Difference of squares

Percentage Accurate: 93.9% → 99.9%

Time: 3.7s

Alternatives: 3

Speedup: 0.5×

• 44.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ a \cdot a - b \cdot b \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (- (* a a) (* b b)))

C

double code(double a, double b) {
	return (a * a) - (b * b);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (a * a) - (b * b)
end function

Java

public static double code(double a, double b) {
	return (a * a) - (b * b);
}

Python

def code(a, b):
	return (a * a) - (b * b)

Julia

function code(a, b)
	return Float64(Float64(a * a) - Float64(b * b))
end

MATLAB

function tmp = code(a, b)
	tmp = (a * a) - (b * b);
end

Wolfram

code[a_, b_] := N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision]

Initial Program: 93.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ a \cdot a - b \cdot b \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (- (* a a) (* b b)))

C

double code(double a, double b) {
	return (a * a) - (b * b);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (a * a) - (b * b)
end function

Java

public static double code(double a, double b) {
	return (a * a) - (b * b);
}

Python

def code(a, b):
	return (a * a) - (b * b)

Julia

function code(a, b)
	return Float64(Float64(a * a) - Float64(b * b))
end

MATLAB

function tmp = code(a, b)
	tmp = (a * a) - (b * b);
end

Wolfram

code[a_, b_] := N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;a\_m \cdot a\_m \leq 3 \cdot 10^{+272}:\\ \;\;\;\;a\_m \cdot a\_m - b\_m \cdot b\_m\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot \left(a\_m + b\_m \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m)
 :precision binary64
 (if (<= (* a_m a_m) 3e+272)
   (- (* a_m a_m) (* b_m b_m))
   (* a_m (+ a_m (* b_m -2.0)))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m) {
	double tmp;
	if ((a_m * a_m) <= 3e+272) {
		tmp = (a_m * a_m) - (b_m * b_m);
	} else {
		tmp = a_m * (a_m + (b_m * -2.0));
	}
	return tmp;
}

Fortran

a_m = abs(a)
b_m = abs(b)
real(8) function code(a_m, b_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8) :: tmp
    if ((a_m * a_m) <= 3d+272) then
        tmp = (a_m * a_m) - (b_m * b_m)
    else
        tmp = a_m * (a_m + (b_m * (-2.0d0)))
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
public static double code(double a_m, double b_m) {
	double tmp;
	if ((a_m * a_m) <= 3e+272) {
		tmp = (a_m * a_m) - (b_m * b_m);
	} else {
		tmp = a_m * (a_m + (b_m * -2.0));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
def code(a_m, b_m):
	tmp = 0
	if (a_m * a_m) <= 3e+272:
		tmp = (a_m * a_m) - (b_m * b_m)
	else:
		tmp = a_m * (a_m + (b_m * -2.0))
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m)
	tmp = 0.0
	if (Float64(a_m * a_m) <= 3e+272)
		tmp = Float64(Float64(a_m * a_m) - Float64(b_m * b_m));
	else
		tmp = Float64(a_m * Float64(a_m + Float64(b_m * -2.0)));
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
function tmp_2 = code(a_m, b_m)
	tmp = 0.0;
	if ((a_m * a_m) <= 3e+272)
		tmp = (a_m * a_m) - (b_m * b_m);
	else
		tmp = a_m * (a_m + (b_m * -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_] := If[LessEqual[N[(a$95$m * a$95$m), $MachinePrecision], 3e+272], N[(N[(a$95$m * a$95$m), $MachinePrecision] - N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision], N[(a$95$m * N[(a$95$m + N[(b$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 3.0000000000000002e272

   1. Initial program 100.0%

      \[a \cdot a - b \cdot b \]
   2. Add Preprocessing

   if 3.0000000000000002e272 < (*.f64 a a)

   1. Initial program 77.9%

      \[a \cdot a - b \cdot b \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 74.0%

         \[\leadsto \color{blue}{\sqrt{a \cdot a - b \cdot b} \cdot \sqrt{a \cdot a - b \cdot b}} \]

      2. pow2 74.0%

         \[\leadsto \color{blue}{{\left(\sqrt{a \cdot a - b \cdot b}\right)}^{2}} \]

      3. difference-of-squares 87.0%

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

      4. sqrt-prod 44.1%

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

      5. add-sqr-sqrt 14.3%

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

      6. sqrt-prod 44.1%

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

      7. sqr-neg 44.1%

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

      8. sqrt-unprod 29.8%

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

      9. add-sqr-sqrt 44.1%

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

      10. sub-neg 44.1%

          \[\leadsto {\left(\sqrt{\color{blue}{a - b}} \cdot \sqrt{a - b}\right)}^{2} \]

