Difference of squares

Percentage Accurate: 94.0% → 100.0%

Time: 5.8s

Alternatives: 5

Speedup: 1.0×

• 42.9% 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: 94.0% 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 92.6%

   \[a \cdot a - b \cdot b \]
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. sub-neg 100.0%

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

   3. add-sqr-sqrt 44.8%

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

   4. sqrt-unprod 71.2%

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

   5. sqr-neg 71.2%

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

   6. sqrt-prod 27.5%

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

   7. add-sqr-sqrt 50.6%

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

4. Applied egg-rr 50.6%

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

   1. add-sqr-sqrt 27.5%

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

   2. add-sqr-sqrt 27.5%

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

   3. sqr-neg 27.5%

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

   4. sqrt-unprod 0.0%

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

   5. add-sqr-sqrt 55.0%

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

   6. cancel-sign-sub-inv 55.0%

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

   7. add-sqr-sqrt 100.0%

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

6. Applied egg-rr 100.0%

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

Alternative 2: 81.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-7}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(a - b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= (* b b) 2e-7) (* a a) (* b (- a b))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e-7) {
		tmp = a * a;
	} else {
		tmp = 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 ((b * b) <= 2d-7) then
        tmp = a * a
    else
        tmp = b * (a - b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e-7) {
		tmp = a * a;
	} else {
		tmp = b * (a - b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2e-7:
		tmp = a * a
	else:
		tmp = b * (a - b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e-7)
		tmp = Float64(a * a);
	else
		tmp = Float64(b * Float64(a - b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 2e-7)
		tmp = a * a;
	else
		tmp = b * (a - b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e-7], N[(a * a), $MachinePrecision], N[(b * N[(a - b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.9999999999999999e-7

   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. sub-neg 100.0%

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

      3. add-sqr-sqrt 44.6%

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

      4. sqrt-unprod 87.4%

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

      5. sqr-neg 87.4%

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

      6. sqrt-prod 42.8%

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

      7. add-sqr-sqrt 80.7%

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

   4. Applied egg-rr 80.7%

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

   5. Taylor expanded in a around inf 81.2%

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

   6. Taylor expanded in a around inf 81.0%

      \[\leadsto \color{blue}{a} \cdot a \]

   if 1.9999999999999999e-7 < (*.f64 b b)

   1. Initial program 84.9%

      \[a \cdot a - b \cdot b \]
   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. sub-neg 100.0%

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

      3. add-sqr-sqrt 45.1%

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

      4. sqrt-unprod 54.5%

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

      5. sqr-neg 54.5%

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

      6. sqrt-prod 11.6%

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

      7. add-sqr-sqrt 19.6%

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

   4. Applied egg-rr 19.6%

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

      1. add-sqr-sqrt 11.6%

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

      2. add-sqr-sqrt 11.6%

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

      3. sqr-neg 11.6%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 54.6%

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

      6. cancel-sign-sub-inv 54.6%

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

      7. add-sqr-sqrt 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in a around 0 86.1%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-7}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(a - b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-7}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(-b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= (* b b) 2e-7) (* a a) (* b (- b))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e-7) {
		tmp = a * a;
	} 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-7) then
        tmp = a * a
    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-7) {
		tmp = a * a;
	} else {
		tmp = b * -b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2e-7:
		tmp = a * a
	else:
		tmp = b * -b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e-7)
		tmp = Float64(a * a);
	else
		tmp = Float64(b * Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 2e-7)
		tmp = a * a;
	else
		tmp = b * -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e-7], N[(a * a), $MachinePrecision], N[(b * (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 1.9999999999999999e-7

   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. sub-neg 100.0%

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

      3. add-sqr-sqrt 44.6%

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

      4. sqrt-unprod 87.4%

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

      5. sqr-neg 87.4%

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

      6. sqrt-prod 42.8%

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

      7. add-sqr-sqrt 80.7%

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

   4. Applied egg-rr 80.7%

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

   5. Taylor expanded in a around inf 81.2%

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

   6. Taylor expanded in a around inf 81.0%

      \[\leadsto \color{blue}{a} \cdot a \]

   if 1.9999999999999999e-7 < (*.f64 b b)

