Radioactive exchange between two surfaces

Percentage Accurate: 85.9% → 99.8%

Time: 12.5s

Alternatives: 5

Speedup: 12.1×

• 60.8% 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: 85.9% 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, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.2%

   \[{x}^{4} - {y}^{4} \]
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 94.0%

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

4. Applied egg-rr 94.0%

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

   1. *-commutative N/A

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

   2. flip-+ N/A

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

   3. associate-*l/ N/A

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

   4. div-inv N/A

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

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

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

6. Applied egg-rr 42.8%

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

   1. *-commutative N/A

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

   2. difference-of-squares N/A

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

   3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   12. associate-*r* N/A

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

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

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

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

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

   15. *-lowering-*.f64 42.8%

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

8. Applied egg-rr 42.8%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. /-rgt-identity N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. associate-*l* N/A

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

   7. div-inv N/A

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

   8. associate-*r* N/A

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

   9. flip-+ N/A

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

   10. +-commutative N/A

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

10. Applied egg-rr 99.9%

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

Alternative 2: 91.0% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ t_1 := \frac{x \cdot x - y \cdot y}{\frac{1}{x \cdot x}}\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{+160}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{-58}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.26 \cdot 10^{-10}:\\ \;\;\;\;0 - y \cdot \left(y \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* x (* x x))))
        (t_1 (/ (- (* x x) (* y y)) (/ 1.0 (* x x)))))
   (if (<= x -1.4e+160)
     t_0
     (if (<= x -6.6e-58)
       t_1
       (if (<= x 1.26e-10)
         (- 0.0 (* y (* y (* y y))))
         (if (<= x 1.35e+154) t_1 t_0))))))

C

double code(double x, double y) {
	double t_0 = x * (x * (x * x));
	double t_1 = ((x * x) - (y * y)) / (1.0 / (x * x));
	double tmp;
	if (x <= -1.4e+160) {
		tmp = t_0;
	} else if (x <= -6.6e-58) {
		tmp = t_1;
	} else if (x <= 1.26e-10) {
		tmp = 0.0 - (y * (y * (y * y)));
	} else if (x <= 1.35e+154) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = x * (x * (x * x))
    t_1 = ((x * x) - (y * y)) / (1.0d0 / (x * x))
    if (x <= (-1.4d+160)) then
        tmp = t_0
    else if (x <= (-6.6d-58)) then
        tmp = t_1
    else if (x <= 1.26d-10) then
        tmp = 0.0d0 - (y * (y * (y * y)))
    else if (x <= 1.35d+154) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (x * (x * x));
	double t_1 = ((x * x) - (y * y)) / (1.0 / (x * x));
	double tmp;
	if (x <= -1.4e+160) {
		tmp = t_0;
	} else if (x <= -6.6e-58) {
		tmp = t_1;
	} else if (x <= 1.26e-10) {
		tmp = 0.0 - (y * (y * (y * y)));
	} else if (x <= 1.35e+154) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (x * (x * x))
	t_1 = ((x * x) - (y * y)) / (1.0 / (x * x))
	tmp = 0
	if x <= -1.4e+160:
		tmp = t_0
	elif x <= -6.6e-58:
		tmp = t_1
	elif x <= 1.26e-10:
		tmp = 0.0 - (y * (y * (y * y)))
	elif x <= 1.35e+154:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(x * Float64(x * x)))
	t_1 = Float64(Float64(Float64(x * x) - Float64(y * y)) / Float64(1.0 / Float64(x * x)))
	tmp = 0.0
	if (x <= -1.4e+160)
		tmp = t_0;
	elseif (x <= -6.6e-58)
		tmp = t_1;
	elseif (x <= 1.26e-10)
		tmp = Float64(0.0 - Float64(y * Float64(y * Float64(y * y))));
	elseif (x <= 1.35e+154)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (x * (x * x));
	t_1 = ((x * x) - (y * y)) / (1.0 / (x * x));
	tmp = 0.0;
	if (x <= -1.4e+160)
		tmp = t_0;
	elseif (x <= -6.6e-58)
		tmp = t_1;
	elseif (x <= 1.26e-10)
		tmp = 0.0 - (y * (y * (y * y)));
	elseif (x <= 1.35e+154)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x * x), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.4e+160], t$95$0, If[LessEqual[x, -6.6e-58], t$95$1, If[LessEqual[x, 1.26e-10], N[(0.0 - N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.35e+154], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4e160 or 1.35000000000000003e154 < x

   1. Initial program 59.3%

      \[{x}^{4} - {y}^{4} \]
   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 74.6%

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

   4. Applied egg-rr 74.6%

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

   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 88.1%

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

   7. Simplified 88.1%

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

   if -1.4e160 < x < -6.60000000000000052e-58 or 1.26000000000000004e-10 < x < 1.35000000000000003e154

   1. Initial program 75.9%

      \[{x}^{4} - {y}^{4} \]
   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 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

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

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

      8. remove-double-div N/A

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

      9. clear-num N/A

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

      10. flip3-+ N/A

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      2. unpow2 N/A

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

      3. *-lowering-*.f64 95.1%

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

   9. Simplified 95.1%

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

   if -6.60000000000000052e-58 < x < 1.26000000000000004e-10

   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 96.8%

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

   5. Simplified 96.8%

      \[\leadsto \color{blue}{0 - {y}^{4}} \]
   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 96.8%

