Radioactive exchange between two surfaces

Percentage Accurate: 86.1% → 99.8%

Time: 14.9s

Alternatives: 7

Speedup: 12.1×

• 60.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ {x}^{4} - {y}^{4} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (pow x 4.0) (pow y 4.0)))

C

double code(double x, double y) {
	return pow(x, 4.0) - pow(y, 4.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x ** 4.0d0) - (y ** 4.0d0)
end function

Java

public static double code(double x, double y) {
	return Math.pow(x, 4.0) - Math.pow(y, 4.0);
}

Python

def code(x, y):
	return math.pow(x, 4.0) - math.pow(y, 4.0)

Julia

function code(x, y)
	return Float64((x ^ 4.0) - (y ^ 4.0))
end

MATLAB

function tmp = code(x, y)
	tmp = (x ^ 4.0) - (y ^ 4.0);
end

Wolfram

code[x_, y_] := N[(N[Power[x, 4.0], $MachinePrecision] - N[Power[y, 4.0], $MachinePrecision]), $MachinePrecision]

Initial Program: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {x}^{4} - {y}^{4} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (pow x 4.0) (pow y 4.0)))

C

double code(double x, double y) {
	return pow(x, 4.0) - pow(y, 4.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x ** 4.0d0) - (y ** 4.0d0)
end function

Java

public static double code(double x, double y) {
	return Math.pow(x, 4.0) - Math.pow(y, 4.0);
}

Python

def code(x, y):
	return math.pow(x, 4.0) - math.pow(y, 4.0)

Julia

function code(x, y)
	return Float64((x ^ 4.0) - (y ^ 4.0))
end

MATLAB

function tmp = code(x, y)
	tmp = (x ^ 4.0) - (y ^ 4.0);
end

Wolfram

code[x_, y_] := N[(N[Power[x, 4.0], $MachinePrecision] - N[Power[y, 4.0], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot x}{\frac{\frac{1}{x + y}}{x - y}}\\ \mathbf{if}\;x \leq -1.75 \cdot 10^{+78}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+63}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot x\right)\right) - y \cdot \left(y \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x x) (/ (/ 1.0 (+ x y)) (- x y)))))
   (if (<= x -1.75e+78)
     t_0
     (if (<= x 5e+63) (- (* x (* x (* x x))) (* y (* y (* y y)))) t_0))))

C

double code(double x, double y) {
	double t_0 = (x * x) / ((1.0 / (x + y)) / (x - y));
	double tmp;
	if (x <= -1.75e+78) {
		tmp = t_0;
	} else if (x <= 5e+63) {
		tmp = (x * (x * (x * x))) - (y * (y * (y * y)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * x) / ((1.0d0 / (x + y)) / (x - y))
    if (x <= (-1.75d+78)) then
        tmp = t_0
    else if (x <= 5d+63) then
        tmp = (x * (x * (x * x))) - (y * (y * (y * y)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * x) / ((1.0 / (x + y)) / (x - y));
	double tmp;
	if (x <= -1.75e+78) {
		tmp = t_0;
	} else if (x <= 5e+63) {
		tmp = (x * (x * (x * x))) - (y * (y * (y * y)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * x) / ((1.0 / (x + y)) / (x - y))
	tmp = 0
	if x <= -1.75e+78:
		tmp = t_0
	elif x <= 5e+63:
		tmp = (x * (x * (x * x))) - (y * (y * (y * y)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * x) / Float64(Float64(1.0 / Float64(x + y)) / Float64(x - y)))
	tmp = 0.0
	if (x <= -1.75e+78)
		tmp = t_0;
	elseif (x <= 5e+63)
		tmp = Float64(Float64(x * Float64(x * Float64(x * x))) - Float64(y * Float64(y * Float64(y * y))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * x) / ((1.0 / (x + y)) / (x - y));
	tmp = 0.0;
	if (x <= -1.75e+78)
		tmp = t_0;
	elseif (x <= 5e+63)
		tmp = (x * (x * (x * x))) - (y * (y * (y * y)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] / N[(N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.75e+78], t$95$0, If[LessEqual[x, 5e+63], N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.7500000000000001e78 or 5.00000000000000011e63 < x

   1. Initial program 70.7%

      \[{x}^{4} - {y}^{4} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sqr-pow N/A

         \[\leadsto {x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)} - {\color{blue}{y}}^{4} \]

      2. sqr-pow N/A

         \[\leadsto {x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)} \cdot \color{blue}{{y}^{\left(\frac{4}{2}\right)}} \]

      3. difference-of-squares N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right)} \]

      4. *-commutative N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \cdot \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right), \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left({x}^{\left(\frac{4}{2}\right)}\right), \left({y}^{\left(\frac{4}{2}\right)}\right)\right), \left(\color{blue}{{x}^{\left(\frac{4}{2}\right)}} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left({x}^{2}\right), \left({y}^{\left(\frac{4}{2}\right)}\right)\right), \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot x\right), \left({y}^{\left(\frac{4}{2}\right)}\right)\right), \left({\color{blue}{x}}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({y}^{\left(\frac{4}{2}\right)}\right)\right), \left({\color{blue}{x}}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({y}^{2}\right)\right), \left({x}^{\left(\frac{4}{\color{blue}{2}}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot y\right)\right), \left({x}^{\color{blue}{\left(\frac{4}{2}\right)}} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \left({x}^{\color{blue}{\left(\frac{4}{2}\right)}} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \left({y}^{\left(\frac{4}{2}\right)} + \color{blue}{{x}^{\left(\frac{4}{2}\right)}}\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\left({y}^{\left(\frac{4}{2}\right)}\right), \color{blue}{\left({x}^{\left(\frac{4}{2}\right)}\right)}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\left({y}^{2}\right), \left({x}^{\left(\frac{4}{2}\right)}\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\left(y \cdot y\right), \left({\color{blue}{x}}^{\left(\frac{4}{2}\right)}\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({\color{blue}{x}}^{\left(\frac{4}{2}\right)}\right)\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({x}^{2}\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      20. *-lowering-*.f64 82.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   4. Applied egg-rr 82.5%

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

      1. flip3-- N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{{\left(x \cdot x\right)}^{3} - {\left(y \cdot y\right)}^{3}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(x \cdot x\right) \cdot \left(y \cdot y\right)\right)}\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(y, y\right)}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(x \cdot x\right) \cdot \left(y \cdot y\right)\right)}{{\left(x \cdot x\right)}^{3} - {\left(y \cdot y\right)}^{3}}}\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(y, y\right)}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(x \cdot x\right) \cdot \left(y \cdot y\right)\right)}{{\left(x \cdot x\right)}^{3} - {\left(y \cdot y\right)}^{3}}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(y, y\right)}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

      4. cube-unmult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(x \cdot x\right) \cdot \left(y \cdot y\right)\right)}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) - {\left(y \cdot y\right)}^{3}}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, x\right)\right)\right) \]

