Difference of squares

Percentage Accurate: 93.3% → 99.8%

Time: 4.5s

Alternatives: 4

Speedup: 0.5×

• 45.2% 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: 93.3% 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: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(1 - \frac{b}{a} \cdot \frac{b}{a}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+300)
   (fma a a (* b (- b)))
   (* a (* a (- 1.0 (* (/ b a) (/ b a)))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+300) {
		tmp = fma(a, a, (b * -b));
	} else {
		tmp = a * (a * (1.0 - ((b / a) * (b / a))));
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+300)
		tmp = fma(a, a, Float64(b * Float64(-b)));
	else
		tmp = Float64(a * Float64(a * Float64(1.0 - Float64(Float64(b / a) * Float64(b / a)))));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+300], N[(a * a + N[(b * (-b)), $MachinePrecision]), $MachinePrecision], N[(a * N[(a * N[(1.0 - N[(N[(b / a), $MachinePrecision] * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2.0000000000000001e300

   1. Initial program 100.0%

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

      1. sqr-neg 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. fma-define 100.0%

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

   3. Simplified 100.0%

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

   if 2.0000000000000001e300 < (*.f64 b b)

   1. Initial program 76.0%

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

   3. Taylor expanded in a around inf 53.5%

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

      1. mul-1-neg 53.5%

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

      2. unsub-neg 53.5%

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

   5. Simplified 53.5%

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

      1. add-sqr-sqrt 1.4%

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

      2. pow2 1.4%

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

      3. sqrt-prod 1.4%

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

      4. sqrt-pow1 1.4%

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

      5. metadata-eval 1.4%

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

      6. pow1 1.4%

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

      7. add-sqr-sqrt 1.4%

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

      8. pow2 1.4%

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

      9. sqrt-div 1.4%

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

      10. sqrt-pow1 11.3%

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

      11. metadata-eval 11.3%

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

      12. pow1 11.3%

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

      13. sqrt-pow1 11.3%

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

      14. metadata-eval 11.3%

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

      15. pow1 11.3%

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

   7. Applied egg-rr 11.3%

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

      1. unpow2 11.3%

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

      2. *-commutative 11.3%

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

      3. *-commutative 11.3%

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

      4. swap-sqr 11.2%

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

      5. pow2 11.2%

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

      6. pow2 11.2%

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

      7. add-sqr-sqrt 77.4%

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

      8. associate-*r* 100.0%

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

      9. pow2 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. pow2 100.0%

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

   11. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(1 - \frac{b}{a} \cdot \frac{b}{a}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+300}:\\ \;\;\;\;a \cdot a - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(1 - \frac{b}{a} \cdot \frac{b}{a}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* b b) 2e+300)
   (- (* a a) (* b b))
   (* a (* a (- 1.0 (* (/ b a) (/ b a)))))))

C

double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+300) {
		tmp = (a * a) - (b * b);
	} else {
		tmp = a * (a * (1.0 - ((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 ((b * b) <= 2d+300) then
        tmp = (a * a) - (b * b)
    else
        tmp = a * (a * (1.0d0 - ((b / a) * (b / a))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((b * b) <= 2e+300) {
		tmp = (a * a) - (b * b);
	} else {
		tmp = a * (a * (1.0 - ((b / a) * (b / a))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (b * b) <= 2e+300:
		tmp = (a * a) - (b * b)
	else:
		tmp = a * (a * (1.0 - ((b / a) * (b / a))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(b * b) <= 2e+300)
		tmp = Float64(Float64(a * a) - Float64(b * b));
	else
		tmp = Float64(a * Float64(a * Float64(1.0 - Float64(Float64(b / a) * Float64(b / a)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((b * b) <= 2e+300)
		tmp = (a * a) - (b * b);
	else
		tmp = a * (a * (1.0 - ((b / a) * (b / a))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(b * b), $MachinePrecision], 2e+300], N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision], N[(a * N[(a * N[(1.0 - N[(N[(b / a), $MachinePrecision] * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 b b) < 2.0000000000000001e300

   1. Initial program 100.0%

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

   if 2.0000000000000001e300 < (*.f64 b b)