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

   7. Applied egg-rr 96.8%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+160}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{-58}:\\ \;\;\;\;\frac{x \cdot x - y \cdot y}{\frac{1}{x \cdot x}}\\ \mathbf{elif}\;x \leq 1.26 \cdot 10^{-10}:\\ \;\;\;\;0 - y \cdot \left(y \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{x \cdot x - y \cdot y}{\frac{1}{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.5% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{+160}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x - y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* x (* x x)))))
   (if (<= x -1.4e+160)
     t_0
     (if (<= x 1.35e+154) (* (+ (* x x) (* y y)) (- (* x x) (* y y))) t_0))))

C

double code(double x, double y) {
	double t_0 = x * (x * (x * x));
	double tmp;
	if (x <= -1.4e+160) {
		tmp = t_0;
	} else if (x <= 1.35e+154) {
		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 * (x * x))
    if (x <= (-1.4d+160)) then
        tmp = t_0
    else if (x <= 1.35d+154) 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 * (x * x));
	double tmp;
	if (x <= -1.4e+160) {
		tmp = t_0;
	} else if (x <= 1.35e+154) {
		tmp = ((x * x) + (y * y)) * ((x * x) - (y * y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (x * (x * x))
	tmp = 0
	if x <= -1.4e+160:
		tmp = t_0
	elif x <= 1.35e+154:
		tmp = ((x * x) + (y * y)) * ((x * x) - (y * y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(x * Float64(x * x)))
	tmp = 0.0
	if (x <= -1.4e+160)
		tmp = t_0;
	elseif (x <= 1.35e+154)
		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 * (x * x));
	tmp = 0.0;
	if (x <= -1.4e+160)
		tmp = t_0;
	elseif (x <= 1.35e+154)
		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[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.4e+160], t$95$0, If[LessEqual[x, 1.35e+154], 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 < -1.4e160 or 1.35000000000000003e154 < x

   1. Initial program 59.3%

      \[{x}^{4} - {y}^{4} \]
   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 74.6%

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

   4. Applied egg-rr 74.6%

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

   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 88.1%

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

   7. Simplified 88.1%

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

   if -1.4e160 < x < 1.35000000000000003e154

   1. Initial program 90.4%

      \[{x}^{4} - {y}^{4} \]
   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 99.8%

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

   4. Applied egg-rr 99.8%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+160}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x - y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.2% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;x \leq -1.3 \cdot 10^{-51}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+85}:\\ \;\;\;\;0 - y \cdot \left(y \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* x (* x x)))))
   (if (<= x -1.3e-51)
     t_0
     (if (<= x 2.25e+85) (- 0.0 (* y (* y (* y y)))) t_0))))

C

double code(double x, double y) {
	double t_0 = x * (x * (x * x));
	double tmp;
	if (x <= -1.3e-51) {
		tmp = t_0;
	} else if (x <= 2.25e+85) {
		tmp = 0.0 - (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 * (x * x))
    if (x <= (-1.3d-51)) then
        tmp = t_0
    else if (x <= 2.25d+85) then
        tmp = 0.0d0 - (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 * (x * x));
	double tmp;
	if (x <= -1.3e-51) {
		tmp = t_0;
	} else if (x <= 2.25e+85) {
		tmp = 0.0 - (y * (y * (y * y)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (x * (x * x))
	tmp = 0
	if x <= -1.3e-51:
		tmp = t_0
	elif x <= 2.25e+85:
		tmp = 0.0 - (y * (y * (y * y)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(x * Float64(x * x)))
	tmp = 0.0
	if (x <= -1.3e-51)
		tmp = t_0;
	elseif (x <= 2.25e+85)
		tmp = Float64(0.0 - 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 * (x * x));
	tmp = 0.0;
	if (x <= -1.3e-51)
		tmp = t_0;
	elseif (x <= 2.25e+85)
		tmp = 0.0 - (y * (y * (y * y)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.3e-51], t$95$0, If[LessEqual[x, 2.25e+85], N[(0.0 - N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3e-51 or 2.25000000000000003e85 < x

   1. Initial program 65.9%

      \[{x}^{4} - {y}^{4} \]
   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 87.7%

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

   4. Applied egg-rr 87.7%

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

   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 75.6%

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

   7. Simplified 75.6%

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

   if -1.3e-51 < x < 2.25000000000000003e85

   1. Initial program 99.2%

      \[{x}^{4} - {y}^{4} \]
   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 92.1%

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

   5. Simplified 92.1%

      \[\leadsto \color{blue}{0 - {y}^{4}} \]
   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 92.0%

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

   7. Applied egg-rr 92.0%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-51}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+85}:\\ \;\;\;\;0 - y \cdot \left(y \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.6% 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 83.2%

   \[{x}^{4} - {y}^{4} \]
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 94.0%

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

4. Applied egg-rr 94.0%

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

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 57.7%

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

7. Simplified 57.7%

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

Reproduce

herbie shell --seed 2024193 
(FPCore (x y)
  :name "Radioactive exchange between two surfaces"
  :precision binary64
  (- (pow x 4.0) (pow y 4.0)))