Difference of squares

Percentage Accurate: 93.6% → 98.3%

Time: 2.9s

Alternatives: 4

Speedup: 0.8×

• 43.9% 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: 93.6% 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: 98.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a_m \leq 10^{+206}:\\ \;\;\;\;\mathsf{fma}\left(a_m, a_m, b \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a_m - b\right) \cdot \left(a_m - b\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b)
 :precision binary64
 (if (<= a_m 1e+206) (fma a_m a_m (* b (- b))) (* (- a_m b) (- a_m b))))

C

a_m = fabs(a);
double code(double a_m, double b) {
	double tmp;
	if (a_m <= 1e+206) {
		tmp = fma(a_m, a_m, (b * -b));
	} else {
		tmp = (a_m - b) * (a_m - b);
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b)
	tmp = 0.0
	if (a_m <= 1e+206)
		tmp = fma(a_m, a_m, Float64(b * Float64(-b)));
	else
		tmp = Float64(Float64(a_m - b) * Float64(a_m - b));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := If[LessEqual[a$95$m, 1e+206], N[(a$95$m * a$95$m + N[(b * (-b)), $MachinePrecision]), $MachinePrecision], N[(N[(a$95$m - b), $MachinePrecision] * N[(a$95$m - b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1e206

   1. Initial program 93.7%

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

      1. sqr-neg 93.7%

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

      2. cancel-sign-sub 93.7%

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

      3. fma-def 97.0%

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

   3. Simplified 97.0%

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

   if 1e206 < a

   1. Initial program 78.9%

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

      1. difference-of-squares 100.0%

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

      2. add-sqr-sqrt 68.4%

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

      3. sqrt-prod 94.7%

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

      4. sqr-neg 94.7%

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

      5. sqrt-unprod 26.3%

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

      6. add-sqr-sqrt 94.7%

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

      7. sub-neg 94.7%

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

      8. pow1 94.7%

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

      9. pow1 94.7%

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

      10. pow-prod-up 94.7%

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

      11. metadata-eval 94.7%

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

      12. add-sqr-sqrt 94.7%

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

      13. add-sqr-sqrt 68.4%

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

      14. difference-of-squares 68.4%

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

      15. unpow-prod-down 68.4%

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

   3. Applied egg-rr 68.4%

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

      1. unpow2 68.4%

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

      2. unpow2 68.4%

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

      3. unswap-sqr 68.4%

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

      4. difference-of-squares 68.4%

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

      5. unpow1/2 68.4%

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

      6. unpow1/2 68.4%

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

      7. pow-sqr 68.4%

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

      8. metadata-eval 68.4%

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

      9. unpow1 68.4%

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

      10. unpow1/2 68.4%

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

      11. unpow1/2 68.4%

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

      12. pow-sqr 68.4%

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

      13. metadata-eval 68.4%

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

      14. unpow1 68.4%

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

      15. difference-of-squares 68.4%

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

      16. unpow1/2 68.4%

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

      17. unpow1/2 68.4%

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

      18. pow-sqr 68.4%

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

      19. metadata-eval 68.4%

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

      20. unpow1 68.4%

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

   5. Simplified 94.7%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 10^{+206}:\\ \;\;\;\;\mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a - b\right) \cdot \left(a - b\right)\\ \end{array} \]

Alternative 2: 76.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a_m \leq 1.9 \cdot 10^{-57}:\\ \;\;\;\;b \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a_m - b\right) \cdot \left(a_m - b\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b)
 :precision binary64
 (if (<= a_m 1.9e-57) (* b (- b)) (* (- a_m b) (- a_m b))))

C

a_m = fabs(a);
double code(double a_m, double b) {
	double tmp;
	if (a_m <= 1.9e-57) {
		tmp = b * -b;
	} else {
		tmp = (a_m - b) * (a_m - b);
	}
	return tmp;
}

