Examples.Basics.BasicTests:f2 from sbv-4.4

Percentage Accurate: 93.7% → 97.0%

Time: 3.6s

Alternatives: 4

Speedup: 0.5×

• 43.4% 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: 97.0% 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 91.0%

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

   1. sqr-neg 91.0%

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

   2. cancel-sign-sub 91.0%

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

   3. fma-def 97.3%

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

3. Simplified 97.3%

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

5. Final simplification 97.3%

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

Alternative 2: 77.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{-216} \lor \neg \left(y \cdot y \leq 10^{-183}\right) \land y \cdot y \leq 10^{+50}:\\ \;\;\;\;\left(x - y\right) \cdot \left(x - y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* y y) 1e-216)
         (and (not (<= (* y y) 1e-183)) (<= (* y y) 1e+50)))
   (* (- x y) (- x y))
   (* y (- y))))

C

double code(double x, double y) {
	double tmp;
	if (((y * y) <= 1e-216) || (!((y * y) <= 1e-183) && ((y * y) <= 1e+50))) {
		tmp = (x - y) * (x - y);
	} else {
		tmp = y * -y;
	}
	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-216) .or. (.not. ((y * y) <= 1d-183)) .and. ((y * y) <= 1d+50)) then
        tmp = (x - y) * (x - y)
    else
        tmp = y * -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((y * y) <= 1e-216) || (!((y * y) <= 1e-183) && ((y * y) <= 1e+50))) {
		tmp = (x - y) * (x - y);
	} else {
		tmp = y * -y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((y * y) <= 1e-216) or (not ((y * y) <= 1e-183) and ((y * y) <= 1e+50)):
		tmp = (x - y) * (x - y)
	else:
		tmp = y * -y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(y * y) <= 1e-216) || (!(Float64(y * y) <= 1e-183) && (Float64(y * y) <= 1e+50)))
		tmp = Float64(Float64(x - y) * Float64(x - y));
	else
		tmp = Float64(y * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((y * y) <= 1e-216) || (~(((y * y) <= 1e-183)) && ((y * y) <= 1e+50)))
		tmp = (x - y) * (x - y);
	else
		tmp = y * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(y * y), $MachinePrecision], 1e-216], And[N[Not[LessEqual[N[(y * y), $MachinePrecision], 1e-183]], $MachinePrecision], LessEqual[N[(y * y), $MachinePrecision], 1e+50]]], N[(N[(x - y), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision], N[(y * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1e-216 or 1.00000000000000001e-183 < (*.f64 y y) < 1.0000000000000001e50

   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. add-sqr-sqrt 49.6%

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

      3. sqrt-prod 92.3%

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

      4. sqr-neg 92.3%

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

      5. sqrt-unprod 42.7%

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

      6. add-sqr-sqrt 85.5%

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

      7. sub-neg 85.5%

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

      8. pow1 85.5%

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

      9. pow1 85.5%

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

      10. pow-prod-up 85.5%

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

      11. add-sqr-sqrt 42.6%

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

      12. add-sqr-sqrt 22.5%

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

      13. difference-of-squares 22.5%

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

      14. metadata-eval 22.5%

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

      15. unpow-prod-down 22.4%

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

   4. Applied egg-rr 22.4%

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

      1. unpow2 22.4%

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

      2. unpow2 22.4%

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

      3. unswap-sqr 22.5%

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

      4. difference-of-squares 22.5%

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

      5. unpow1/2 22.5%

         \[\leadsto \left(\color{blue}{{x}^{0.5}} \cdot \sqrt{x} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \left(\left(\sqrt{x} + \sqrt{y}\right) \cdot \left(\sqrt{x} - \sqrt{y}\right)\right) \]

      6. unpow1/2 22.5%

         \[\leadsto \left({x}^{0.5} \cdot \color{blue}{{x}^{0.5}} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \left(\left(\sqrt{x} + \sqrt{y}\right) \cdot \left(\sqrt{x} - \sqrt{y}\right)\right) \]

      7. pow-sqr 22.5%

         \[\leadsto \left(\color{blue}{{x}^{\left(2 \cdot 0.5\right)}} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \left(\left(\sqrt{x} + \sqrt{y}\right) \cdot \left(\sqrt{x} - \sqrt{y}\right)\right) \]