      5. swap-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(x \cdot x\right) \cdot \left(y \cdot y\right)\right)}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) - {\left(y \cdot y\right)}^{3}}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, x\right)\right)\right) \]

      6. cube-unmult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(x \cdot x\right) \cdot \left(y \cdot y\right)\right)}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) - \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, x\right)\right)\right) \]

      7. swap-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(x \cdot x\right) \cdot \left(y \cdot y\right)\right)}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) - \left(y \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot \left(y \cdot y\right)\right)}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, x\right)\right)\right) \]

      8. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{1}{\frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) - \left(y \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot \left(y \cdot y\right)\right)}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(x \cdot x\right) \cdot \left(y \cdot y\right)\right)}}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \color{blue}{y}\right), \mathsf{*.f64}\left(x, x\right)\right)\right) \]

   6. Applied egg-rr 82.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{x \cdot x - y \cdot y}}} \cdot \left(y \cdot y + x \cdot x\right) \]
   7. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{1 \cdot \left(y \cdot y + x \cdot x\right)}{\color{blue}{\frac{1}{x \cdot x - y \cdot y}}} \]

      2. *-lft-identity N/A

         \[\leadsto \frac{y \cdot y + x \cdot x}{\frac{\color{blue}{1}}{x \cdot x - y \cdot y}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot y + x \cdot x\right), \color{blue}{\left(\frac{1}{x \cdot x - y \cdot y}\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot x + y \cdot y\right), \left(\frac{\color{blue}{1}}{x \cdot x - y \cdot y}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), \left(y \cdot y\right)\right), \left(\frac{\color{blue}{1}}{x \cdot x - y \cdot y}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot y\right)\right), \left(\frac{1}{x \cdot x - y \cdot y}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{1}{x \cdot x - y \cdot y}\right)\right) \]

      8. difference-of-squares N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{1}{\left(x + y\right) \cdot \color{blue}{\left(x - y\right)}}\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{\frac{1}{x + y}}{\color{blue}{x - y}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{/.f64}\left(\left(\frac{1}{x + y}\right), \color{blue}{\left(x - y\right)}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(x + y\right)\right), \left(\color{blue}{x} - y\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, y\right)\right), \left(x - y\right)\right)\right) \]

      13. --lowering--.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   8. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{x \cdot x + y \cdot y}{\frac{\frac{1}{x + y}}{x - y}}} \]

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left({x}^{2}\right)}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(x, y\right)\right)\right) \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(x \cdot x\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, y\right)\right)}, \mathsf{\_.f64}\left(x, y\right)\right)\right) \]

       2. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, y\right)\right)}, \mathsf{\_.f64}\left(x, y\right)\right)\right) \]

   11. Simplified 100.0%

       \[\leadsto \frac{\color{blue}{x \cdot x}}{\frac{\frac{1}{x + y}}{x - y}} \]

   if -1.7500000000000001e78 < x < 5.00000000000000011e63

   1. Initial program 100.0%

      \[{x}^{4} - {y}^{4} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({x}^{4}\right), \color{blue}{\left({y}^{4}\right)}\right) \]

      2. sqr-pow N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)}\right), \left({\color{blue}{y}}^{4}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left({x}^{2} \cdot {x}^{\left(\frac{4}{2}\right)}\right), \left({y}^{4}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(x \cdot x\right) \cdot {x}^{\left(\frac{4}{2}\right)}\right), \left({y}^{4}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \left(x \cdot {x}^{\left(\frac{4}{2}\right)}\right)\right), \left({\color{blue}{y}}^{4}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \left(x \cdot {x}^{2}\right)\right), \left({y}^{4}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), \left({y}^{4}\right)\right) \]

      8. cube-unmult N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot {x}^{3}\right), \left({y}^{4}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{3}\right)\right), \left({\color{blue}{y}}^{4}\right)\right) \]

      10. cube-unmult N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot x\right)\right)\right), \left({y}^{4}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot {x}^{2}\right)\right), \left({y}^{4}\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot {x}^{\left(\frac{4}{2}\right)}\right)\right), \left({y}^{4}\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{\left(\frac{4}{2}\right)}\right)\right)\right), \left({y}^{4}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2}\right)\right)\right), \left({y}^{4}\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right), \left({y}^{4}\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left({y}^{4}\right)\right) \]

      17. sqr-pow N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left({y}^{\left(\frac{4}{2}\right)} \cdot \color{blue}{{y}^{\left(\frac{4}{2}\right)}}\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left({y}^{2} \cdot {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\left(y \cdot y\right) \cdot {\color{blue}{y}}^{\left(\frac{4}{2}\right)}\right)\right) \]

      20. associate-*l* N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(y \cdot \color{blue}{\left(y \cdot {y}^{\left(\frac{4}{2}\right)}\right)}\right)\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(y \cdot \left(y \cdot {y}^{2}\right)\right)\right) \]

      22. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(y \cdot \left(y \cdot \left(y \cdot \color{blue}{y}\right)\right)\right)\right) \]

      23. cube-unmult N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(y \cdot {y}^{\color{blue}{3}}\right)\right) \]

      24. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{3}\right)}\right)\right) \]

      25. cube-unmult N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right) \]

      26. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \left(y \cdot {y}^{\color{blue}{2}}\right)\right)\right) \]

      27. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \left(y \cdot {y}^{\left(\frac{4}{\color{blue}{2}}\right)}\right)\right)\right) \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(x \cdot x\right)\right) - y \cdot \left(y \cdot \left(y \cdot y\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.7% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot x}{\frac{\frac{1}{x + y}}{x - y}}\\ \mathbf{if}\;x \leq -1 \cdot 10^{+158}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+63}:\\ \;\;\;\;\left(x \cdot x - y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x x) (/ (/ 1.0 (+ x y)) (- x y)))))
   (if (<= x -1e+158)
     t_0
     (if (<= x 5.6e+63) (* (- (* x x) (* y y)) (+ (* x x) (* y y))) t_0))))

