Difference of squares

Percentage Accurate: 94.0% → 100.0%

Time: 4.7s

Alternatives: 6

Speedup: 0.6×

• 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: 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, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+307}:\\ \;\;\;\;a \cdot a - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} \cdot \mathsf{fma}\left(a, \frac{1}{b} \cdot \frac{a}{b}, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+307)
   (- (* a a) (* b b))
   (* (pow b 2.0) (fma a (* (/ 1.0 b) (/ a b)) -1.0))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+307) {
		tmp = (a * a) - (b * b);
	} else {
		tmp = pow(b, 2.0) * fma(a, ((1.0 / b) * (a / b)), -1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+307)
		tmp = Float64(Float64(a * a) - Float64(b * b));
	else
		tmp = Float64((b ^ 2.0) * fma(a, Float64(Float64(1.0 / b) * Float64(a / b)), -1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+307], N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] * N[(a * N[(N[(1.0 / b), $MachinePrecision] * N[(a / b), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5e307

   1. Initial program 100.0%

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

   if 5e307 < (*.f64 b b)

   1. Initial program 71.7%

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

   3. Taylor expanded in b around inf 71.7%

      \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{{a}^{2}}{{b}^{2}} - 1\right)} \]
   4. Step-by-step derivation

      1. unpow2 71.7%

         \[\leadsto {b}^{2} \cdot \left(\frac{\color{blue}{a \cdot a}}{{b}^{2}} - 1\right) \]

      2. associate-/l* 81.7%

         \[\leadsto {b}^{2} \cdot \left(\color{blue}{a \cdot \frac{a}{{b}^{2}}} - 1\right) \]

      3. fma-neg 81.7%

         \[\leadsto {b}^{2} \cdot \color{blue}{\mathsf{fma}\left(a, \frac{a}{{b}^{2}}, -1\right)} \]

      4. metadata-eval 81.7%

         \[\leadsto {b}^{2} \cdot \mathsf{fma}\left(a, \frac{a}{{b}^{2}}, \color{blue}{-1}\right) \]

   5. Simplified 81.7%

      \[\leadsto \color{blue}{{b}^{2} \cdot \mathsf{fma}\left(a, \frac{a}{{b}^{2}}, -1\right)} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 81.7%

         \[\leadsto {b}^{2} \cdot \mathsf{fma}\left(a, \frac{\color{blue}{1 \cdot a}}{{b}^{2}}, -1\right) \]

      2. unpow2 81.7%

         \[\leadsto {b}^{2} \cdot \mathsf{fma}\left(a, \frac{1 \cdot a}{\color{blue}{b \cdot b}}, -1\right) \]

      3. times-frac 100.0%

         \[\leadsto {b}^{2} \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{b} \cdot \frac{a}{b}}, -1\right) \]

   7. Applied egg-rr 100.0%

      \[\leadsto {b}^{2} \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{b} \cdot \frac{a}{b}}, -1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 100.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+307}:\\ \;\;\;\;a \cdot a - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} \cdot \left(-1 + {\left(\frac{a}{b}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+307)
   (- (* a a) (* b b))
   (* (pow b 2.0) (+ -1.0 (pow (/ a b) 2.0)))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+307) {
		tmp = (a * a) - (b * b);
	} else {
		tmp = pow(b, 2.0) * (-1.0 + pow((a / b), 2.0));
	}
	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+307) then
        tmp = (a * a) - (b * b)
    else
        tmp = (b ** 2.0d0) * ((-1.0d0) + ((a / b) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+307) {
		tmp = (a * a) - (b * b);
	} else {
		tmp = Math.pow(b, 2.0) * (-1.0 + Math.pow((a / b), 2.0));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e+307:
		tmp = (a * a) - (b * b)
	else:
		tmp = math.pow(b, 2.0) * (-1.0 + math.pow((a / b), 2.0))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+307)
		tmp = Float64(Float64(a * a) - Float64(b * b));
	else
		tmp = Float64((b ^ 2.0) * Float64(-1.0 + (Float64(a / b) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e+307)
		tmp = (a * a) - (b * b);
	else
		tmp = (b ^ 2.0) * (-1.0 + ((a / b) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+307], N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] * N[(-1.0 + N[Power[N[(a / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5e307