   1. Initial program 76.0%

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

   3. Taylor expanded in a around inf 53.5%

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

      1. mul-1-neg 53.5%

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

      2. unsub-neg 53.5%

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

   5. Simplified 53.5%

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

      1. add-sqr-sqrt 1.4%

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

      2. pow2 1.4%

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

      3. sqrt-prod 1.4%

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

      4. sqrt-pow1 1.4%

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

      5. metadata-eval 1.4%

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

      6. pow1 1.4%

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

      7. add-sqr-sqrt 1.4%

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

      8. pow2 1.4%

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

      9. sqrt-div 1.4%

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

      10. sqrt-pow1 11.3%

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

      11. metadata-eval 11.3%

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

      12. pow1 11.3%

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

      13. sqrt-pow1 11.3%

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

      14. metadata-eval 11.3%

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

      15. pow1 11.3%

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

   7. Applied egg-rr 11.3%

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

      1. unpow2 11.3%

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

      2. *-commutative 11.3%

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

      3. *-commutative 11.3%

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

      4. swap-sqr 11.2%

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

      5. pow2 11.2%

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

      6. pow2 11.2%

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

      7. add-sqr-sqrt 77.4%

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

      8. associate-*r* 100.0%

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

      9. pow2 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. pow2 100.0%

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

   11. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \leq 2 \cdot 10^{+300}:\\ \;\;\;\;a \cdot a - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(1 - \frac{b}{a} \cdot \frac{b}{a}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot a \leq 5 \cdot 10^{+302}:\\ \;\;\;\;a \cdot a - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (* a a) 5e+302) (- (* a a) (* b b)) (* a a)))

C

double code(double a, double b) {
	double tmp;
	if ((a * a) <= 5e+302) {
		tmp = (a * a) - (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) <= 5d+302) then
        tmp = (a * a) - (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) <= 5e+302) {
		tmp = (a * a) - (b * b);
	} else {
		tmp = a * a;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (a * a) <= 5e+302:
		tmp = (a * a) - (b * b)
	else:
		tmp = a * a
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(a * a) <= 5e+302)
		tmp = Float64(Float64(a * a) - Float64(b * b));
	else
		tmp = Float64(a * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((a * a) <= 5e+302)
		tmp = (a * a) - (b * b);
	else
		tmp = a * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(a * a), $MachinePrecision], 5e+302], N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision], N[(a * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a a) < 5e302

   1. Initial program 100.0%

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

   if 5e302 < (*.f64 a a)

   1. Initial program 73.8%

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

   3. Taylor expanded in a around inf 73.8%

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

      1. mul-1-neg 73.8%

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

      2. unsub-neg 73.8%

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

   5. Simplified 73.8%

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

      1. add-sqr-sqrt 73.8%

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

      2. pow2 73.8%

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

      3. sqrt-prod 73.8%

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

      4. sqrt-pow1 73.8%

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

      5. metadata-eval 73.8%

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

      6. pow1 73.8%

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

      7. add-sqr-sqrt 73.8%

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

      8. pow2 73.8%

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

      9. sqrt-div 73.8%

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

      10. sqrt-pow1 84.6%

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

      11. metadata-eval 84.6%

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

      12. pow1 84.6%

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

      13. sqrt-pow1 84.6%

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

      14. metadata-eval 84.6%

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

      15. pow1 84.6%

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

   7. Applied egg-rr 84.6%

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

      1. unpow2 84.6%

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

      2. *-commutative 84.6%

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

      3. *-commutative 84.6%

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

      4. swap-sqr 84.6%

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

      5. pow2 84.6%

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

      6. pow2 84.6%

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

      7. add-sqr-sqrt 100.0%

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

      8. associate-*r* 100.0%

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

      9. pow2 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in b around 0 84.6%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 5 \cdot 10^{+302}:\\ \;\;\;\;a \cdot a - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.5% 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 93.3%

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

3. Taylor expanded in a around inf 66.9%

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

   1. mul-1-neg 66.9%

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

   2. unsub-neg 66.9%

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

5. Simplified 66.9%

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

   1. add-sqr-sqrt 40.6%

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

   2. pow2 40.6%

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

   3. sqrt-prod 40.6%

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

   4. sqrt-pow1 40.6%

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

   5. metadata-eval 40.6%

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

   6. pow1 40.6%

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

   7. add-sqr-sqrt 40.6%

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

   8. pow2 40.6%

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

   9. sqrt-div 40.6%

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

   10. sqrt-pow1 43.4%

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

   11. metadata-eval 43.4%

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

   12. pow1 43.4%

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

   13. sqrt-pow1 45.3%

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

   14. metadata-eval 45.3%

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

   15. pow1 45.3%

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

7. Applied egg-rr 45.3%

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

   1. unpow2 45.3%

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

   2. *-commutative 45.3%

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

   3. *-commutative 45.3%

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

   4. swap-sqr 45.3%

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

   5. pow2 45.3%

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

   6. pow2 45.3%

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

   7. add-sqr-sqrt 78.0%

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

   8. associate-*r* 87.9%

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

   9. pow2 87.9%

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

9. Applied egg-rr 87.9%

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

10. Taylor expanded in b around 0 48.0%

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

11. Final simplification 48.0%

    \[\leadsto a \cdot a \]
12. 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 2024075 
(FPCore (a b)
  :name "Difference of squares"
  :precision binary64

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

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