C

double code(double x, double y) {
	double t_0 = (x * x) / ((1.0 / (x + y)) / (x - y));
	double tmp;
	if (x <= -1e+158) {
		tmp = t_0;
	} else if (x <= 5.6e+63) {
		tmp = ((x * x) - (y * y)) * ((x * x) + (y * y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * x) / ((1.0d0 / (x + y)) / (x - y))
    if (x <= (-1d+158)) then
        tmp = t_0
    else if (x <= 5.6d+63) then
        tmp = ((x * x) - (y * y)) * ((x * x) + (y * y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * x) / ((1.0 / (x + y)) / (x - y));
	double tmp;
	if (x <= -1e+158) {
		tmp = t_0;
	} else if (x <= 5.6e+63) {
		tmp = ((x * x) - (y * y)) * ((x * x) + (y * y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * x) / ((1.0 / (x + y)) / (x - y))
	tmp = 0
	if x <= -1e+158:
		tmp = t_0
	elif x <= 5.6e+63:
		tmp = ((x * x) - (y * y)) * ((x * x) + (y * y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * x) / Float64(Float64(1.0 / Float64(x + y)) / Float64(x - y)))
	tmp = 0.0
	if (x <= -1e+158)
		tmp = t_0;
	elseif (x <= 5.6e+63)
		tmp = Float64(Float64(Float64(x * x) - Float64(y * y)) * Float64(Float64(x * x) + Float64(y * y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * x) / ((1.0 / (x + y)) / (x - y));
	tmp = 0.0;
	if (x <= -1e+158)
		tmp = t_0;
	elseif (x <= 5.6e+63)
		tmp = ((x * x) - (y * y)) * ((x * x) + (y * y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] / N[(N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1e+158], t$95$0, If[LessEqual[x, 5.6e+63], N[(N[(N[(x * x), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.99999999999999953e157 or 5.59999999999999974e63 < x

   1. Initial program 69.2%

      \[{x}^{4} - {y}^{4} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sqr-pow N/A

         \[\leadsto {x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)} - {\color{blue}{y}}^{4} \]

      2. sqr-pow N/A

         \[\leadsto {x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)} \cdot \color{blue}{{y}^{\left(\frac{4}{2}\right)}} \]

      3. difference-of-squares N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right)} \]

      4. *-commutative N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \cdot \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right), \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left({x}^{\left(\frac{4}{2}\right)}\right), \left({y}^{\left(\frac{4}{2}\right)}\right)\right), \left(\color{blue}{{x}^{\left(\frac{4}{2}\right)}} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left({x}^{2}\right), \left({y}^{\left(\frac{4}{2}\right)}\right)\right), \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot x\right), \left({y}^{\left(\frac{4}{2}\right)}\right)\right), \left({\color{blue}{x}}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({y}^{\left(\frac{4}{2}\right)}\right)\right), \left({\color{blue}{x}}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({y}^{2}\right)\right), \left({x}^{\left(\frac{4}{\color{blue}{2}}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot y\right)\right), \left({x}^{\color{blue}{\left(\frac{4}{2}\right)}} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \left({x}^{\color{blue}{\left(\frac{4}{2}\right)}} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \left({y}^{\left(\frac{4}{2}\right)} + \color{blue}{{x}^{\left(\frac{4}{2}\right)}}\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\left({y}^{\left(\frac{4}{2}\right)}\right), \color{blue}{\left({x}^{\left(\frac{4}{2}\right)}\right)}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\left({y}^{2}\right), \left({x}^{\left(\frac{4}{2}\right)}\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\left(y \cdot y\right), \left({\color{blue}{x}}^{\left(\frac{4}{2}\right)}\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({\color{blue}{x}}^{\left(\frac{4}{2}\right)}\right)\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({x}^{2}\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      20. *-lowering-*.f64 79.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   4. Applied egg-rr 79.4%

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

      1. flip3-- N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{{\left(x \cdot x\right)}^{3} - {\left(y \cdot y\right)}^{3}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(x \cdot x\right) \cdot \left(y \cdot y\right)\right)}\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(y, y\right)}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(x \cdot x\right) \cdot \left(y \cdot y\right)\right)}{{\left(x \cdot x\right)}^{3} - {\left(y \cdot y\right)}^{3}}}\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(y, y\right)}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(x \cdot x\right) \cdot \left(y \cdot y\right)\right)}{{\left(x \cdot x\right)}^{3} - {\left(y \cdot y\right)}^{3}}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(y, y\right)}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

      4. cube-unmult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(x \cdot x\right) \cdot \left(y \cdot y\right)\right)}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) - {\left(y \cdot y\right)}^{3}}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, x\right)\right)\right) \]

      5. swap-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(x \cdot x\right) \cdot \left(y \cdot y\right)\right)}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) - {\left(y \cdot y\right)}^{3}}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, x\right)\right)\right) \]

      6. cube-unmult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(x \cdot x\right) \cdot \left(y \cdot y\right)\right)}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) - \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, x\right)\right)\right) \]

      7. swap-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(x \cdot x\right) \cdot \left(y \cdot y\right)\right)}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) - \left(y \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot \left(y \cdot y\right)\right)}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, x\right)\right)\right) \]

      8. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{1}{\frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) - \left(y \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot \left(y \cdot y\right)\right)}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(x \cdot x\right) \cdot \left(y \cdot y\right)\right)}}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \color{blue}{y}\right), \mathsf{*.f64}\left(x, x\right)\right)\right) \]

   6. Applied egg-rr 79.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{x \cdot x - y \cdot y}}} \cdot \left(y \cdot y + x \cdot x\right) \]
   7. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{1 \cdot \left(y \cdot y + x \cdot x\right)}{\color{blue}{\frac{1}{x \cdot x - y \cdot y}}} \]

      2. *-lft-identity N/A

         \[\leadsto \frac{y \cdot y + x \cdot x}{\frac{\color{blue}{1}}{x \cdot x - y \cdot y}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot y + x \cdot x\right), \color{blue}{\left(\frac{1}{x \cdot x - y \cdot y}\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot x + y \cdot y\right), \left(\frac{\color{blue}{1}}{x \cdot x - y \cdot y}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), \left(y \cdot y\right)\right), \left(\frac{\color{blue}{1}}{x \cdot x - y \cdot y}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot y\right)\right), \left(\frac{1}{x \cdot x - y \cdot y}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{1}{x \cdot x - y \cdot y}\right)\right) \]

      8. difference-of-squares N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{1}{\left(x + y\right) \cdot \color{blue}{\left(x - y\right)}}\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{\frac{1}{x + y}}{\color{blue}{x - y}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{/.f64}\left(\left(\frac{1}{x + y}\right), \color{blue}{\left(x - y\right)}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(x + y\right)\right), \left(\color{blue}{x} - y\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, y\right)\right), \left(x - y\right)\right)\right) \]

      13. --lowering--.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   8. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{x \cdot x + y \cdot y}{\frac{\frac{1}{x + y}}{x - y}}} \]

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left({x}^{2}\right)}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(x, y\right)\right)\right) \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(x \cdot x\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, y\right)\right)}, \mathsf{\_.f64}\left(x, y\right)\right)\right) \]

       2. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, y\right)\right)}, \mathsf{\_.f64}\left(x, y\right)\right)\right) \]

   11. Simplified 100.0%

       \[\leadsto \frac{\color{blue}{x \cdot x}}{\frac{\frac{1}{x + y}}{x - y}} \]

   if -9.99999999999999953e157 < x < 5.59999999999999974e63

   1. Initial program 98.3%

      \[{x}^{4} - {y}^{4} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sqr-pow N/A

         \[\leadsto {x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)} - {\color{blue}{y}}^{4} \]