   1. Initial program 100.0%

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

   if 5e307 < (*.f64 b b)

   1. Initial program 71.7%

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

   3. Taylor expanded in b around inf 71.7%

      \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{{a}^{2}}{{b}^{2}} - 1\right)} \]
   4. Step-by-step derivation

      1. unpow2 71.7%

         \[\leadsto {b}^{2} \cdot \left(\frac{\color{blue}{a \cdot a}}{{b}^{2}} - 1\right) \]

      2. associate-/l* 81.7%

         \[\leadsto {b}^{2} \cdot \left(\color{blue}{a \cdot \frac{a}{{b}^{2}}} - 1\right) \]

      3. fma-neg 81.7%

         \[\leadsto {b}^{2} \cdot \color{blue}{\mathsf{fma}\left(a, \frac{a}{{b}^{2}}, -1\right)} \]

      4. metadata-eval 81.7%

         \[\leadsto {b}^{2} \cdot \mathsf{fma}\left(a, \frac{a}{{b}^{2}}, \color{blue}{-1}\right) \]

   5. Simplified 81.7%

      \[\leadsto \color{blue}{{b}^{2} \cdot \mathsf{fma}\left(a, \frac{a}{{b}^{2}}, -1\right)} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 81.7%

         \[\leadsto {b}^{2} \cdot \mathsf{fma}\left(a, \frac{\color{blue}{1 \cdot a}}{{b}^{2}}, -1\right) \]

      2. unpow2 81.7%

         \[\leadsto {b}^{2} \cdot \mathsf{fma}\left(a, \frac{1 \cdot a}{\color{blue}{b \cdot b}}, -1\right) \]

      3. times-frac 100.0%

         \[\leadsto {b}^{2} \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{b} \cdot \frac{a}{b}}, -1\right) \]

   7. Applied egg-rr 100.0%

      \[\leadsto {b}^{2} \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{b} \cdot \frac{a}{b}}, -1\right) \]
   8. Step-by-step derivation

      1. fma-undefine 100.0%

         \[\leadsto {b}^{2} \cdot \color{blue}{\left(a \cdot \left(\frac{1}{b} \cdot \frac{a}{b}\right) + -1\right)} \]

      2. associate-*r* 100.0%

         \[\leadsto {b}^{2} \cdot \left(\color{blue}{\left(a \cdot \frac{1}{b}\right) \cdot \frac{a}{b}} + -1\right) \]

      3. div-inv 100.0%

         \[\leadsto {b}^{2} \cdot \left(\color{blue}{\frac{a}{b}} \cdot \frac{a}{b} + -1\right) \]

      4. pow2 100.0%

         \[\leadsto {b}^{2} \cdot \left(\color{blue}{{\left(\frac{a}{b}\right)}^{2}} + -1\right) \]

   9. Applied egg-rr 100.0%

      \[\leadsto {b}^{2} \cdot \color{blue}{\left({\left(\frac{a}{b}\right)}^{2} + -1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+307}:\\ \;\;\;\;a \cdot a - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} \cdot \left(-1 + {\left(\frac{a}{b}\right)}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+307}:\\ \;\;\;\;a \cdot a - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} \cdot \left(-1 + \frac{\frac{a}{b}}{\frac{b}{a}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 5e+307)
   (- (* a a) (* b b))
   (* (pow b 2.0) (+ -1.0 (/ (/ a b) (/ b a))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+307) {
		tmp = (a * a) - (b * b);
	} else {
		tmp = pow(b, 2.0) * (-1.0 + ((a / b) / (b / a)));
	}
	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+307) then
        tmp = (a * a) - (b * b)
    else
        tmp = (b ** 2.0d0) * ((-1.0d0) + ((a / b) / (b / a)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 5e+307) {
		tmp = (a * a) - (b * b);
	} else {
		tmp = Math.pow(b, 2.0) * (-1.0 + ((a / b) / (b / a)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 5e+307:
		tmp = (a * a) - (b * b)
	else:
		tmp = math.pow(b, 2.0) * (-1.0 + ((a / b) / (b / a)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 5e+307)
		tmp = Float64(Float64(a * a) - Float64(b * b));
	else
		tmp = Float64((b ^ 2.0) * Float64(-1.0 + Float64(Float64(a / b) / Float64(b / a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 5e+307)
		tmp = (a * a) - (b * b);
	else
		tmp = (b ^ 2.0) * (-1.0 + ((a / b) / (b / a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 5e+307], N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] * N[(-1.0 + N[(N[(a / b), $MachinePrecision] / N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 5e307