Fortran

a_m = abs(a)
real(8) function code(a_m, b)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a_m <= 1.9d-57) then
        tmp = b * -b
    else
        tmp = (a_m - b) * (a_m - b)
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b) {
	double tmp;
	if (a_m <= 1.9e-57) {
		tmp = b * -b;
	} else {
		tmp = (a_m - b) * (a_m - b);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b):
	tmp = 0
	if a_m <= 1.9e-57:
		tmp = b * -b
	else:
		tmp = (a_m - b) * (a_m - b)
	return tmp

Julia

a_m = abs(a)
function code(a_m, b)
	tmp = 0.0
	if (a_m <= 1.9e-57)
		tmp = Float64(b * Float64(-b));
	else
		tmp = Float64(Float64(a_m - b) * Float64(a_m - b));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b)
	tmp = 0.0;
	if (a_m <= 1.9e-57)
		tmp = b * -b;
	else
		tmp = (a_m - b) * (a_m - b);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := If[LessEqual[a$95$m, 1.9e-57], N[(b * (-b)), $MachinePrecision], N[(N[(a$95$m - b), $MachinePrecision] * N[(a$95$m - b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.8999999999999999e-57

   1. Initial program 93.5%

      \[a \cdot a - b \cdot b \]

   2. Taylor expanded in a around 0 62.0%

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

      1. mul-1-neg 62.0%

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

   4. Simplified 62.0%

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

      1. unpow2 62.0%

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

   6. Applied egg-rr 62.0%

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

   if 1.8999999999999999e-57 < a

   1. Initial program 90.1%

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

      1. difference-of-squares 100.0%

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

      2. add-sqr-sqrt 52.1%

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

      3. sqrt-prod 88.5%

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

      4. sqr-neg 88.5%

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

      5. sqrt-unprod 36.4%

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

      6. add-sqr-sqrt 74.8%

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

      7. sub-neg 74.8%

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

      8. pow1 74.8%

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

      9. pow1 74.8%

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

      10. pow-prod-up 74.8%

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

      11. metadata-eval 74.8%

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

      12. add-sqr-sqrt 74.4%

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

      13. add-sqr-sqrt 38.2%

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

      14. difference-of-squares 38.2%

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

      15. unpow-prod-down 38.2%

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

   3. Applied egg-rr 38.2%

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

      1. unpow2 38.2%

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

      2. unpow2 38.2%

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

      3. unswap-sqr 38.2%

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

      4. difference-of-squares 38.2%

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

      5. unpow1/2 38.2%

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

      6. unpow1/2 38.2%

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

      7. pow-sqr 38.3%

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

      8. metadata-eval 38.3%

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

      9. unpow1 38.3%

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

      10. unpow1/2 38.3%

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

      11. unpow1/2 38.3%

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

      12. pow-sqr 38.3%

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

      13. metadata-eval 38.3%

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

      14. unpow1 38.3%

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

      15. difference-of-squares 38.3%

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

      16. unpow1/2 38.3%

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

      17. unpow1/2 38.3%

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

      18. pow-sqr 38.4%

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

      19. metadata-eval 38.4%

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

      20. unpow1 38.4%

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

   5. Simplified 74.8%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.9 \cdot 10^{-57}:\\ \;\;\;\;b \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a - b\right) \cdot \left(a - b\right)\\ \end{array} \]

Alternative 3: 96.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;a_m \leq 1.18 \cdot 10^{+147}:\\ \;\;\;\;a_m \cdot a_m - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(a_m - b\right) \cdot \left(a_m - b\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b)
 :precision binary64
 (if (<= a_m 1.18e+147) (- (* a_m a_m) (* b b)) (* (- a_m b) (- a_m b))))

C

a_m = fabs(a);
double code(double a_m, double b) {
	double tmp;
	if (a_m <= 1.18e+147) {
		tmp = (a_m * a_m) - (b * b);
	} else {
		tmp = (a_m - b) * (a_m - b);
	}
	return tmp;
}