      2. sqr-pow N/A

         \[\leadsto {x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)} \cdot \color{blue}{{y}^{\left(\frac{4}{2}\right)}} \]

      3. difference-of-squares N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right)} \]

      4. *-commutative N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \cdot \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right), \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left({x}^{\left(\frac{4}{2}\right)}\right), \left({y}^{\left(\frac{4}{2}\right)}\right)\right), \left(\color{blue}{{x}^{\left(\frac{4}{2}\right)}} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left({x}^{2}\right), \left({y}^{\left(\frac{4}{2}\right)}\right)\right), \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot x\right), \left({y}^{\left(\frac{4}{2}\right)}\right)\right), \left({\color{blue}{x}}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({y}^{\left(\frac{4}{2}\right)}\right)\right), \left({\color{blue}{x}}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({y}^{2}\right)\right), \left({x}^{\left(\frac{4}{\color{blue}{2}}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot y\right)\right), \left({x}^{\color{blue}{\left(\frac{4}{2}\right)}} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \left({x}^{\color{blue}{\left(\frac{4}{2}\right)}} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \left({y}^{\left(\frac{4}{2}\right)} + \color{blue}{{x}^{\left(\frac{4}{2}\right)}}\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\left({y}^{\left(\frac{4}{2}\right)}\right), \color{blue}{\left({x}^{\left(\frac{4}{2}\right)}\right)}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\left({y}^{2}\right), \left({x}^{\left(\frac{4}{2}\right)}\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\left(y \cdot y\right), \left({\color{blue}{x}}^{\left(\frac{4}{2}\right)}\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({\color{blue}{x}}^{\left(\frac{4}{2}\right)}\right)\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({x}^{2}\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      20. *-lowering-*.f64 99.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   4. Applied egg-rr 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+158}:\\ \;\;\;\;\frac{x \cdot x}{\frac{\frac{1}{x + y}}{x - y}}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+63}:\\ \;\;\;\;\left(x \cdot x - y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{\frac{\frac{1}{x + y}}{x - y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.5% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot x}{\frac{\frac{1}{x + y}}{x - y}}\\ \mathbf{if}\;x \leq -6.4 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-66}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot \left(0 - y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x x) (/ (/ 1.0 (+ x y)) (- x y)))))
   (if (<= x -6.4e-15)
     t_0
     (if (<= x 9.5e-66) (* y (* (* y y) (- 0.0 y))) t_0))))

C

double code(double x, double y) {
	double t_0 = (x * x) / ((1.0 / (x + y)) / (x - y));
	double tmp;
	if (x <= -6.4e-15) {
		tmp = t_0;
	} else if (x <= 9.5e-66) {
		tmp = y * ((y * y) * (0.0 - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * x) / ((1.0d0 / (x + y)) / (x - y))
    if (x <= (-6.4d-15)) then
        tmp = t_0
    else if (x <= 9.5d-66) then
        tmp = y * ((y * y) * (0.0d0 - y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * x) / ((1.0 / (x + y)) / (x - y));
	double tmp;
	if (x <= -6.4e-15) {
		tmp = t_0;
	} else if (x <= 9.5e-66) {
		tmp = y * ((y * y) * (0.0 - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * x) / ((1.0 / (x + y)) / (x - y))
	tmp = 0
	if x <= -6.4e-15:
		tmp = t_0
	elif x <= 9.5e-66:
		tmp = y * ((y * y) * (0.0 - y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * x) / Float64(Float64(1.0 / Float64(x + y)) / Float64(x - y)))
	tmp = 0.0
	if (x <= -6.4e-15)
		tmp = t_0;
	elseif (x <= 9.5e-66)
		tmp = Float64(y * Float64(Float64(y * y) * Float64(0.0 - y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * x) / ((1.0 / (x + y)) / (x - y));
	tmp = 0.0;
	if (x <= -6.4e-15)
		tmp = t_0;
	elseif (x <= 9.5e-66)
		tmp = y * ((y * y) * (0.0 - y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] / N[(N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.4e-15], t$95$0, If[LessEqual[x, 9.5e-66], N[(y * N[(N[(y * y), $MachinePrecision] * N[(0.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.3999999999999999e-15 or 9.5000000000000004e-66 < x

   1. Initial program 81.6%

      \[{x}^{4} - {y}^{4} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sqr-pow N/A

         \[\leadsto {x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)} - {\color{blue}{y}}^{4} \]

      2. sqr-pow N/A

         \[\leadsto {x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)} \cdot \color{blue}{{y}^{\left(\frac{4}{2}\right)}} \]

      3. difference-of-squares N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right)} \]

      4. *-commutative N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \cdot \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right), \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left({x}^{\left(\frac{4}{2}\right)}\right), \left({y}^{\left(\frac{4}{2}\right)}\right)\right), \left(\color{blue}{{x}^{\left(\frac{4}{2}\right)}} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left({x}^{2}\right), \left({y}^{\left(\frac{4}{2}\right)}\right)\right), \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot x\right), \left({y}^{\left(\frac{4}{2}\right)}\right)\right), \left({\color{blue}{x}}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({y}^{\left(\frac{4}{2}\right)}\right)\right), \left({\color{blue}{x}}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({y}^{2}\right)\right), \left({x}^{\left(\frac{4}{\color{blue}{2}}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot y\right)\right), \left({x}^{\color{blue}{\left(\frac{4}{2}\right)}} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \left({x}^{\color{blue}{\left(\frac{4}{2}\right)}} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \left({y}^{\left(\frac{4}{2}\right)} + \color{blue}{{x}^{\left(\frac{4}{2}\right)}}\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\left({y}^{\left(\frac{4}{2}\right)}\right), \color{blue}{\left({x}^{\left(\frac{4}{2}\right)}\right)}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\left({y}^{2}\right), \left({x}^{\left(\frac{4}{2}\right)}\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\left(y \cdot y\right), \left({\color{blue}{x}}^{\left(\frac{4}{2}\right)}\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({\color{blue}{x}}^{\left(\frac{4}{2}\right)}\right)\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({x}^{2}\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      20. *-lowering-*.f64 88.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   4. Applied egg-rr 88.9%

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

      1. flip3-- N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{{\left(x \cdot x\right)}^{3} - {\left(y \cdot y\right)}^{3}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(x \cdot x\right) \cdot \left(y \cdot y\right)\right)}\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(y, y\right)}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(x \cdot x\right) \cdot \left(y \cdot y\right)\right)}{{\left(x \cdot x\right)}^{3} - {\left(y \cdot y\right)}^{3}}}\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(y, y\right)}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(x \cdot x\right) \cdot \left(y \cdot y\right)\right)}{{\left(x \cdot x\right)}^{3} - {\left(y \cdot y\right)}^{3}}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(y, y\right)}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