   1. Initial program 100.0%

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

   if 5e307 < (*.f64 b b)

   1. Initial program 71.7%

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

   3. Taylor expanded in b around inf 71.7%

      \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{{a}^{2}}{{b}^{2}} - 1\right)} \]
   4. Step-by-step derivation

      1. unpow2 71.7%

         \[\leadsto {b}^{2} \cdot \left(\frac{\color{blue}{a \cdot a}}{{b}^{2}} - 1\right) \]

      2. associate-/l* 81.7%

         \[\leadsto {b}^{2} \cdot \left(\color{blue}{a \cdot \frac{a}{{b}^{2}}} - 1\right) \]

      3. fma-neg 81.7%

         \[\leadsto {b}^{2} \cdot \color{blue}{\mathsf{fma}\left(a, \frac{a}{{b}^{2}}, -1\right)} \]

      4. metadata-eval 81.7%

         \[\leadsto {b}^{2} \cdot \mathsf{fma}\left(a, \frac{a}{{b}^{2}}, \color{blue}{-1}\right) \]

   5. Simplified 81.7%

      \[\leadsto \color{blue}{{b}^{2} \cdot \mathsf{fma}\left(a, \frac{a}{{b}^{2}}, -1\right)} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 81.7%

         \[\leadsto {b}^{2} \cdot \mathsf{fma}\left(a, \frac{\color{blue}{1 \cdot a}}{{b}^{2}}, -1\right) \]

      2. unpow2 81.7%

         \[\leadsto {b}^{2} \cdot \mathsf{fma}\left(a, \frac{1 \cdot a}{\color{blue}{b \cdot b}}, -1\right) \]

      3. times-frac 100.0%

         \[\leadsto {b}^{2} \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{b} \cdot \frac{a}{b}}, -1\right) \]

   7. Applied egg-rr 100.0%

      \[\leadsto {b}^{2} \cdot \mathsf{fma}\left(a, \color{blue}{\frac{1}{b} \cdot \frac{a}{b}}, -1\right) \]
   8. Step-by-step derivation

      1. fma-undefine 100.0%

         \[\leadsto {b}^{2} \cdot \color{blue}{\left(a \cdot \left(\frac{1}{b} \cdot \frac{a}{b}\right) + -1\right)} \]

      2. associate-*r* 100.0%

         \[\leadsto {b}^{2} \cdot \left(\color{blue}{\left(a \cdot \frac{1}{b}\right) \cdot \frac{a}{b}} + -1\right) \]

      3. div-inv 100.0%

         \[\leadsto {b}^{2} \cdot \left(\color{blue}{\frac{a}{b}} \cdot \frac{a}{b} + -1\right) \]

      4. pow2 100.0%

         \[\leadsto {b}^{2} \cdot \left(\color{blue}{{\left(\frac{a}{b}\right)}^{2}} + -1\right) \]

   9. Applied egg-rr 100.0%

      \[\leadsto {b}^{2} \cdot \color{blue}{\left({\left(\frac{a}{b}\right)}^{2} + -1\right)} \]
   10. Step-by-step derivation

       1. unpow2 100.0%

          \[\leadsto {b}^{2} \cdot \left(\color{blue}{\frac{a}{b} \cdot \frac{a}{b}} + -1\right) \]

       2. clear-num 100.0%

          \[\leadsto {b}^{2} \cdot \left(\frac{a}{b} \cdot \color{blue}{\frac{1}{\frac{b}{a}}} + -1\right) \]