      11. add-sqr-sqrt 87.0%

          \[\leadsto {\color{blue}{\left(a - b\right)}}^{2} \]

      12. add-sqr-sqrt 44.1%

          \[\leadsto {\left(\color{blue}{\sqrt{a} \cdot \sqrt{a}} - b\right)}^{2} \]

      13. add-sqr-sqrt 14.3%

          \[\leadsto {\left(\sqrt{a} \cdot \sqrt{a} - \color{blue}{\sqrt{b} \cdot \sqrt{b}}\right)}^{2} \]

      14. difference-of-squares 14.3%

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

      15. unpow-prod-down 14.3%

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

   4. Applied egg-rr 14.3%

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

      1. unpow2 14.3%

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

      2. unpow2 14.3%

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

      3. unswap-sqr 14.3%

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

      4. difference-of-squares 14.3%

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

      5. rem-square-sqrt 14.3%

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

      6. rem-square-sqrt 14.3%

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

      7. difference-of-squares 14.3%

         \[\leadsto \left(a - b\right) \cdot \color{blue}{\left(\sqrt{a} \cdot \sqrt{a} - \sqrt{b} \cdot \sqrt{b}\right)} \]

      8. rem-square-sqrt 39.0%

         \[\leadsto \left(a - b\right) \cdot \left(\color{blue}{a} - \sqrt{b} \cdot \sqrt{b}\right) \]

      9. rem-square-sqrt 87.0%

         \[\leadsto \left(a - b\right) \cdot \left(a - \color{blue}{b}\right) \]

   6. Simplified 87.0%

      \[\leadsto \color{blue}{\left(a - b\right) \cdot \left(a - b\right)} \]

   7. Taylor expanded in a around inf 77.9%

      \[\leadsto \color{blue}{-2 \cdot \left(a \cdot b\right) + {a}^{2}} \]
   8. Step-by-step derivation

      1. *-commutative 77.9%

         \[\leadsto \color{blue}{\left(a \cdot b\right) \cdot -2} + {a}^{2} \]

      2. associate-*l* 77.9%

         \[\leadsto \color{blue}{a \cdot \left(b \cdot -2\right)} + {a}^{2} \]

      3. unpow2 77.9%

         \[\leadsto a \cdot \left(b \cdot -2\right) + \color{blue}{a \cdot a} \]

      4. distribute-lft-out 96.1%

         \[\leadsto \color{blue}{a \cdot \left(b \cdot -2 + a\right)} \]

   9. Simplified 96.1%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 3 \cdot 10^{+272}:\\ \;\;\;\;a \cdot a - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a + b \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 60.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ a\_m \cdot \left(a\_m + b\_m \cdot -2\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m) :precision binary64 (* a_m (+ a_m (* b_m -2.0))))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m) {
	return a_m * (a_m + (b_m * -2.0));
}

Fortran

a_m = abs(a)
b_m = abs(b)
real(8) function code(a_m, b_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    code = a_m * (a_m + (b_m * (-2.0d0)))
end function

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
public static double code(double a_m, double b_m) {
	return a_m * (a_m + (b_m * -2.0));
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
def code(a_m, b_m):
	return a_m * (a_m + (b_m * -2.0))

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m)
	return Float64(a_m * Float64(a_m + Float64(b_m * -2.0)))
end

MATLAB

a_m = abs(a);
b_m = abs(b);
function tmp = code(a_m, b_m)
	tmp = a_m * (a_m + (b_m * -2.0));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_] := N[(a$95$m * N[(a$95$m + N[(b$95$m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.4%

   \[a \cdot a - b \cdot b \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 49.2%

      \[\leadsto \color{blue}{\sqrt{a \cdot a - b \cdot b} \cdot \sqrt{a \cdot a - b \cdot b}} \]

   2. pow2 49.2%

      \[\leadsto \color{blue}{{\left(\sqrt{a \cdot a - b \cdot b}\right)}^{2}} \]

   3. difference-of-squares 53.1%

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

   4. sqrt-prod 26.0%

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

   5. add-sqr-sqrt 10.5%

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

   6. sqrt-prod 26.9%

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

   7. sqr-neg 26.9%

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

   8. sqrt-unprod 16.5%

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

   9. add-sqr-sqrt 26.7%

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

   10. sub-neg 26.7%

       \[\leadsto {\left(\sqrt{\color{blue}{a - b}} \cdot \sqrt{a - b}\right)}^{2} \]