      4. cube-unmult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(x \cdot x\right) \cdot \left(y \cdot y\right)\right)}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) - {\left(y \cdot y\right)}^{3}}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, x\right)\right)\right) \]

      5. swap-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(x \cdot x\right) \cdot \left(y \cdot y\right)\right)}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) - {\left(y \cdot y\right)}^{3}}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, x\right)\right)\right) \]

      6. cube-unmult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(x \cdot x\right) \cdot \left(y \cdot y\right)\right)}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) - \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, x\right)\right)\right) \]

      7. swap-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(x \cdot x\right) \cdot \left(y \cdot y\right)\right)}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) - \left(y \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot \left(y \cdot y\right)\right)}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, x\right)\right)\right) \]

      8. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{1}{\frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) - \left(y \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot \left(y \cdot y\right)\right)}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(x \cdot x\right) \cdot \left(y \cdot y\right)\right)}}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \color{blue}{y}\right), \mathsf{*.f64}\left(x, x\right)\right)\right) \]

   6. Applied egg-rr 88.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{x \cdot x - y \cdot y}}} \cdot \left(y \cdot y + x \cdot x\right) \]
   7. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{1 \cdot \left(y \cdot y + x \cdot x\right)}{\color{blue}{\frac{1}{x \cdot x - y \cdot y}}} \]

      2. *-lft-identity N/A

         \[\leadsto \frac{y \cdot y + x \cdot x}{\frac{\color{blue}{1}}{x \cdot x - y \cdot y}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(y \cdot y + x \cdot x\right), \color{blue}{\left(\frac{1}{x \cdot x - y \cdot y}\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot x + y \cdot y\right), \left(\frac{\color{blue}{1}}{x \cdot x - y \cdot y}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), \left(y \cdot y\right)\right), \left(\frac{\color{blue}{1}}{x \cdot x - y \cdot y}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot y\right)\right), \left(\frac{1}{x \cdot x - y \cdot y}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{1}{x \cdot x - y \cdot y}\right)\right) \]

      8. difference-of-squares N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{1}{\left(x + y\right) \cdot \color{blue}{\left(x - y\right)}}\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{\frac{1}{x + y}}{\color{blue}{x - y}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{/.f64}\left(\left(\frac{1}{x + y}\right), \color{blue}{\left(x - y\right)}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(x + y\right)\right), \left(\color{blue}{x} - y\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, y\right)\right), \left(x - y\right)\right)\right) \]

      13. --lowering--.f64 99.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   8. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{x \cdot x + y \cdot y}{\frac{\frac{1}{x + y}}{x - y}}} \]

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left({x}^{2}\right)}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(x, y\right)\right)\right) \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(x \cdot x\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, y\right)\right)}, \mathsf{\_.f64}\left(x, y\right)\right)\right) \]

       2. *-lowering-*.f64 95.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, y\right)\right)}, \mathsf{\_.f64}\left(x, y\right)\right)\right) \]

   11. Simplified 95.5%

       \[\leadsto \frac{\color{blue}{x \cdot x}}{\frac{\frac{1}{x + y}}{x - y}} \]

   if -6.3999999999999999e-15 < x < 9.5000000000000004e-66

   1. Initial program 100.0%

      \[{x}^{4} - {y}^{4} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot {y}^{4}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({y}^{4}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{{y}^{4}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left({y}^{4}\right)}\right) \]

      4. pow-lowering-pow.f64 98.9%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(y, \color{blue}{4}\right)\right) \]

   5. Simplified 98.9%

      \[\leadsto \color{blue}{0 - {y}^{4}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left({y}^{4}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left({y}^{4}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{neg.f64}\left(\left({y}^{\left(3 + 1\right)}\right)\right) \]

      4. pow-plus N/A

         \[\leadsto \mathsf{neg.f64}\left(\left({y}^{3} \cdot y\right)\right) \]

      5. cube-unmult N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\left(y \cdot \left(y \cdot y\right)\right) \cdot y\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(y, \left(y \cdot \left(y \cdot y\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(y \cdot y\right)\right)\right)\right) \]

      9. *-lowering-*.f64 98.9%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right)\right) \]

   7. Applied egg-rr 98.9%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.4 \cdot 10^{-15}:\\ \;\;\;\;\frac{x \cdot x}{\frac{\frac{1}{x + y}}{x - y}}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-66}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot \left(0 - y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{\frac{\frac{1}{x + y}}{x - y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.1% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(\left(y \cdot y\right) \cdot \left(0 - y\right)\right)\\ \mathbf{if}\;y \leq -4 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.36 \cdot 10^{+19}:\\ \;\;\;\;\frac{x \cdot x}{\frac{1}{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* (* y y) (- 0.0 y)))))
   (if (<= y -4e-18)
     t_0
     (if (<= y 1.36e+19) (/ (* x x) (/ 1.0 (* x x))) t_0))))

C

double code(double x, double y) {
	double t_0 = y * ((y * y) * (0.0 - y));
	double tmp;
	if (y <= -4e-18) {
		tmp = t_0;
	} else if (y <= 1.36e+19) {
		tmp = (x * x) / (1.0 / (x * x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * ((y * y) * (0.0d0 - y))
    if (y <= (-4d-18)) then
        tmp = t_0
    else if (y <= 1.36d+19) then
        tmp = (x * x) / (1.0d0 / (x * x))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * ((y * y) * (0.0 - y));
	double tmp;
	if (y <= -4e-18) {
		tmp = t_0;
	} else if (y <= 1.36e+19) {
		tmp = (x * x) / (1.0 / (x * x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * ((y * y) * (0.0 - y))
	tmp = 0
	if y <= -4e-18:
		tmp = t_0
	elif y <= 1.36e+19:
		tmp = (x * x) / (1.0 / (x * x))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(Float64(y * y) * Float64(0.0 - y)))
	tmp = 0.0
	if (y <= -4e-18)
		tmp = t_0;
	elseif (y <= 1.36e+19)
		tmp = Float64(Float64(x * x) / Float64(1.0 / Float64(x * x)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * ((y * y) * (0.0 - y));
	tmp = 0.0;
	if (y <= -4e-18)
		tmp = t_0;
	elseif (y <= 1.36e+19)
		tmp = (x * x) / (1.0 / (x * x));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(N[(y * y), $MachinePrecision] * N[(0.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4e-18], t$95$0, If[LessEqual[y, 1.36e+19], N[(N[(x * x), $MachinePrecision] / N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.0000000000000003e-18 or 1.36e19 < y

   1. Initial program 78.9%

      \[{x}^{4} - {y}^{4} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot {y}^{4}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({y}^{4}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{{y}^{4}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left({y}^{4}\right)}\right) \]

      4. pow-lowering-pow.f64 82.8%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(y, \color{blue}{4}\right)\right) \]

