4.4

Percentage Accurate: 93.6% → 98.7%

Time: 3.9s

Alternatives: 4

Speedup: 0.6×

• 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: 93.6% 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: 98.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x - y \cdot y \leq -2 \cdot 10^{-239}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{2} \cdot \left(1 - \frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- (* x x) (* y y)) -2e-239)
   (fma x x (* y (- y)))
   (* (pow x 2.0) (- 1.0 (* (/ y x) (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (((x * x) - (y * y)) <= -2e-239) {
		tmp = fma(x, x, (y * -y));
	} else {
		tmp = pow(x, 2.0) * (1.0 - ((y / x) * (y / x)));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * x) - Float64(y * y)) <= -2e-239)
		tmp = fma(x, x, Float64(y * Float64(-y)));
	else
		tmp = Float64((x ^ 2.0) * Float64(1.0 - Float64(Float64(y / x) * Float64(y / x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x * x), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision], -2e-239], N[(x * x + N[(y * (-y)), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, 2.0], $MachinePrecision] * N[(1.0 - N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x x) (*.f64 y y)) < -2.0000000000000002e-239

   1. Initial program 100.0%

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

      1. sqr-neg 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. fma-define 100.0%

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

   3. Simplified 100.0%

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

   if -2.0000000000000002e-239 < (-.f64 (*.f64 x x) (*.f64 y y))

   1. Initial program 83.2%

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

   3. Taylor expanded in x around inf 73.8%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + -1 \cdot \frac{{y}^{2}}{{x}^{2}}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg 73.8%

         \[\leadsto {x}^{2} \cdot \left(1 + \color{blue}{\left(-\frac{{y}^{2}}{{x}^{2}}\right)}\right) \]

      2. unsub-neg 73.8%

         \[\leadsto {x}^{2} \cdot \color{blue}{\left(1 - \frac{{y}^{2}}{{x}^{2}}\right)} \]

   5. Simplified 73.8%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 - \frac{{y}^{2}}{{x}^{2}}\right)} \]
   6. Step-by-step derivation

      1. unpow2 73.8%

         \[\leadsto {x}^{2} \cdot \left(1 - \frac{\color{blue}{y \cdot y}}{{x}^{2}}\right) \]

      2. unpow2 73.8%

         \[\leadsto {x}^{2} \cdot \left(1 - \frac{y \cdot y}{\color{blue}{x \cdot x}}\right) \]

      3. times-frac 100.0%

         \[\leadsto {x}^{2} \cdot \left(1 - \color{blue}{\frac{y}{x} \cdot \frac{y}{x}}\right) \]

   7. Applied egg-rr 100.0%

      \[\leadsto {x}^{2} \cdot \left(1 - \color{blue}{\frac{y}{x} \cdot \frac{y}{x}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 97.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.1 \cdot 10^{+250}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 4.1e+250) (fma x x (* y (- y))) (* (+ x y) (+ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4.1e+250) {
		tmp = fma(x, x, (y * -y));
	} else {
		tmp = (x + y) * (x + y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4.1e+250)
		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, 4.1e+250], 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 < 4.09999999999999999e250

   1. Initial program 91.7%

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

      1. sqr-neg 91.7%

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

      2. cancel-sign-sub 91.7%

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

      3. fma-define 96.3%

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

   3. Simplified 96.3%

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

   if 4.09999999999999999e250 < x

   1. Initial program 64.3%

      \[x \cdot x - y \cdot y \]
   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 50.0%

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

      4. sqrt-unprod 92.9%

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

      5. sqr-neg 92.9%

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

      6. sqrt-prod 42.9%

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

      7. add-sqr-sqrt 92.9%

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

   4. Applied egg-rr 92.9%

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

Alternative 3: 94.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{+132}:\\ \;\;\;\;x \cdot x - y \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot \left(x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 8e+132) (- (* x x) (* y y)) (* (+ x y) (+ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 8e+132) {
		tmp = (x * x) - (y * y);
	} 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) :: tmp
    if (x <= 8d+132) then
        tmp = (x * x) - (y * y)
    else
        tmp = (x + y) * (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 8e+132) {
		tmp = (x * x) - (y * y);
	} else {
		tmp = (x + y) * (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 8e+132:
		tmp = (x * x) - (y * y)
	else:
		tmp = (x + y) * (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 8e+132)
		tmp = Float64(Float64(x * x) - Float64(y * y));
	else
		tmp = Float64(Float64(x + y) * Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 8e+132)
		tmp = (x * x) - (y * y);
	else
		tmp = (x + y) * (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 8e+132], N[(N[(x * x), $MachinePrecision] - 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 < 7.99999999999999993e132

   1. Initial program 94.4%

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

   if 7.99999999999999993e132 < x

   1. Initial program 69.8%

      \[x \cdot x - y \cdot y \]
   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 58.1%

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

      4. sqrt-unprod 93.0%

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

      5. sqr-neg 93.0%

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

      6. sqrt-prod 34.9%

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

      7. add-sqr-sqrt 83.7%

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

   4. Applied egg-rr 83.7%

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

Alternative 4: 53.1% 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 90.2%

   \[x \cdot x - y \cdot y \]
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 47.2%

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

   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 27.7%

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

   7. add-sqr-sqrt 52.8%

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

4. Applied egg-rr 52.8%

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

Reproduce

herbie shell --seed 2024085 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f2 from sbv-4.4"
  :precision binary64
  (- (* x x) (* y y)))