Difference of squares

Percentage Accurate: 93.7% → 97.0%

Time: 3.6s

Alternatives: 4

Speedup: 0.5×

• 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: 93.7% 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 94.1%

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

   1. sqr-neg 94.1%

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

   2. cancel-sign-sub 94.1%

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

   3. fma-def 98.0%

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

3. Simplified 98.0%

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

5. Final simplification 98.0%

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

Alternative 2: 77.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+44}:\\ \;\;\;\;\left(a - b\right) \cdot \left(a - b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(-b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 1e+44) (* (- a b) (- a b)) (* b (- b))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 1e+44) {
		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+44) 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+44) {
		tmp = (a - b) * (a - b);
	} else {
		tmp = b * -b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 1e+44:
		tmp = (a - b) * (a - b)
	else:
		tmp = b * -b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 1e+44)
		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+44)
		tmp = (a - b) * (a - b);
	else
		tmp = b * -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 1e+44], 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) < 1.0000000000000001e44

   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 44.6%

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

      3. sqrt-prod 91.1%

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

      4. sqr-neg 91.1%

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

      5. sqrt-unprod 46.4%

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

      6. add-sqr-sqrt 83.0%

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

      7. sub-neg 83.0%

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

      8. pow1 83.0%

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

      9. pow1 83.0%

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

      10. pow-prod-up 83.0%

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

      11. add-sqr-sqrt 46.1%

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

      12. add-sqr-sqrt 19.7%

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

      13. difference-of-squares 19.7%

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

      14. metadata-eval 19.7%

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

      15. unpow-prod-down 19.7%

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

   4. Applied egg-rr 19.7%

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

      1. unpow2 19.7%

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

      2. unpow2 19.7%

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

      3. unswap-sqr 19.7%

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

      4. difference-of-squares 19.7%

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

      5. unpow1/2 19.7%

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

      6. unpow1/2 19.7%

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

      7. pow-sqr 19.8%

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

      8. metadata-eval 19.8%

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

      9. unpow1 19.8%

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

      10. unpow1/2 19.8%

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

      11. unpow1/2 19.8%

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

      12. pow-sqr 19.8%

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

      13. metadata-eval 19.8%

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

      14. unpow1 19.8%

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

      15. difference-of-squares 19.8%

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

      16. unpow1/2 19.8%

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

      17. unpow1/2 19.8%

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

      18. pow-sqr 36.6%

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

      19. metadata-eval 36.6%

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

      20. unpow1 36.6%

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

   6. Simplified 83.0%

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

   if 1.0000000000000001e44 < (*.f64 b b)

   1. Initial program 87.9%

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

   3. Taylor expanded in a around 0 83.5%

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

      1. mul-1-neg 83.5%

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

   5. Simplified 83.5%

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

      1. unpow2 83.5%

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

   7. Applied egg-rr 83.5%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 10^{+44}:\\ \;\;\;\;\left(a - b\right) \cdot \left(a - b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(-b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+291}:\\ \;\;\;\;a \cdot a - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(-b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+291) (- (* a a) (* b b)) (* b (- b))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+291) {
		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) <= 5d+291) 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) <= 5e+291) {
		tmp = (a * a) - (b * b);
	} else {
		tmp = b * -b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e+291:
		tmp = (a * a) - (b * b)
	else:
		tmp = b * -b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+291)
		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) <= 5e+291)
		tmp = (a * a) - (b * b);
	else
		tmp = b * -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+291], 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) < 5.0000000000000001e291

   1. Initial program 100.0%

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

   if 5.0000000000000001e291 < (*.f64 b b)

   1. Initial program 78.3%

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

   3. Taylor expanded in a around 0 92.7%

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

      1. mul-1-neg 92.7%

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

   5. Simplified 92.7%

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

      1. unpow2 92.8%

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

   7. Applied egg-rr 92.8%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+291}:\\ \;\;\;\;a \cdot a - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(-b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.3% 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 94.1%

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

3. Taylor expanded in a around 0 56.3%

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

   1. mul-1-neg 56.3%

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

5. Simplified 56.3%

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

   1. unpow2 56.3%

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

7. Applied egg-rr 56.3%

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

8. Final simplification 56.3%

   \[\leadsto b \cdot \left(-b\right) \]
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 2024020 
(FPCore (a b)
  :name "Difference of squares"
  :precision binary64

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

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