   11. add-sqr-sqrt 53.6%

       \[\leadsto {\color{blue}{\left(a - b\right)}}^{2} \]

   12. add-sqr-sqrt 27.9%

       \[\leadsto {\left(\color{blue}{\sqrt{a} \cdot \sqrt{a}} - b\right)}^{2} \]

   13. add-sqr-sqrt 11.5%

       \[\leadsto {\left(\sqrt{a} \cdot \sqrt{a} - \color{blue}{\sqrt{b} \cdot \sqrt{b}}\right)}^{2} \]

   14. difference-of-squares 11.5%

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

   15. unpow-prod-down 11.5%

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

4. Applied egg-rr 11.5%

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

   1. unpow2 11.5%

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

   2. unpow2 11.5%

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

   3. unswap-sqr 11.5%

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

   4. difference-of-squares 11.5%

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

   5. rem-square-sqrt 11.6%

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

   6. rem-square-sqrt 11.6%

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

   7. difference-of-squares 11.6%

      \[\leadsto \left(a - b\right) \cdot \color{blue}{\left(\sqrt{a} \cdot \sqrt{a} - \sqrt{b} \cdot \sqrt{b}\right)} \]

   8. rem-square-sqrt 24.2%

      \[\leadsto \left(a - b\right) \cdot \left(\color{blue}{a} - \sqrt{b} \cdot \sqrt{b}\right) \]

   9. rem-square-sqrt 53.6%

      \[\leadsto \left(a - b\right) \cdot \left(a - \color{blue}{b}\right) \]

6. Simplified 53.6%

   \[\leadsto \color{blue}{\left(a - b\right) \cdot \left(a - b\right)} \]

7. Taylor expanded in a around inf 53.9%

   \[\leadsto \color{blue}{-2 \cdot \left(a \cdot b\right) + {a}^{2}} \]
8. Step-by-step derivation

   1. *-commutative 53.9%

      \[\leadsto \color{blue}{\left(a \cdot b\right) \cdot -2} + {a}^{2} \]

   2. associate-*l* 53.9%

      \[\leadsto \color{blue}{a \cdot \left(b \cdot -2\right)} + {a}^{2} \]

   3. unpow2 53.9%

      \[\leadsto a \cdot \left(b \cdot -2\right) + \color{blue}{a \cdot a} \]

   4. distribute-lft-out 59.4%

      \[\leadsto \color{blue}{a \cdot \left(b \cdot -2 + a\right)} \]

9. Simplified 59.4%

   \[\leadsto \color{blue}{a \cdot \left(b \cdot -2 + a\right)} \]

10. Final simplification 59.4%

    \[\leadsto a \cdot \left(a + b \cdot -2\right) \]
11. Add Preprocessing

Alternative 3: 15.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ b\_m \cdot \left(a\_m \cdot -2\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m) :precision binary64 (* b_m (* a_m -2.0)))

C

a_m = fabs(a);
b_m = fabs(b);
double code(double a_m, double b_m) {
	return b_m * (a_m * -2.0);
}

Fortran

a_m = abs(a)
b_m = abs(b)
real(8) function code(a_m, b_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    code = b_m * (a_m * (-2.0d0))
end function

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
public static double code(double a_m, double b_m) {
	return b_m * (a_m * -2.0);
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
def code(a_m, b_m):
	return b_m * (a_m * -2.0)

Julia

a_m = abs(a)
b_m = abs(b)
function code(a_m, b_m)
	return Float64(b_m * Float64(a_m * -2.0))
end

MATLAB

a_m = abs(a);
b_m = abs(b);
function tmp = code(a_m, b_m)
	tmp = b_m * (a_m * -2.0);
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
code[a$95$m_, b$95$m_] := N[(b$95$m * N[(a$95$m * -2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.4%

   \[a \cdot a - b \cdot b \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 49.2%

      \[\leadsto \color{blue}{\sqrt{a \cdot a - b \cdot b} \cdot \sqrt{a \cdot a - b \cdot b}} \]

   2. pow2 49.2%

      \[\leadsto \color{blue}{{\left(\sqrt{a \cdot a - b \cdot b}\right)}^{2}} \]

