Difference of squares

Percentage Accurate: 93.7% → 97.0%

Time: 3.4s

Alternatives: 4

Speedup: 0.5×

• 43.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 93.7% 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: 97.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.0%

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

   1. sqr-neg 91.0%

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

   2. cancel-sign-sub 91.0%

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

   3. fma-def 97.3%

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

3. Simplified 97.3%

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

5. Final simplification 97.3%

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

Alternative 2: 77.8% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (((b * b) <= 1d-216) .or. (.not. ((b * b) <= 1d-183)) .and. ((b * b) <= 1d+50)) then
        tmp = (a - b) * (a - b)
    else
        tmp = b * -b
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := If[Or[LessEqual[N[(b * b), $MachinePrecision], 1e-216], And[N[Not[LessEqual[N[(b * b), $MachinePrecision], 1e-183]], $MachinePrecision], LessEqual[N[(b * b), $MachinePrecision], 1e+50]]], N[(N[(a - b), $MachinePrecision] * N[(a - b), $MachinePrecision]), $MachinePrecision], N[(b * (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

      \[a \cdot a - b \cdot b \]
   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 49.6%

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

      3. sqrt-prod 92.3%

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

      4. sqr-neg 92.3%

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

      5. sqrt-unprod 42.7%

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

      6. add-sqr-sqrt 85.5%

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

      7. sub-neg 85.5%

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

      8. pow1 85.5%

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

      9. pow1 85.5%

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

      10. pow-prod-up 85.5%

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

      11. add-sqr-sqrt 42.6%

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

      12. add-sqr-sqrt 22.5%

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

      13. difference-of-squares 22.5%

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

      14. metadata-eval 22.5%

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

      15. unpow-prod-down 22.4%

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

   4. Applied egg-rr 22.4%

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

      1. unpow2 22.4%

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

      2. unpow2 22.4%

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

      3. unswap-sqr 22.5%

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

      4. difference-of-squares 22.5%

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

      5. unpow1/2 22.5%

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

      6. unpow1/2 22.5%

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

      7. pow-sqr 22.5%

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

      8. metadata-eval 22.5%

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

      9. unpow1 22.5%

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

      10. unpow1/2 22.5%

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

      11. unpow1/2 22.5%

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

      12. pow-sqr 22.5%

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

      13. metadata-eval 22.5%

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

      14. unpow1 22.5%

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

      15. difference-of-squares 22.5%

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

      16. unpow1/2 22.5%

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

      17. unpow1/2 22.5%

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

      18. pow-sqr 42.8%

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

      19. metadata-eval 42.8%

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

      20. unpow1 42.8%

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

   6. Simplified 85.5%

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

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

   1. Initial program 81.9%

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

   3. Taylor expanded in a around 0 81.4%

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

      1. mul-1-neg 81.4%

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

   5. Simplified 81.4%

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

      1. unpow2 81.4%

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

   7. Applied egg-rr 81.4%

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

4. Final simplification 83.5%

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

Alternative 3: 96.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+292) (- (* a a) (* b b)) (* b (- b))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+292) {
		tmp = (a * a) - (b * b);
	} else {
		tmp = b * -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b * b) <= 2d+292) then
        tmp = (a * a) - (b * b)
    else
        tmp = b * -b
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+292], N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision], N[(b * (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   if 2e292 < (*.f64 b b)

   1. Initial program 66.7%

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

   3. Taylor expanded in a around 0 89.9%

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

      1. mul-1-neg 89.9%

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

   5. Simplified 89.9%

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

      1. unpow2 89.9%

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

   7. Applied egg-rr 89.9%

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

4. Final simplification 97.3%

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

Alternative 4: 54.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.0%

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

3. Taylor expanded in a around 0 56.4%

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

   1. mul-1-neg 56.4%

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

5. Simplified 56.4%

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

   1. unpow2 56.4%

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

7. Applied egg-rr 56.4%

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

8. Final simplification 56.4%

   \[\leadsto b \cdot \left(-b\right) \]
9. 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 2024031 
(FPCore (a b)
  :name "Difference of squares"
  :precision binary64

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

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