Radioactive exchange between two surfaces

Percentage Accurate: 85.5% → 95.2%

Time: 25.4s

Alternatives: 7

Speedup: 10.2×

• 61.4% 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: 85.5% 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: 95.2% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(x \cdot x - y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \frac{1}{\frac{-1}{y \cdot y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.35e+154)
   (* (- (* x x) (* y y)) (+ (* x x) (* y y)))
   (* (* y y) (/ 1.0 (/ -1.0 (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.35e+154) {
		tmp = ((x * x) - (y * y)) * ((x * x) + (y * y));
	} else {
		tmp = (y * y) * (1.0 / (-1.0 / (y * y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.35d+154) then
        tmp = ((x * x) - (y * y)) * ((x * x) + (y * y))
    else
        tmp = (y * y) * (1.0d0 / ((-1.0d0) / (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.35e+154) {
		tmp = ((x * x) - (y * y)) * ((x * x) + (y * y));
	} else {
		tmp = (y * y) * (1.0 / (-1.0 / (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.35e+154:
		tmp = ((x * x) - (y * y)) * ((x * x) + (y * y))
	else:
		tmp = (y * y) * (1.0 / (-1.0 / (y * y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.35e+154)
		tmp = Float64(Float64(Float64(x * x) - Float64(y * y)) * Float64(Float64(x * x) + Float64(y * y)));
	else
		tmp = Float64(Float64(y * y) * Float64(1.0 / Float64(-1.0 / Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.35e+154)
		tmp = ((x * x) - (y * y)) * ((x * x) + (y * y));
	else
		tmp = (y * y) * (1.0 / (-1.0 / (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 1.35000000000000003e154

   1. Initial program 87.6%

      \[{x}^{4} - {y}^{4} \]
   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. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. fmm-def N/A

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

      7. metadata-eval N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \mathsf{fma}\left(x, x, \mathsf{neg}\left({y}^{2}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \mathsf{fma}\left(x, x, \mathsf{neg}\left(y \cdot y\right)\right) \]

      9. *-commutative N/A

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

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

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

   4. Applied egg-rr 96.6%

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

   if 1.35000000000000003e154 < y

   1. Initial program 53.3%

      \[{x}^{4} - {y}^{4} \]
   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. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. fmm-def N/A

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

      7. metadata-eval N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \mathsf{fma}\left(x, x, \mathsf{neg}\left({y}^{2}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \mathsf{fma}\left(x, x, \mathsf{neg}\left(y \cdot y\right)\right) \]

      9. *-commutative N/A

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

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

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

   4. Applied egg-rr 66.7%

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

      1. flip-- N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. remove-double-div N/A

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

      5. +-commutative N/A

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

      6. associate-/l/ N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 86.7%

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

   9. Simplified 86.7%

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

   10. Taylor expanded in y around inf

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

       1. unpow2 N/A

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

       2. *-lowering-*.f64 86.7%

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

   12. Simplified 86.7%

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

4. Final simplification 95.4%

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

Alternative 2: 90.7% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ t_1 := \left(x \cdot x\right) \cdot \left(x \cdot x - y \cdot y\right)\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{+178}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 26.5:\\ \;\;\;\;y \cdot \left(y \cdot \left(0 - y \cdot y\right)\right)\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+145}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* x (* x x)))) (t_1 (* (* x x) (- (* x x) (* y y)))))
   (if (<= x -1.8e+178)
     t_0
     (if (<= x -1.05e-72)
       t_1
       (if (<= x 26.5)
         (* y (* y (- 0.0 (* y y))))
         (if (<= x 3.8e+145) t_1 t_0))))))

C

double code(double x, double y) {
	double t_0 = x * (x * (x * x));
	double t_1 = (x * x) * ((x * x) - (y * y));
	double tmp;
	if (x <= -1.8e+178) {
		tmp = t_0;
	} else if (x <= -1.05e-72) {
		tmp = t_1;
	} else if (x <= 26.5) {
		tmp = y * (y * (0.0 - (y * y)));
	} else if (x <= 3.8e+145) {
		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) * ((x * x) - (y * y))
    if (x <= (-1.8d+178)) then
        tmp = t_0
    else if (x <= (-1.05d-72)) then
        tmp = t_1
    else if (x <= 26.5d0) then
        tmp = y * (y * (0.0d0 - (y * y)))
    else if (x <= 3.8d+145) 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) * ((x * x) - (y * y));
	double tmp;
	if (x <= -1.8e+178) {
		tmp = t_0;
	} else if (x <= -1.05e-72) {
		tmp = t_1;
	} else if (x <= 26.5) {
		tmp = y * (y * (0.0 - (y * y)));
	} else if (x <= 3.8e+145) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (x * (x * x))
	t_1 = (x * x) * ((x * x) - (y * y))
	tmp = 0
	if x <= -1.8e+178:
		tmp = t_0
	elif x <= -1.05e-72:
		tmp = t_1
	elif x <= 26.5:
		tmp = y * (y * (0.0 - (y * y)))
	elif x <= 3.8e+145:
		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(x * x) * Float64(Float64(x * x) - Float64(y * y)))
	tmp = 0.0
	if (x <= -1.8e+178)
		tmp = t_0;
	elseif (x <= -1.05e-72)
		tmp = t_1;
	elseif (x <= 26.5)
		tmp = Float64(y * Float64(y * Float64(0.0 - Float64(y * y))));
	elseif (x <= 3.8e+145)
		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) * ((x * x) - (y * y));
	tmp = 0.0;
	if (x <= -1.8e+178)
		tmp = t_0;
	elseif (x <= -1.05e-72)
		tmp = t_1;
	elseif (x <= 26.5)
		tmp = y * (y * (0.0 - (y * y)));
	elseif (x <= 3.8e+145)
		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[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.8e+178], t$95$0, If[LessEqual[x, -1.05e-72], t$95$1, If[LessEqual[x, 26.5], N[(y * N[(y * N[(0.0 - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.8e+145], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.7999999999999999e178 or 3.80000000000000012e145 < x

