Difference of squares

Percentage Accurate: 93.5% → 100.0%

Time: 6.7s

Alternatives: 4

Speedup: 1.0×

• 44.8% 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: 93.5% 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 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. sub-neg 100.0%

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

   3. add-sqr-sqrt 48.3%

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

   4. sqrt-unprod 75.5%

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

   5. sqr-neg 75.5%

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

   6. sqrt-prod 28.2%

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

   7. add-sqr-sqrt 55.2%

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

4. Applied egg-rr 55.2%

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

   1. add-sqr-sqrt 28.2%

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

   2. sqrt-prod 75.5%

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

   3. add-sqr-sqrt 28.2%

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

   4. add-sqr-sqrt 28.2%

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

   5. sqr-neg 28.2%

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

   6. swap-sqr 28.2%

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

   7. sqrt-unprod 0.0%

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

   8. add-sqr-sqrt 51.4%

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

   9. distribute-rgt-neg-out 51.4%

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

   10. add-sqr-sqrt 100.0%

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

   11. sub-neg 100.0%

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

6. Applied egg-rr 100.0%

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

Alternative 2: 80.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-185}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \cdot b \leq 2 \cdot 10^{-133}:\\ \;\;\;\;b \cdot \left(-b\right)\\ \mathbf{elif}\;b \cdot b \leq 2 \cdot 10^{-61}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(a - b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e-185)
   (* a a)
   (if (<= (* b b) 2e-133)
     (* b (- b))
     (if (<= (* b b) 2e-61) (* a a) (* b (- a b))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e-185) {
		tmp = a * a;
	} else if ((b * b) <= 2e-133) {
		tmp = b * -b;
	} else if ((b * b) <= 2e-61) {
		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-185) then
        tmp = a * a
    else if ((b * b) <= 2d-133) then
        tmp = b * -b
    else if ((b * b) <= 2d-61) 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-185) {
		tmp = a * a;
	} else if ((b * b) <= 2e-133) {
		tmp = b * -b;
	} else if ((b * b) <= 2e-61) {
		tmp = a * a;
	} else {
		tmp = b * (a - b);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2e-185:
		tmp = a * a
	elif (b * b) <= 2e-133:
		tmp = b * -b
	elif (b * b) <= 2e-61:
		tmp = a * a
	else:
		tmp = b * (a - b)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e-185)
		tmp = Float64(a * a);
	elseif (Float64(b * b) <= 2e-133)
		tmp = Float64(b * Float64(-b));
	elseif (Float64(b * b) <= 2e-61)
		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-185)
		tmp = a * a;
	elseif ((b * b) <= 2e-133)
		tmp = b * -b;
	elseif ((b * b) <= 2e-61)
		tmp = a * a;
	else
		tmp = b * (a - b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e-185], N[(a * a), $MachinePrecision], If[LessEqual[N[(b * b), $MachinePrecision], 2e-133], N[(b * (-b)), $MachinePrecision], If[LessEqual[N[(b * b), $MachinePrecision], 2e-61], N[(a * a), $MachinePrecision], N[(b * N[(a - b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 b b) < 2e-185 or 2.0000000000000001e-133 < (*.f64 b b) < 2.0000000000000001e-61

   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 45.1%

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

      4. sqrt-unprod 93.7%

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

      5. sqr-neg 93.7%

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

      6. sqrt-prod 48.6%

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

      7. add-sqr-sqrt 91.2%

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

   4. Applied egg-rr 91.2%

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

   5. Taylor expanded in a around inf 91.4%

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

   6. Taylor expanded in a around inf 91.9%

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

   if 2e-185 < (*.f64 b b) < 2.0000000000000001e-133

   1. Initial program 99.9%

      \[a \cdot a - b \cdot b \]
   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 43.4%

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

      4. sqrt-unprod 57.7%

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

      5. sqr-neg 57.7%

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

      6. sqrt-prod 13.9%

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

      7. add-sqr-sqrt 27.4%

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

   4. Applied egg-rr 27.4%

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

      1. add-sqr-sqrt 13.9%

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

      2. sqrt-prod 57.7%

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

      3. add-sqr-sqrt 13.9%

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

      4. add-sqr-sqrt 13.9%

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

      5. sqr-neg 13.9%

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

      6. swap-sqr 13.9%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 55.8%

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

      9. distribute-rgt-neg-out 55.8%

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

      10. add-sqr-sqrt 100.0%

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

      11. sub-neg 100.0%

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

   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 72.5%

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

   8. Taylor expanded in a around 0 74.2%

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

      1. neg-mul-1 74.2%

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

   10. Simplified 74.2%

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

   if 2.0000000000000001e-61 < (*.f64 b b)

   1. Initial program 88.2%

      \[a \cdot a - b \cdot b \]
   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 51.8%