       3. un-div-inv 100.0%

          \[\leadsto {b}^{2} \cdot \left(\color{blue}{\frac{\frac{a}{b}}{\frac{b}{a}}} + -1\right) \]

   11. Applied egg-rr 100.0%

       \[\leadsto {b}^{2} \cdot \left(\color{blue}{\frac{\frac{a}{b}}{\frac{b}{a}}} + -1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 5 \cdot 10^{+307}:\\ \;\;\;\;a \cdot a - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} \cdot \left(-1 + \frac{\frac{a}{b}}{\frac{b}{a}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.2 \cdot 10^{+208}:\\ \;\;\;\;\mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(b + a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 3.2e+208) (fma a a (* b (- b))) (* (+ b a) (+ b a))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 3.2e+208) {
		tmp = fma(a, a, (b * -b));
	} else {
		tmp = (b + a) * (b + a);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 3.2e+208)
		tmp = fma(a, a, Float64(b * Float64(-b)));
	else
		tmp = Float64(Float64(b + a) * Float64(b + a));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, 3.2e+208], N[(a * a + N[(b * (-b)), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.2000000000000001e208

   1. Initial program 94.5%

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

      1. sqr-neg 94.5%

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

      2. cancel-sign-sub 94.5%

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

      3. fma-define 96.6%

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

   3. Simplified 96.6%

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

   if 3.2000000000000001e208 < a

   1. Initial program 77.8%

      \[a \cdot a - b \cdot b \]
   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 44.4%

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

      4. sqrt-unprod 94.4%

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

      5. sqr-neg 94.4%

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

      6. sqrt-prod 50.0%

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

      7. add-sqr-sqrt 94.4%

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

   4. Applied egg-rr 94.4%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.2 \cdot 10^{+208}:\\ \;\;\;\;\mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(b + a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 7.5 \cdot 10^{+161}:\\ \;\;\;\;a \cdot a - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(b + a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a 7.5e+161) (- (* a a) (* b b)) (* (+ b a) (+ b a))))

C

double code(double a, double b) {
	double tmp;
	if (a <= 7.5e+161) {
		tmp = (a * a) - (b * b);
	} else {
		tmp = (b + a) * (b + 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 <= 7.5d+161) then
        tmp = (a * a) - (b * b)
    else
        tmp = (b + a) * (b + a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= 7.5e+161) {
		tmp = (a * a) - (b * b);
	} else {
		tmp = (b + a) * (b + a);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= 7.5e+161:
		tmp = (a * a) - (b * b)
	else:
		tmp = (b + a) * (b + a)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= 7.5e+161)
		tmp = Float64(Float64(a * a) - Float64(b * b));
	else
		tmp = Float64(Float64(b + a) * Float64(b + a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= 7.5e+161)
		tmp = (a * a) - (b * b);
	else
		tmp = (b + a) * (b + a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, 7.5e+161], N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 7.4999999999999995e161

   1. Initial program 94.4%

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

   if 7.4999999999999995e161 < a

   1. Initial program 82.6%

      \[a \cdot a - b \cdot b \]
   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 43.5%

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

      4. sqrt-unprod 95.7%

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

      5. sqr-neg 95.7%

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

      6. sqrt-prod 52.2%

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

      7. add-sqr-sqrt 95.7%

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

   4. Applied egg-rr 95.7%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 7.5 \cdot 10^{+161}:\\ \;\;\;\;a \cdot a - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(b + a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.4%

   \[a \cdot a - b \cdot b \]
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 49.5%

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

   4. sqrt-unprod 71.9%

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

   5. sqr-neg 71.9%

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

   6. sqrt-prod 25.4%

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

   7. add-sqr-sqrt 54.3%

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

4. Applied egg-rr 54.3%

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

5. Final simplification 54.3%

   \[\leadsto \left(b + a\right) \cdot \left(b + a\right) \]
6. 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 2024103 
(FPCore (a b)
  :name "Difference of squares"
  :precision binary64

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

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