   1. Initial program 61.2%

      \[{x}^{4} - {y}^{4} \]
   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. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. fmm-def N/A

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

      7. metadata-eval N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \mathsf{fma}\left(x, x, \mathsf{neg}\left({y}^{2}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \mathsf{fma}\left(x, x, \mathsf{neg}\left(y \cdot y\right)\right) \]

      9. *-commutative N/A

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

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

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

   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. +-lft-identity N/A

         \[\leadsto {x}^{2} \cdot \left(0 + \color{blue}{{x}^{2}}\right) \]

      4. unpow2 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{0} + {x}^{2}\right) \]

      5. mul0-lft N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(0 \cdot {y}^{2} + {\color{blue}{x}}^{2}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(-1 + 1\right) \cdot {y}^{2} + {x}^{2}\right) \]

      7. distribute-lft1-in N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(-1 \cdot {y}^{2} + {y}^{2}\right) + {\color{blue}{x}}^{2}\right) \]

      8. associate-+r+ N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(-1 \cdot {y}^{2} + \color{blue}{\left({y}^{2} + {x}^{2}\right)}\right) \]

      9. +-commutative N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(-1 \cdot {y}^{2} + \left({x}^{2} + \color{blue}{{y}^{2}}\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(-1 \cdot {y}^{2} + \left({x}^{2} + {y}^{2}\right)\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto x \cdot \left(x \cdot \left(-1 \cdot {y}^{2} + \left({y}^{2} + \color{blue}{{x}^{2}}\right)\right)\right) \]

      12. associate-+r+ N/A

          \[\leadsto x \cdot \left(x \cdot \left(\left(-1 \cdot {y}^{2} + {y}^{2}\right) + \color{blue}{{x}^{2}}\right)\right) \]

      13. distribute-lft1-in N/A

          \[\leadsto x \cdot \left(x \cdot \left(\left(-1 + 1\right) \cdot {y}^{2} + {\color{blue}{x}}^{2}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto x \cdot \left(x \cdot \left(0 \cdot {y}^{2} + {x}^{2}\right)\right) \]

      15. mul0-lft N/A

          \[\leadsto x \cdot \left(x \cdot \left(0 + {\color{blue}{x}}^{2}\right)\right) \]

      16. +-lft-identity N/A

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

      17. unpow2 N/A

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

      18. cube-mult N/A

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

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

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

      20. cube-mult N/A

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

      21. unpow2 N/A

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

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

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

      23. unpow2 N/A

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

   7. Simplified 86.6%

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

   if -1.7999999999999999e178 < x < -1.05e-72 or 26.5 < x < 3.80000000000000012e145

   1. Initial program 81.4%

      \[{x}^{4} - {y}^{4} \]
   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. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. fmm-def N/A

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

      7. metadata-eval N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \mathsf{fma}\left(x, x, \mathsf{neg}\left({y}^{2}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \mathsf{fma}\left(x, x, \mathsf{neg}\left(y \cdot y\right)\right) \]

      9. *-commutative N/A

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

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

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 92.5%

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

   7. Simplified 92.5%

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

   if -1.05e-72 < x < 26.5

   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. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. fmm-def N/A

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

      7. metadata-eval N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \mathsf{fma}\left(x, x, \mathsf{neg}\left({y}^{2}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \mathsf{fma}\left(x, x, \mathsf{neg}\left(y \cdot y\right)\right) \]

      9. *-commutative N/A

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

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

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

   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 y around inf

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

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

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

      2. unpow2 N/A

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

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

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

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

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 76.5%

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

   7. Simplified 76.5%

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

   8. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. mul-1-neg N/A

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

      10. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(0 - \color{blue}{{y}^{2}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(0, \color{blue}{\left({y}^{2}\right)}\right)\right)\right) \]