   5. Simplified 82.8%

      \[\leadsto \color{blue}{0 - {y}^{4}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left({y}^{4}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left({y}^{4}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{neg.f64}\left(\left({y}^{\left(3 + 1\right)}\right)\right) \]

      4. pow-plus N/A

         \[\leadsto \mathsf{neg.f64}\left(\left({y}^{3} \cdot y\right)\right) \]

      5. cube-unmult N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\left(y \cdot \left(y \cdot y\right)\right) \cdot y\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(y, \left(y \cdot \left(y \cdot y\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(y \cdot y\right)\right)\right)\right) \]

      9. *-lowering-*.f64 82.7%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right)\right) \]

   7. Applied egg-rr 82.7%

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

   if -4.0000000000000003e-18 < y < 1.36e19

   1. Initial program 100.0%

      \[{x}^{4} - {y}^{4} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sqr-pow N/A

         \[\leadsto {x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)} - {\color{blue}{y}}^{4} \]

      2. sqr-pow N/A

         \[\leadsto {x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)} \cdot \color{blue}{{y}^{\left(\frac{4}{2}\right)}} \]

      3. difference-of-squares N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right)} \]

      4. *-commutative N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \cdot \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right), \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left({x}^{\left(\frac{4}{2}\right)}\right), \left({y}^{\left(\frac{4}{2}\right)}\right)\right), \left(\color{blue}{{x}^{\left(\frac{4}{2}\right)}} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left({x}^{2}\right), \left({y}^{\left(\frac{4}{2}\right)}\right)\right), \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot x\right), \left({y}^{\left(\frac{4}{2}\right)}\right)\right), \left({\color{blue}{x}}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({y}^{\left(\frac{4}{2}\right)}\right)\right), \left({\color{blue}{x}}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({y}^{2}\right)\right), \left({x}^{\left(\frac{4}{\color{blue}{2}}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot y\right)\right), \left({x}^{\color{blue}{\left(\frac{4}{2}\right)}} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \left({x}^{\color{blue}{\left(\frac{4}{2}\right)}} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \left({y}^{\left(\frac{4}{2}\right)} + \color{blue}{{x}^{\left(\frac{4}{2}\right)}}\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\left({y}^{\left(\frac{4}{2}\right)}\right), \color{blue}{\left({x}^{\left(\frac{4}{2}\right)}\right)}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\left({y}^{2}\right), \left({x}^{\left(\frac{4}{2}\right)}\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\left(y \cdot y\right), \left({\color{blue}{x}}^{\left(\frac{4}{2}\right)}\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({\color{blue}{x}}^{\left(\frac{4}{2}\right)}\right)\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({x}^{2}\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      20. *-lowering-*.f64 99.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\left(x \cdot x - y \cdot y\right) \cdot \left(y \cdot y + x \cdot x\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4}} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto {x}^{\left(2 \cdot \color{blue}{2}\right)} \]

      2. pow-sqr N/A

         \[\leadsto {x}^{2} \cdot \color{blue}{{x}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \left(x \cdot x\right) \cdot {\color{blue}{x}}^{2} \]

      4. associate-*r* N/A

         \[\leadsto x \cdot \color{blue}{\left(x \cdot {x}^{2}\right)} \]

      5. unpow2 N/A

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

      6. cube-mult N/A

         \[\leadsto x \cdot {x}^{\color{blue}{3}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{3}\right)}\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot {x}^{\color{blue}{2}}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      12. *-lowering-*.f64 93.1%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   7. Simplified 93.1%

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot x\right), \color{blue}{\left(x \cdot x\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{x} \cdot x\right)\right) \]

      4. *-lowering-*.f64 93.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   9. Applied egg-rr 93.1%

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} \]
   10. Step-by-step derivation

       1. remove-double-div N/A

          \[\leadsto \left(x \cdot x\right) \cdot \frac{1}{\color{blue}{\frac{1}{x \cdot x}}} \]

       2. un-div-inv N/A

          \[\leadsto \frac{x \cdot x}{\color{blue}{\frac{1}{x \cdot x}}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(x \cdot x\right), \color{blue}{\left(\frac{1}{x \cdot x}\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{\color{blue}{1}}{x \cdot x}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot x\right)}\right)\right) \]

       6. *-lowering-*.f64 93.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   11. Applied egg-rr 93.1%

       \[\leadsto \color{blue}{\frac{x \cdot x}{\frac{1}{x \cdot x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-18}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot \left(0 - y\right)\right)\\ \mathbf{elif}\;y \leq 1.36 \cdot 10^{+19}:\\ \;\;\;\;\frac{x \cdot x}{\frac{1}{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot \left(0 - y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.2% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(\left(y \cdot y\right) \cdot \left(0 - y\right)\right)\\ \mathbf{if}\;y \leq -8.8 \cdot 10^{-16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+19}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* (* y y) (- 0.0 y)))))
   (if (<= y -8.8e-16) t_0 (if (<= y 5.6e+19) (* x (* x (* x x))) t_0))))

C

double code(double x, double y) {
	double t_0 = y * ((y * y) * (0.0 - y));
	double tmp;
	if (y <= -8.8e-16) {
		tmp = t_0;
	} else if (y <= 5.6e+19) {
		tmp = x * (x * (x * x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * ((y * y) * (0.0d0 - y))
    if (y <= (-8.8d-16)) then
        tmp = t_0
    else if (y <= 5.6d+19) then
        tmp = x * (x * (x * x))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * ((y * y) * (0.0 - y));
	double tmp;
	if (y <= -8.8e-16) {
		tmp = t_0;
	} else if (y <= 5.6e+19) {
		tmp = x * (x * (x * x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * ((y * y) * (0.0 - y))
	tmp = 0
	if y <= -8.8e-16:
		tmp = t_0
	elif y <= 5.6e+19:
		tmp = x * (x * (x * x))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(Float64(y * y) * Float64(0.0 - y)))
	tmp = 0.0
	if (y <= -8.8e-16)
		tmp = t_0;
	elseif (y <= 5.6e+19)
		tmp = Float64(x * Float64(x * Float64(x * x)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * ((y * y) * (0.0 - y));
	tmp = 0.0;
	if (y <= -8.8e-16)
		tmp = t_0;
	elseif (y <= 5.6e+19)
		tmp = x * (x * (x * x));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(N[(y * y), $MachinePrecision] * N[(0.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8.8e-16], t$95$0, If[LessEqual[y, 5.6e+19], N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.80000000000000001e-16 or 5.6e19 < y

   1. Initial program 78.9%

      \[{x}^{4} - {y}^{4} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot {y}^{4}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({y}^{4}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{{y}^{4}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left({y}^{4}\right)}\right) \]

      4. pow-lowering-pow.f64 82.8%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(y, \color{blue}{4}\right)\right) \]

   5. Simplified 82.8%

      \[\leadsto \color{blue}{0 - {y}^{4}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left({y}^{4}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left({y}^{4}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{neg.f64}\left(\left({y}^{\left(3 + 1\right)}\right)\right) \]

