Difference of squares

Percentage Accurate: 93.8% → 99.9%

Time: 3.7s

Alternatives: 3

Speedup: 0.5×

• 43.5% 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: 93.8% 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

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;a_m \cdot a_m \leq 5 \cdot 10^{+287}:\\ \;\;\;\;a_m \cdot a_m - b_m \cdot b_m\\ \mathbf{else}:\\ \;\;\;\;a_m \cdot \left(a_m + b_m \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
(FPCore (a_m b_m)
 :precision binary64
 (if (<= (* a_m a_m) 5e+287)
   (- (* 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) <= 5e+287) {
		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) <= 5d+287) 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) <= 5e+287) {
		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) <= 5e+287:
		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) <= 5e+287)
		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) <= 5e+287)
		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], 5e+287], 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) < 5e287

   1. Initial program 100.0%

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

   if 5e287 < (*.f64 a a)

   1. Initial program 80.3%

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

      1. difference-of-squares 100.0%

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

      2. add-sqr-sqrt 55.3%

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

      3. sqrt-prod 89.5%

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

      4. sqr-neg 89.5%

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

      5. sqrt-unprod 42.1%

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

      6. add-sqr-sqrt 92.1%

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

      7. sub-neg 92.1%

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

      8. pow1 92.1%

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

      9. pow1 92.1%

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

      10. pow-prod-up 92.1%

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

      11. add-sqr-sqrt 42.1%

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

      12. add-sqr-sqrt 23.7%

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

      13. difference-of-squares 23.7%

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

      14. metadata-eval 23.7%

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

      15. unpow-prod-down 23.7%

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

   4. Applied egg-rr 23.7%

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

      1. unpow2 23.7%

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

      2. unpow2 23.7%

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

      3. unswap-sqr 23.7%

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

      4. difference-of-squares 23.7%

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

      5. unpow1/2 23.7%

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

      6. unpow1/2 23.7%

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

      7. pow-sqr 23.7%

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

      8. metadata-eval 23.7%

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

      9. unpow1 23.7%

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

      10. unpow1/2 23.7%

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

      11. unpow1/2 23.7%

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

      12. pow-sqr 23.7%

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

      13. metadata-eval 23.7%

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

      14. unpow1 23.7%

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

      15. difference-of-squares 23.7%

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

      16. unpow1/2 23.7%

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

      17. unpow1/2 23.7%

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

      18. pow-sqr 50.0%

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

      19. metadata-eval 50.0%

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

      20. unpow1 50.0%

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

   6. Simplified 92.1%

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

   7. Taylor expanded in a around inf 82.9%

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

      1. *-commutative 82.9%

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

      2. associate-*l* 82.9%

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

      3. unpow2 82.9%

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

      4. distribute-lft-out 94.7%

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

   9. Simplified 94.7%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 5 \cdot 10^{+287}:\\ \;\;\;\;a \cdot a - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a + b \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 61.2% 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 94.1%

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

   1. difference-of-squares 100.0%

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

   2. add-sqr-sqrt 51.9%

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

   3. sqrt-prod 79.2%

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

   4. sqr-neg 79.2%

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

   5. sqrt-unprod 29.6%

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

   6. add-sqr-sqrt 58.9%

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

   7. sub-neg 58.9%

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

   8. pow1 58.9%

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

   9. pow1 58.9%

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

   10. pow-prod-up 58.9%

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

   11. add-sqr-sqrt 28.3%

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

   12. add-sqr-sqrt 15.8%

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

   13. difference-of-squares 15.8%

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

   14. metadata-eval 15.8%

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

   15. unpow-prod-down 15.8%

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

4. Applied egg-rr 15.8%

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

   1. unpow2 15.8%

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

   2. unpow2 15.8%

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

   3. unswap-sqr 15.8%

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

   4. difference-of-squares 15.8%

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

   5. unpow1/2 15.8%

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

   6. unpow1/2 15.8%

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

   7. pow-sqr 15.8%

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

   8. metadata-eval 15.8%

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

   9. unpow1 15.8%

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

   10. unpow1/2 15.8%

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

   11. unpow1/2 15.8%

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

   12. pow-sqr 15.8%

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

   13. metadata-eval 15.8%

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

   14. unpow1 15.8%

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

   15. difference-of-squares 15.8%

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

   16. unpow1/2 15.8%

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

   17. unpow1/2 15.8%

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

   18. pow-sqr 29.3%

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

   19. metadata-eval 29.3%

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

   20. unpow1 29.3%

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

6. Simplified 58.9%

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

7. Taylor expanded in a around inf 58.3%

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

   1. *-commutative 58.3%

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

   2. associate-*l* 58.3%

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

   3. unpow2 58.3%

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

   4. distribute-lft-out 61.8%

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

9. Simplified 61.8%

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

10. Final simplification 61.8%

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

Alternative 3: 15.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

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 = (-2.0d0) * (a_m * b_m)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

   1. difference-of-squares 100.0%

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

   2. add-sqr-sqrt 51.9%

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

   3. sqrt-prod 79.2%

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

   4. sqr-neg 79.2%

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

   5. sqrt-unprod 29.6%

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

   6. add-sqr-sqrt 58.9%

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

   7. sub-neg 58.9%

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

   8. pow1 58.9%

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

   9. pow1 58.9%

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

   10. pow-prod-up 58.9%

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

   11. add-sqr-sqrt 28.3%

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

   12. add-sqr-sqrt 15.8%

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

   13. difference-of-squares 15.8%

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

   14. metadata-eval 15.8%

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

   15. unpow-prod-down 15.8%

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

4. Applied egg-rr 15.8%

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

   1. unpow2 15.8%

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

   2. unpow2 15.8%

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

   3. unswap-sqr 15.8%

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

   4. difference-of-squares 15.8%

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

   5. unpow1/2 15.8%

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

   6. unpow1/2 15.8%

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

   7. pow-sqr 15.8%

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

   8. metadata-eval 15.8%

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

   9. unpow1 15.8%

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

   10. unpow1/2 15.8%

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

   11. unpow1/2 15.8%

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

   12. pow-sqr 15.8%

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

   13. metadata-eval 15.8%

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

   14. unpow1 15.8%

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

   15. difference-of-squares 15.8%

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

   16. unpow1/2 15.8%

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

   17. unpow1/2 15.8%

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

   18. pow-sqr 29.3%

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

   19. metadata-eval 29.3%

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

   20. unpow1 29.3%

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

6. Simplified 58.9%

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

7. Taylor expanded in a around inf 58.3%

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

   1. *-commutative 58.3%

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

   2. associate-*l* 58.3%

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

   3. unpow2 58.3%

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

   4. distribute-lft-out 61.8%

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

9. Simplified 61.8%

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

10. Taylor expanded in a around 0 14.2%

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

11. Final simplification 14.2%

    \[\leadsto -2 \cdot \left(a \cdot b\right) \]
12. 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 2024017 
(FPCore (a b)
  :name "Difference of squares"
  :precision binary64

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

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