   1. Initial program 84.9%

      \[a \cdot a - b \cdot b \]
   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. sub-neg 100.0%

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

      3. add-sqr-sqrt 45.1%

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

      4. sqrt-unprod 54.5%

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

      5. sqr-neg 54.5%

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

      6. sqrt-prod 11.6%

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

      7. add-sqr-sqrt 19.6%

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

   4. Applied egg-rr 19.6%

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

      1. add-sqr-sqrt 11.6%

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

      2. add-sqr-sqrt 11.6%

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

      3. sqr-neg 11.6%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 54.6%

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

      6. cancel-sign-sub-inv 54.6%

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

      7. add-sqr-sqrt 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in a around 0 83.7%

      \[\leadsto \color{blue}{\left(-1 \cdot b\right)} \cdot \left(a + b\right) \]
   8. Step-by-step derivation

      1. neg-mul-1 83.7%

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

   9. Simplified 83.7%

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

   10. Taylor expanded in a around 0 79.7%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-7}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(-b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.8 \cdot 10^{+241}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b 4.8e+241) (* a a) (* a (- b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 4.8e+241) {
		tmp = a * a;
	} else {
		tmp = 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 (b <= 4.8d+241) then
        tmp = a * a
    else
        tmp = a * -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 4.8e+241) {
		tmp = a * a;
	} else {
		tmp = a * -b;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 4.8e+241:
		tmp = a * a
	else:
		tmp = a * -b
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 4.8e+241)
		tmp = Float64(a * a);
	else
		tmp = Float64(a * Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 4.8e+241)
		tmp = a * a;
	else
		tmp = a * -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 4.8e+241], N[(a * a), $MachinePrecision], N[(a * (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.7999999999999998e241

   1. Initial program 94.9%

      \[a \cdot a - b \cdot b \]
   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. sub-neg 100.0%

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

      3. add-sqr-sqrt 48.9%

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

      4. sqrt-unprod 77.2%

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

      5. sqr-neg 77.2%

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

      6. sqrt-prod 29.5%

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

      7. add-sqr-sqrt 54.7%

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

   4. Applied egg-rr 54.7%

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

   5. Taylor expanded in a around inf 57.6%

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

   6. Taylor expanded in a around inf 55.4%

      \[\leadsto \color{blue}{a} \cdot a \]

   if 4.7999999999999998e241 < b

   1. Initial program 66.7%

      \[a \cdot a - b \cdot b \]
   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. sub-neg 100.0%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 4.8%

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

      5. sqr-neg 4.8%

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

      6. sqrt-prod 4.8%

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

      7. add-sqr-sqrt 4.8%

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

   4. Applied egg-rr 4.8%

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

      1. add-sqr-sqrt 4.8%

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

      2. add-sqr-sqrt 4.8%

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

      3. sqr-neg 4.8%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. cancel-sign-sub-inv 100.0%

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

      7. add-sqr-sqrt 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in a around 0 95.2%

      \[\leadsto \color{blue}{\left(-1 \cdot b\right)} \cdot \left(a + b\right) \]
   8. Step-by-step derivation

      1. neg-mul-1 95.2%

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

   9. Simplified 95.2%

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

   10. Taylor expanded in a around inf 34.1%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.8 \cdot 10^{+241}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.9% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.6%

   \[a \cdot a - b \cdot b \]
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. sub-neg 100.0%

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

   3. add-sqr-sqrt 44.8%

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

   4. sqrt-unprod 71.2%

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

   5. sqr-neg 71.2%

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

   6. sqrt-prod 27.5%

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

   7. add-sqr-sqrt 50.6%

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

4. Applied egg-rr 50.6%

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

5. Taylor expanded in a around inf 54.6%

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

6. Taylor expanded in a around inf 51.3%

   \[\leadsto \color{blue}{a} \cdot a \]
7. Add Preprocessing

Developer Target 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]

Reproduce

herbie shell --seed 2024121 
(FPCore (a b)
  :name "Difference of squares"
  :precision binary64

  :alt
  (! :herbie-platform default (* (+ a b) (- a b)))

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