Fortran

a_m = abs(a)
real(8) function code(a_m, b)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a_m <= 1.18d+147) then
        tmp = (a_m * a_m) - (b * b)
    else
        tmp = (a_m - b) * (a_m - b)
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b) {
	double tmp;
	if (a_m <= 1.18e+147) {
		tmp = (a_m * a_m) - (b * b);
	} else {
		tmp = (a_m - b) * (a_m - b);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b):
	tmp = 0
	if a_m <= 1.18e+147:
		tmp = (a_m * a_m) - (b * b)
	else:
		tmp = (a_m - b) * (a_m - b)
	return tmp

Julia

a_m = abs(a)
function code(a_m, b)
	tmp = 0.0
	if (a_m <= 1.18e+147)
		tmp = Float64(Float64(a_m * a_m) - Float64(b * b));
	else
		tmp = Float64(Float64(a_m - b) * Float64(a_m - b));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b)
	tmp = 0.0;
	if (a_m <= 1.18e+147)
		tmp = (a_m * a_m) - (b * b);
	else
		tmp = (a_m - b) * (a_m - b);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := If[LessEqual[a$95$m, 1.18e+147], N[(N[(a$95$m * a$95$m), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(a$95$m - b), $MachinePrecision] * N[(a$95$m - b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.18000000000000006e147

   1. Initial program 94.6%

      \[a \cdot a - b \cdot b \]

   if 1.18000000000000006e147 < a

   1. Initial program 78.8%

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

      1. difference-of-squares 100.0%

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

      2. add-sqr-sqrt 51.5%

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

      3. sqrt-prod 90.9%

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

      4. sqr-neg 90.9%

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

      5. sqrt-unprod 39.4%

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

      6. add-sqr-sqrt 87.9%

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

      7. sub-neg 87.9%

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

      8. pow1 87.9%

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

      9. pow1 87.9%

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

      10. pow-prod-up 87.9%

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

      11. metadata-eval 87.9%

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

      12. add-sqr-sqrt 87.9%

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

      13. add-sqr-sqrt 48.5%

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

      14. difference-of-squares 48.5%

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

      15. unpow-prod-down 48.5%

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

   3. Applied egg-rr 48.5%

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

      1. unpow2 48.5%

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

      2. unpow2 48.5%

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

      3. unswap-sqr 48.5%

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

      4. difference-of-squares 48.5%

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

      5. unpow1/2 48.5%

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

      6. unpow1/2 48.5%

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

      7. pow-sqr 48.5%

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

      8. metadata-eval 48.5%

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

      9. unpow1 48.5%

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

      10. unpow1/2 48.5%

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

      11. unpow1/2 48.5%

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

      12. pow-sqr 48.5%

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

      13. metadata-eval 48.5%

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

      14. unpow1 48.5%

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

      15. difference-of-squares 48.5%

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

      16. unpow1/2 48.5%

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

      17. unpow1/2 48.5%

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

      18. pow-sqr 48.5%

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

      19. metadata-eval 48.5%

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

      20. unpow1 48.5%

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

   5. Simplified 87.9%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.18 \cdot 10^{+147}:\\ \;\;\;\;a \cdot a - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(a - b\right) \cdot \left(a - b\right)\\ \end{array} \]

Alternative 4: 52.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b \cdot \left(-b\right) \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b) :precision binary64 (* b (- b)))

C

a_m = fabs(a);
double code(double a_m, double b) {
	return b * -b;
}

Fortran

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

Java

a_m = Math.abs(a);
public static double code(double a_m, double b) {
	return b * -b;
}

Python

a_m = math.fabs(a)
def code(a_m, b):
	return b * -b

Julia

a_m = abs(a)
function code(a_m, b)
	return Float64(b * Float64(-b))
end

MATLAB

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

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_] := N[(b * (-b)), $MachinePrecision]

Derivation

1. Initial program 92.6%

   \[a \cdot a - b \cdot b \]

2. Taylor expanded in a around 0 51.8%

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

   1. mul-1-neg 51.8%

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

4. Simplified 51.8%

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

   1. unpow2 51.8%

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

6. Applied egg-rr 51.8%

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

7. Final simplification 51.8%

   \[\leadsto b \cdot \left(-b\right) \]

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 2023315 
(FPCore (a b)
  :name "Difference of squares"
  :precision binary64

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

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