   3. difference-of-squares 53.1%

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

   4. sqrt-prod 26.0%

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

   5. add-sqr-sqrt 10.5%

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

   6. sqrt-prod 26.9%

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

   7. sqr-neg 26.9%

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

   8. sqrt-unprod 16.5%

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

   9. add-sqr-sqrt 26.7%

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

   10. sub-neg 26.7%

       \[\leadsto {\left(\sqrt{\color{blue}{a - b}} \cdot \sqrt{a - b}\right)}^{2} \]

   11. add-sqr-sqrt 53.6%

       \[\leadsto {\color{blue}{\left(a - b\right)}}^{2} \]

   12. add-sqr-sqrt 27.9%

       \[\leadsto {\left(\color{blue}{\sqrt{a} \cdot \sqrt{a}} - b\right)}^{2} \]

   13. add-sqr-sqrt 11.5%

       \[\leadsto {\left(\sqrt{a} \cdot \sqrt{a} - \color{blue}{\sqrt{b} \cdot \sqrt{b}}\right)}^{2} \]

   14. difference-of-squares 11.5%

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

   15. unpow-prod-down 11.5%

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

4. Applied egg-rr 11.5%

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

   1. unpow2 11.5%

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

   2. unpow2 11.5%

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

   3. unswap-sqr 11.5%

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

   4. difference-of-squares 11.5%

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

   5. rem-square-sqrt 11.6%

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

   6. rem-square-sqrt 11.6%

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

   7. difference-of-squares 11.6%

      \[\leadsto \left(a - b\right) \cdot \color{blue}{\left(\sqrt{a} \cdot \sqrt{a} - \sqrt{b} \cdot \sqrt{b}\right)} \]

   8. rem-square-sqrt 24.2%

      \[\leadsto \left(a - b\right) \cdot \left(\color{blue}{a} - \sqrt{b} \cdot \sqrt{b}\right) \]

   9. rem-square-sqrt 53.6%

      \[\leadsto \left(a - b\right) \cdot \left(a - \color{blue}{b}\right) \]

6. Simplified 53.6%

   \[\leadsto \color{blue}{\left(a - b\right) \cdot \left(a - b\right)} \]

7. Taylor expanded in a around inf 53.9%

   \[\leadsto \color{blue}{-2 \cdot \left(a \cdot b\right) + {a}^{2}} \]
8. Step-by-step derivation

   1. *-commutative 53.9%

      \[\leadsto \color{blue}{\left(a \cdot b\right) \cdot -2} + {a}^{2} \]

   2. associate-*l* 53.9%

      \[\leadsto \color{blue}{a \cdot \left(b \cdot -2\right)} + {a}^{2} \]

   3. unpow2 53.9%

      \[\leadsto a \cdot \left(b \cdot -2\right) + \color{blue}{a \cdot a} \]

   4. distribute-lft-out 59.4%

      \[\leadsto \color{blue}{a \cdot \left(b \cdot -2 + a\right)} \]

9. Simplified 59.4%

   \[\leadsto \color{blue}{a \cdot \left(b \cdot -2 + a\right)} \]

10. Taylor expanded in a around 0 18.1%

    \[\leadsto \color{blue}{-2 \cdot \left(a \cdot b\right)} \]
11. Step-by-step derivation

    1. associate-*r* 18.1%

       \[\leadsto \color{blue}{\left(-2 \cdot a\right) \cdot b} \]

    2. *-commutative 18.1%

       \[\leadsto \color{blue}{b \cdot \left(-2 \cdot a\right)} \]

12. Simplified 18.1%

    \[\leadsto \color{blue}{b \cdot \left(-2 \cdot a\right)} \]

13. Final simplification 18.1%

    \[\leadsto b \cdot \left(a \cdot -2\right) \]
14. Add Preprocessing

Developer target: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(a + b\right) \cdot \left(a - b\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (* (+ a b) (- a b)))

C

double code(double a, double b) {
	return (a + b) * (a - b);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (a + b) * (a - b)
end function

Java

public static double code(double a, double b) {
	return (a + b) * (a - b);
}

Python

def code(a, b):
	return (a + b) * (a - b)

Julia

function code(a, b)
	return Float64(Float64(a + b) * Float64(a - b))
end

MATLAB

function tmp = code(a, b)
	tmp = (a + b) * (a - b);
end

Wolfram

code[a_, b_] := N[(N[(a + b), $MachinePrecision] * N[(a - b), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024043 
(FPCore (a b)
  :name "Difference of squares"
  :precision binary64

  :herbie-target
  (* (+ a b) (- a b))

  (- (* a a) (* b b)))