Difference of squares

Percentage Accurate: 93.9% → 100.0%

Time: 4.0s

Alternatives: 3

Speedup: 1.0×

• 43.0% 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: 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]

Derivation

1. Initial program 95.3%

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

   1. add-sqr-sqrt 51.4%

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

   2. associate-*l* 51.5%

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

   3. prod-diff 40.5%

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

4. Applied egg-rr 40.5%

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

5. Taylor expanded in b around 0 52.2%

   \[\leadsto \mathsf{fma}\left(a, a, -\left(\sqrt{b} \cdot b\right) \cdot \sqrt{b}\right) + \color{blue}{0} \]
6. Step-by-step derivation

   1. fma-neg 51.5%

      \[\leadsto \color{blue}{\left(a \cdot a - \left(\sqrt{b} \cdot b\right) \cdot \sqrt{b}\right)} + 0 \]

   2. *-commutative 51.5%

      \[\leadsto \left(a \cdot a - \color{blue}{\left(b \cdot \sqrt{b}\right)} \cdot \sqrt{b}\right) + 0 \]

   3. associate-*r* 51.4%

      \[\leadsto \left(a \cdot a - \color{blue}{b \cdot \left(\sqrt{b} \cdot \sqrt{b}\right)}\right) + 0 \]

   4. add-sqr-sqrt 95.3%

      \[\leadsto \left(a \cdot a - b \cdot \color{blue}{b}\right) + 0 \]

   5. difference-of-squares 100.0%

      \[\leadsto \color{blue}{\left(a + b\right) \cdot \left(a - b\right)} + 0 \]

7. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left(a + b\right) \cdot \left(a - b\right)} + 0 \]

8. Final simplification 100.0%

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

Alternative 2: 97.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 10^{+297}:\\ \;\;\;\;a \cdot a - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(a - b\right) \cdot \left(a - b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 1e+297) (- (* a a) (* b b)) (* (- a b) (- a b))))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 1e+297) {
		tmp = (a * a) - (b * b);
	} else {
		tmp = (a - b) * (a - b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a * a) <= 1d+297) then
        tmp = (a * a) - (b * b)
    else
        tmp = (a - b) * (a - b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((a * a) <= 1e+297) {
		tmp = (a * a) - (b * b);
	} else {
		tmp = (a - b) * (a - b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a * a) <= 1e+297:
		tmp = (a * a) - (b * b)
	else:
		tmp = (a - b) * (a - b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 1e+297)
		tmp = Float64(Float64(a * a) - Float64(b * b));
	else
		tmp = Float64(Float64(a - b) * Float64(a - b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a * a) <= 1e+297)
		tmp = (a * a) - (b * b);
	else
		tmp = (a - b) * (a - b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 1e+297], N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(a - b), $MachinePrecision] * N[(a - b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 1e297

   1. Initial program 100.0%

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

   if 1e297 < (*.f64 a a)

   1. Initial program 80.3%

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

      1. add-sqr-sqrt 80.3%

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

      2. pow2 80.3%

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

      3. difference-of-squares 88.5%

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

      4. sqrt-prod 41.0%

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

      5. add-sqr-sqrt 16.4%

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

      6. sqrt-prod 41.0%

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

      7. sqr-neg 41.0%

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

      8. sqrt-unprod 24.6%

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

      9. add-sqr-sqrt 41.0%

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

      10. sub-neg 41.0%

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

      11. add-sqr-sqrt 88.5%

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

      12. add-sqr-sqrt 41.0%

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

      13. add-sqr-sqrt 16.4%

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

      14. difference-of-squares 16.4%

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

      15. unpow-prod-down 16.4%

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

   4. Applied egg-rr 16.4%

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

      1. unpow2 16.4%

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

      2. unpow2 16.4%

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

      3. unswap-sqr 16.4%

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

      4. difference-of-squares 16.4%

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

      5. rem-square-sqrt 16.4%

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

      6. rem-square-sqrt 16.4%

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

      7. difference-of-squares 16.4%

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

      8. rem-square-sqrt 42.6%

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

      9. rem-square-sqrt 88.5%

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

   6. Simplified 88.5%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 10^{+297}:\\ \;\;\;\;a \cdot a - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(a - b\right) \cdot \left(a - b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.6% 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]

Derivation

1. Initial program 95.3%

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

   1. add-sqr-sqrt 50.8%

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

   2. pow2 50.8%

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

   3. difference-of-squares 52.7%

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

   4. sqrt-prod 26.8%

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

   5. add-sqr-sqrt 14.0%

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

   6. sqrt-prod 27.1%

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

   7. sqr-neg 27.1%

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

   8. sqrt-unprod 13.5%

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

   9. add-sqr-sqrt 27.3%

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

   10. sub-neg 27.3%

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

   11. add-sqr-sqrt 52.7%

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

   12. add-sqr-sqrt 26.9%

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

   13. add-sqr-sqrt 13.9%

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

   14. difference-of-squares 13.9%

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

   15. unpow-prod-down 13.9%

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

4. Applied egg-rr 13.9%

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

   1. unpow2 13.9%

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

   2. unpow2 13.9%

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

   3. unswap-sqr 13.9%

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

   4. difference-of-squares 13.9%

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

   5. rem-square-sqrt 13.9%

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

   6. rem-square-sqrt 13.9%

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

   7. difference-of-squares 13.9%

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

   8. rem-square-sqrt 27.0%

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

   9. rem-square-sqrt 52.7%

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

6. Simplified 52.7%

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

7. Final simplification 52.7%

   \[\leadsto \left(a - b\right) \cdot \left(a - b\right) \]
8. 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 2024046 
(FPCore (a b)
  :name "Difference of squares"
  :precision binary64

  :alt
  (* (+ a b) (- a b))

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