      4. pow-plus N/A

         \[\leadsto \mathsf{neg.f64}\left(\left({y}^{3} \cdot y\right)\right) \]

      5. cube-unmult N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\left(y \cdot \left(y \cdot y\right)\right) \cdot y\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(y, \left(y \cdot \left(y \cdot y\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(y \cdot y\right)\right)\right)\right) \]

      9. *-lowering-*.f64 82.7%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right)\right) \]

   7. Applied egg-rr 82.7%

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

   if -8.80000000000000001e-16 < y < 5.6e19

   1. Initial program 100.0%

      \[{x}^{4} - {y}^{4} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sqr-pow N/A

         \[\leadsto {x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)} - {\color{blue}{y}}^{4} \]

      2. sqr-pow N/A

         \[\leadsto {x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)} \cdot \color{blue}{{y}^{\left(\frac{4}{2}\right)}} \]

      3. difference-of-squares N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right)} \]

      4. *-commutative N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \cdot \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right), \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left({x}^{\left(\frac{4}{2}\right)}\right), \left({y}^{\left(\frac{4}{2}\right)}\right)\right), \left(\color{blue}{{x}^{\left(\frac{4}{2}\right)}} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left({x}^{2}\right), \left({y}^{\left(\frac{4}{2}\right)}\right)\right), \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot x\right), \left({y}^{\left(\frac{4}{2}\right)}\right)\right), \left({\color{blue}{x}}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({y}^{\left(\frac{4}{2}\right)}\right)\right), \left({\color{blue}{x}}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({y}^{2}\right)\right), \left({x}^{\left(\frac{4}{\color{blue}{2}}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot y\right)\right), \left({x}^{\color{blue}{\left(\frac{4}{2}\right)}} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \left({x}^{\color{blue}{\left(\frac{4}{2}\right)}} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \left({y}^{\left(\frac{4}{2}\right)} + \color{blue}{{x}^{\left(\frac{4}{2}\right)}}\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\left({y}^{\left(\frac{4}{2}\right)}\right), \color{blue}{\left({x}^{\left(\frac{4}{2}\right)}\right)}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\left({y}^{2}\right), \left({x}^{\left(\frac{4}{2}\right)}\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\left(y \cdot y\right), \left({\color{blue}{x}}^{\left(\frac{4}{2}\right)}\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({\color{blue}{x}}^{\left(\frac{4}{2}\right)}\right)\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({x}^{2}\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      20. *-lowering-*.f64 99.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\left(x \cdot x - y \cdot y\right) \cdot \left(y \cdot y + x \cdot x\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4}} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto {x}^{\left(2 \cdot \color{blue}{2}\right)} \]

      2. pow-sqr N/A

         \[\leadsto {x}^{2} \cdot \color{blue}{{x}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \left(x \cdot x\right) \cdot {\color{blue}{x}}^{2} \]

      4. associate-*r* N/A

         \[\leadsto x \cdot \color{blue}{\left(x \cdot {x}^{2}\right)} \]

      5. unpow2 N/A

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

      6. cube-mult N/A

         \[\leadsto x \cdot {x}^{\color{blue}{3}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{3}\right)}\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot {x}^{\color{blue}{2}}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      12. *-lowering-*.f64 93.1%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   7. Simplified 93.1%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{-16}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot \left(0 - y\right)\right)\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+19}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot y\right) \cdot \left(0 - y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.8% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot x + y \cdot y}{\frac{\frac{1}{x + y}}{x - y}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (+ (* x x) (* y y)) (/ (/ 1.0 (+ x y)) (- x y))))

C

double code(double x, double y) {
	return ((x * x) + (y * y)) / ((1.0 / (x + y)) / (x - y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * x) + (y * y)) / ((1.0d0 / (x + y)) / (x - y))
end function

Java

public static double code(double x, double y) {
	return ((x * x) + (y * y)) / ((1.0 / (x + y)) / (x - y));
}

Python

def code(x, y):
	return ((x * x) + (y * y)) / ((1.0 / (x + y)) / (x - y))

Julia

function code(x, y)
	return Float64(Float64(Float64(x * x) + Float64(y * y)) / Float64(Float64(1.0 / Float64(x + y)) / Float64(x - y)))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * x) + (y * y)) / ((1.0 / (x + y)) / (x - y));
end

Wolfram

code[x_, y_] := N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.5%

   \[{x}^{4} - {y}^{4} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sqr-pow N/A

      \[\leadsto {x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)} - {\color{blue}{y}}^{4} \]

   2. sqr-pow N/A

      \[\leadsto {x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)} \cdot \color{blue}{{y}^{\left(\frac{4}{2}\right)}} \]

   3. difference-of-squares N/A

      \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right)} \]

   4. *-commutative N/A

      \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \cdot \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right), \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)}\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left({x}^{\left(\frac{4}{2}\right)}\right), \left({y}^{\left(\frac{4}{2}\right)}\right)\right), \left(\color{blue}{{x}^{\left(\frac{4}{2}\right)}} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left({x}^{2}\right), \left({y}^{\left(\frac{4}{2}\right)}\right)\right), \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot x\right), \left({y}^{\left(\frac{4}{2}\right)}\right)\right), \left({\color{blue}{x}}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({y}^{\left(\frac{4}{2}\right)}\right)\right), \left({\color{blue}{x}}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({y}^{2}\right)\right), \left({x}^{\left(\frac{4}{\color{blue}{2}}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot y\right)\right), \left({x}^{\color{blue}{\left(\frac{4}{2}\right)}} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \left({x}^{\color{blue}{\left(\frac{4}{2}\right)}} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \left({y}^{\left(\frac{4}{2}\right)} + \color{blue}{{x}^{\left(\frac{4}{2}\right)}}\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\left({y}^{\left(\frac{4}{2}\right)}\right), \color{blue}{\left({x}^{\left(\frac{4}{2}\right)}\right)}\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\left({y}^{2}\right), \left({x}^{\left(\frac{4}{2}\right)}\right)\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\left(y \cdot y\right), \left({\color{blue}{x}}^{\left(\frac{4}{2}\right)}\right)\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({\color{blue}{x}}^{\left(\frac{4}{2}\right)}\right)\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({x}^{2}\right)\right)\right) \]

   19. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(x \cdot \color{blue}{x}\right)\right)\right) \]

   20. *-lowering-*.f64 93.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

4. Applied egg-rr 93.5%

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

   1. flip3-- N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{{\left(x \cdot x\right)}^{3} - {\left(y \cdot y\right)}^{3}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(x \cdot x\right) \cdot \left(y \cdot y\right)\right)}\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(y, y\right)}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(x \cdot x\right) \cdot \left(y \cdot y\right)\right)}{{\left(x \cdot x\right)}^{3} - {\left(y \cdot y\right)}^{3}}}\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(y, y\right)}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(x \cdot x\right) \cdot \left(y \cdot y\right)\right)}{{\left(x \cdot x\right)}^{3} - {\left(y \cdot y\right)}^{3}}\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{*.f64}\left(y, y\right)}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