      8. metadata-eval 22.5%

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

      9. unpow1 22.5%

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

      10. unpow1/2 22.5%

          \[\leadsto \left(x - \color{blue}{{y}^{0.5}} \cdot \sqrt{y}\right) \cdot \left(\left(\sqrt{x} + \sqrt{y}\right) \cdot \left(\sqrt{x} - \sqrt{y}\right)\right) \]

      11. unpow1/2 22.5%

          \[\leadsto \left(x - {y}^{0.5} \cdot \color{blue}{{y}^{0.5}}\right) \cdot \left(\left(\sqrt{x} + \sqrt{y}\right) \cdot \left(\sqrt{x} - \sqrt{y}\right)\right) \]

      12. pow-sqr 22.5%

          \[\leadsto \left(x - \color{blue}{{y}^{\left(2 \cdot 0.5\right)}}\right) \cdot \left(\left(\sqrt{x} + \sqrt{y}\right) \cdot \left(\sqrt{x} - \sqrt{y}\right)\right) \]

      13. metadata-eval 22.5%

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

      14. unpow1 22.5%

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

      15. difference-of-squares 22.5%

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

      16. unpow1/2 22.5%

          \[\leadsto \left(x - y\right) \cdot \left(\color{blue}{{x}^{0.5}} \cdot \sqrt{x} - \sqrt{y} \cdot \sqrt{y}\right) \]

      17. unpow1/2 22.5%

          \[\leadsto \left(x - y\right) \cdot \left({x}^{0.5} \cdot \color{blue}{{x}^{0.5}} - \sqrt{y} \cdot \sqrt{y}\right) \]

      18. pow-sqr 42.8%

          \[\leadsto \left(x - y\right) \cdot \left(\color{blue}{{x}^{\left(2 \cdot 0.5\right)}} - \sqrt{y} \cdot \sqrt{y}\right) \]

      19. metadata-eval 42.8%

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

      20. unpow1 42.8%

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

   6. Simplified 85.5%

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

   if 1e-216 < (*.f64 y y) < 1.00000000000000001e-183 or 1.0000000000000001e50 < (*.f64 y y)

   1. Initial program 81.9%

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

   3. Taylor expanded in x around 0 81.4%

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

      1. mul-1-neg 81.4%

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

   5. Simplified 81.4%

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

      1. unpow2 81.4%

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

   7. Applied egg-rr 81.4%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{-216} \lor \neg \left(y \cdot y \leq 10^{-183}\right) \land y \cdot y \leq 10^{+50}:\\ \;\;\;\;\left(x - y\right) \cdot \left(x - y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 2e+292) (- (* x x) (* y y)) (* y (- y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 2e+292) {
		tmp = (x * x) - (y * y);
	} else {
		tmp = y * -y;
	}
	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) <= 2d+292) then
        tmp = (x * x) - (y * y)
    else
        tmp = y * -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 2e+292) {
		tmp = (x * x) - (y * y);
	} else {
		tmp = y * -y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 2e+292:
		tmp = (x * x) - (y * y)
	else:
		tmp = y * -y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 2e+292)
		tmp = Float64(Float64(x * x) - Float64(y * y));
	else
		tmp = Float64(y * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 2e+292)
		tmp = (x * x) - (y * y);
	else
		tmp = y * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 2e+292], N[(N[(x * x), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision], N[(y * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 2e292

   1. Initial program 100.0%

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

   if 2e292 < (*.f64 y y)

   1. Initial program 66.7%

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

   3. Taylor expanded in x around 0 89.9%

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

      1. mul-1-neg 89.9%

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

   5. Simplified 89.9%

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

      1. unpow2 89.9%

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

   7. Applied egg-rr 89.9%

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

4. Final simplification 97.3%

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

Alternative 4: 54.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.0%

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

3. Taylor expanded in x around 0 56.4%

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

   1. mul-1-neg 56.4%

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

5. Simplified 56.4%

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

   1. unpow2 56.4%

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

7. Applied egg-rr 56.4%

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

8. Final simplification 56.4%

   \[\leadsto y \cdot \left(-y\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f2 from sbv-4.4"
  :precision binary64
  (- (* x x) (* y y)))