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

      4. sqrt-unprod 61.5%

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

      5. sqr-neg 61.5%

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

      6. sqrt-prod 11.9%

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

      7. add-sqr-sqrt 26.8%

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

   4. Applied egg-rr 26.8%

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

      1. add-sqr-sqrt 11.9%

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

      2. sqrt-prod 61.5%

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

      3. add-sqr-sqrt 11.9%

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

      4. add-sqr-sqrt 11.9%

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

      5. sqr-neg 11.9%

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

      6. swap-sqr 11.9%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 47.9%

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

      9. distribute-rgt-neg-out 47.9%

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

      10. add-sqr-sqrt 100.0%

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

      11. sub-neg 100.0%

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

   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 78.8%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{-185}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \cdot b \leq 2 \cdot 10^{-133}:\\ \;\;\;\;b \cdot \left(-b\right)\\ \mathbf{elif}\;b \cdot b \leq 2 \cdot 10^{-61}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(a - b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 6 \cdot 10^{-72} \lor \neg \left(a \cdot a \leq 1.5 \cdot 10^{+61}\right) \land a \cdot a \leq 10^{+167}:\\ \;\;\;\;b \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (or (<= (* a a) 6e-72)
         (and (not (<= (* a a) 1.5e+61)) (<= (* a a) 1e+167)))
   (* b (- b))
   (* a a)))

C

double code(double a, double b) {
	double tmp;
	if (((a * a) <= 6e-72) || (!((a * a) <= 1.5e+61) && ((a * a) <= 1e+167))) {
		tmp = b * -b;
	} else {
		tmp = a * a;
	}
	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) <= 6d-72) .or. (.not. ((a * a) <= 1.5d+61)) .and. ((a * a) <= 1d+167)) then
        tmp = b * -b
    else
        tmp = a * a
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (((a * a) <= 6e-72) || (!((a * a) <= 1.5e+61) && ((a * a) <= 1e+167))) {
		tmp = b * -b;
	} else {
		tmp = a * a;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if ((a * a) <= 6e-72) or (not ((a * a) <= 1.5e+61) and ((a * a) <= 1e+167)):
		tmp = b * -b
	else:
		tmp = a * a
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if ((Float64(a * a) <= 6e-72) || (!(Float64(a * a) <= 1.5e+61) && (Float64(a * a) <= 1e+167)))
		tmp = Float64(b * Float64(-b));
	else
		tmp = Float64(a * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (((a * a) <= 6e-72) || (~(((a * a) <= 1.5e+61)) && ((a * a) <= 1e+167)))
		tmp = b * -b;
	else
		tmp = a * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[Or[LessEqual[N[(a * a), $MachinePrecision], 6e-72], And[N[Not[LessEqual[N[(a * a), $MachinePrecision], 1.5e+61]], $MachinePrecision], LessEqual[N[(a * a), $MachinePrecision], 1e+167]]], N[(b * (-b)), $MachinePrecision], N[(a * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 6e-72 or 1.5e61 < (*.f64 a a) < 1e167

   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 46.1%

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

      4. sqrt-unprod 65.1%

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

      5. sqr-neg 65.1%

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

      6. sqrt-prod 18.8%

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

      7. add-sqr-sqrt 29.9%

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

   4. Applied egg-rr 29.9%

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

      1. add-sqr-sqrt 18.8%

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

      2. sqrt-prod 65.1%

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

      3. add-sqr-sqrt 18.8%

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

      4. add-sqr-sqrt 18.8%

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

      5. sqr-neg 18.8%

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

      6. swap-sqr 18.8%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 53.5%

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

      9. distribute-rgt-neg-out 53.5%

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

      10. add-sqr-sqrt 100.0%

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

      11. sub-neg 100.0%

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

   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 83.2%

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

   8. Taylor expanded in a around 0 83.7%

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

      1. neg-mul-1 83.7%

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

   10. Simplified 83.7%

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

   if 6e-72 < (*.f64 a a) < 1.5e61 or 1e167 < (*.f64 a a)

   1. Initial program 87.7%

      \[a \cdot a - b \cdot b \]
   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 50.8%

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

      4. sqrt-unprod 86.9%

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

      5. sqr-neg 86.9%

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

      6. sqrt-prod 38.5%

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

      7. add-sqr-sqrt 83.0%

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

   4. Applied egg-rr 83.0%

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

   5. Taylor expanded in a around inf 89.2%

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

   6. Taylor expanded in a around inf 83.7%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 6 \cdot 10^{-72} \lor \neg \left(a \cdot a \leq 1.5 \cdot 10^{+61}\right) \land a \cdot a \leq 10^{+167}:\\ \;\;\;\;b \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.0% 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 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. sub-neg 100.0%

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

   3. add-sqr-sqrt 48.3%

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

   4. sqrt-unprod 75.5%

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

   5. sqr-neg 75.5%

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

   6. sqrt-prod 28.2%

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

   7. add-sqr-sqrt 55.2%

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

4. Applied egg-rr 55.2%

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

5. Taylor expanded in a around inf 59.1%

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

6. Taylor expanded in a around inf 56.1%

   \[\leadsto \color{blue}{a} \cdot a \]
7. 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 2024110 
(FPCore (a b)
  :name "Difference of squares"
  :precision binary64

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

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