   4. cube-unmult N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(x \cdot x\right) \cdot \left(y \cdot y\right)\right)}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) - {\left(y \cdot y\right)}^{3}}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, x\right)\right)\right) \]

   5. swap-sqr N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(x \cdot x\right) \cdot \left(y \cdot y\right)\right)}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) - {\left(y \cdot y\right)}^{3}}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, x\right)\right)\right) \]

   6. cube-unmult N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(x \cdot x\right) \cdot \left(y \cdot y\right)\right)}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) - \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, x\right)\right)\right) \]

   7. swap-sqr N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(x \cdot x\right) \cdot \left(y \cdot y\right)\right)}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) - \left(y \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot \left(y \cdot y\right)\right)}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, x\right)\right)\right) \]

   8. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{1}{\frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) - \left(y \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot \left(y \cdot y\right)\right)}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(x \cdot x\right) \cdot \left(y \cdot y\right)\right)}}\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \color{blue}{y}\right), \mathsf{*.f64}\left(x, x\right)\right)\right) \]

6. Applied egg-rr 93.5%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{x \cdot x - y \cdot y}}} \cdot \left(y \cdot y + x \cdot x\right) \]
7. Step-by-step derivation

   1. associate-*l/ N/A

      \[\leadsto \frac{1 \cdot \left(y \cdot y + x \cdot x\right)}{\color{blue}{\frac{1}{x \cdot x - y \cdot y}}} \]

   2. *-lft-identity N/A

      \[\leadsto \frac{y \cdot y + x \cdot x}{\frac{\color{blue}{1}}{x \cdot x - y \cdot y}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(y \cdot y + x \cdot x\right), \color{blue}{\left(\frac{1}{x \cdot x - y \cdot y}\right)}\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot x + y \cdot y\right), \left(\frac{\color{blue}{1}}{x \cdot x - y \cdot y}\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(x \cdot x\right), \left(y \cdot y\right)\right), \left(\frac{\color{blue}{1}}{x \cdot x - y \cdot y}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot y\right)\right), \left(\frac{1}{x \cdot x - y \cdot y}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{1}{x \cdot x - y \cdot y}\right)\right) \]

   8. difference-of-squares N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{1}{\left(x + y\right) \cdot \color{blue}{\left(x - y\right)}}\right)\right) \]

   9. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \left(\frac{\frac{1}{x + y}}{\color{blue}{x - y}}\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{/.f64}\left(\left(\frac{1}{x + y}\right), \color{blue}{\left(x - y\right)}\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(x + y\right)\right), \left(\color{blue}{x} - y\right)\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, y\right)\right), \left(x - y\right)\right)\right) \]

   13. --lowering--.f64 99.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, y\right)\right), \mathsf{\_.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

8. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{x \cdot x + y \cdot y}{\frac{\frac{1}{x + y}}{x - y}}} \]
9. Add Preprocessing

Alternative 7: 57.3% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot \left(x \cdot x\right)\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (* x (* x x))))

C

double code(double x, double y) {
	return x * (x * (x * x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (x * (x * x))
end function

Java

public static double code(double x, double y) {
	return x * (x * (x * x));
}

Python

def code(x, y):
	return x * (x * (x * x))

Julia

function code(x, y)
	return Float64(x * Float64(x * Float64(x * x)))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (x * (x * x));
end

Wolfram

code[x_, y_] := N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.5%

   \[{x}^{4} - {y}^{4} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sqr-pow N/A

      \[\leadsto {x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)} - {\color{blue}{y}}^{4} \]

   2. sqr-pow N/A

      \[\leadsto {x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)} \cdot \color{blue}{{y}^{\left(\frac{4}{2}\right)}} \]

   3. difference-of-squares N/A

      \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right)} \]

   4. *-commutative N/A

      \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \cdot \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right), \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)}\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left({x}^{\left(\frac{4}{2}\right)}\right), \left({y}^{\left(\frac{4}{2}\right)}\right)\right), \left(\color{blue}{{x}^{\left(\frac{4}{2}\right)}} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left({x}^{2}\right), \left({y}^{\left(\frac{4}{2}\right)}\right)\right), \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot x\right), \left({y}^{\left(\frac{4}{2}\right)}\right)\right), \left({\color{blue}{x}}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({y}^{\left(\frac{4}{2}\right)}\right)\right), \left({\color{blue}{x}}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({y}^{2}\right)\right), \left({x}^{\left(\frac{4}{\color{blue}{2}}\right)} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(y \cdot y\right)\right), \left({x}^{\color{blue}{\left(\frac{4}{2}\right)}} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \left({x}^{\color{blue}{\left(\frac{4}{2}\right)}} + {y}^{\left(\frac{4}{2}\right)}\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \left({y}^{\left(\frac{4}{2}\right)} + \color{blue}{{x}^{\left(\frac{4}{2}\right)}}\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\left({y}^{\left(\frac{4}{2}\right)}\right), \color{blue}{\left({x}^{\left(\frac{4}{2}\right)}\right)}\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\left({y}^{2}\right), \left({x}^{\left(\frac{4}{2}\right)}\right)\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\left(y \cdot y\right), \left({\color{blue}{x}}^{\left(\frac{4}{2}\right)}\right)\right)\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({\color{blue}{x}}^{\left(\frac{4}{2}\right)}\right)\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({x}^{2}\right)\right)\right) \]

   19. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(x \cdot \color{blue}{x}\right)\right)\right) \]

   20. *-lowering-*.f64 93.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

4. Applied egg-rr 93.5%

   \[\leadsto \color{blue}{\left(x \cdot x - y \cdot y\right) \cdot \left(y \cdot y + x \cdot x\right)} \]

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{{x}^{4}} \]
6. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto {x}^{\left(2 \cdot \color{blue}{2}\right)} \]

   2. pow-sqr N/A

      \[\leadsto {x}^{2} \cdot \color{blue}{{x}^{2}} \]

   3. unpow2 N/A

      \[\leadsto \left(x \cdot x\right) \cdot {\color{blue}{x}}^{2} \]

   4. associate-*r* N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot {x}^{2}\right)} \]

   5. unpow2 N/A

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

   6. cube-mult N/A

      \[\leadsto x \cdot {x}^{\color{blue}{3}} \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{3}\right)}\right) \]

   8. cube-mult N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot {x}^{\color{blue}{2}}\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

   12. *-lowering-*.f64 55.7%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

7. Simplified 55.7%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024192 
(FPCore (x y)
  :name "Radioactive exchange between two surfaces"
  :precision binary64
  (- (pow x 4.0) (pow y 4.0)))