      12. unpow2 N/A

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

      13. *-lowering-*.f64 94.0%

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

   10. Simplified 94.0%

       \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(0 - y \cdot y\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+178}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-72}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot x - y \cdot y\right)\\ \mathbf{elif}\;x \leq 26.5:\\ \;\;\;\;y \cdot \left(y \cdot \left(0 - y \cdot y\right)\right)\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+145}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot x - y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.2% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot x - y \cdot y}{\frac{1}{y \cdot y}}\\ \mathbf{if}\;y \leq -950000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-36}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \frac{1}{\frac{-1}{y \cdot y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- (* x x) (* y y)) (/ 1.0 (* y y)))))
   (if (<= y -950000000.0)
     t_0
     (if (<= y 9.5e-36)
       (* x (* x (* x x)))
       (if (<= y 1.35e+154) t_0 (* (* y y) (/ 1.0 (/ -1.0 (* y y)))))))))

C

double code(double x, double y) {
	double t_0 = ((x * x) - (y * y)) / (1.0 / (y * y));
	double tmp;
	if (y <= -950000000.0) {
		tmp = t_0;
	} else if (y <= 9.5e-36) {
		tmp = x * (x * (x * x));
	} else if (y <= 1.35e+154) {
		tmp = t_0;
	} else {
		tmp = (y * y) * (1.0 / (-1.0 / (y * y)));
	}
	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) - (y * y)) / (1.0d0 / (y * y))
    if (y <= (-950000000.0d0)) then
        tmp = t_0
    else if (y <= 9.5d-36) then
        tmp = x * (x * (x * x))
    else if (y <= 1.35d+154) then
        tmp = t_0
    else
        tmp = (y * y) * (1.0d0 / ((-1.0d0) / (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((x * x) - (y * y)) / (1.0 / (y * y));
	double tmp;
	if (y <= -950000000.0) {
		tmp = t_0;
	} else if (y <= 9.5e-36) {
		tmp = x * (x * (x * x));
	} else if (y <= 1.35e+154) {
		tmp = t_0;
	} else {
		tmp = (y * y) * (1.0 / (-1.0 / (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((x * x) - (y * y)) / (1.0 / (y * y))
	tmp = 0
	if y <= -950000000.0:
		tmp = t_0
	elif y <= 9.5e-36:
		tmp = x * (x * (x * x))
	elif y <= 1.35e+154:
		tmp = t_0
	else:
		tmp = (y * y) * (1.0 / (-1.0 / (y * y)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x * x) - Float64(y * y)) / Float64(1.0 / Float64(y * y)))
	tmp = 0.0
	if (y <= -950000000.0)
		tmp = t_0;
	elseif (y <= 9.5e-36)
		tmp = Float64(x * Float64(x * Float64(x * x)));
	elseif (y <= 1.35e+154)
		tmp = t_0;
	else
		tmp = Float64(Float64(y * y) * Float64(1.0 / Float64(-1.0 / Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((x * x) - (y * y)) / (1.0 / (y * y));
	tmp = 0.0;
	if (y <= -950000000.0)
		tmp = t_0;
	elseif (y <= 9.5e-36)
		tmp = x * (x * (x * x));
	elseif (y <= 1.35e+154)
		tmp = t_0;
	else
		tmp = (y * y) * (1.0 / (-1.0 / (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x * x), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -950000000.0], t$95$0, If[LessEqual[y, 9.5e-36], N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e+154], t$95$0, N[(N[(y * y), $MachinePrecision] * N[(1.0 / N[(-1.0 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.5e8 or 9.5000000000000003e-36 < y < 1.35000000000000003e154

   1. Initial program 71.4%

      \[{x}^{4} - {y}^{4} \]
   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. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. fmm-def N/A

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

      7. metadata-eval N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \mathsf{fma}\left(x, x, \mathsf{neg}\left({y}^{2}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \mathsf{fma}\left(x, x, \mathsf{neg}\left(y \cdot y\right)\right) \]

      9. *-commutative N/A

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

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

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

   4. Applied egg-rr 92.6%

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

      1. flip-- N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. remove-double-div N/A

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

      5. +-commutative N/A

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

      6. associate-/l/ N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

   6. Applied egg-rr 54.8%

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

      1. associate-/r* N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. frac-times N/A

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

   8. Applied egg-rr 92.6%

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

   9. Taylor expanded in x around 0

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 86.6%

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

   11. Simplified 86.6%

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

   if -9.5e8 < y < 9.5000000000000003e-36

   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. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. fmm-def N/A

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

      7. metadata-eval N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \mathsf{fma}\left(x, x, \mathsf{neg}\left({y}^{2}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \mathsf{fma}\left(x, x, \mathsf{neg}\left(y \cdot y\right)\right) \]

      9. *-commutative N/A

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

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

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

   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. +-lft-identity N/A

         \[\leadsto {x}^{2} \cdot \left(0 + \color{blue}{{x}^{2}}\right) \]

      4. unpow2 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{0} + {x}^{2}\right) \]

      5. mul0-lft N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(0 \cdot {y}^{2} + {\color{blue}{x}}^{2}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(-1 + 1\right) \cdot {y}^{2} + {x}^{2}\right) \]

      7. distribute-lft1-in N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(-1 \cdot {y}^{2} + {y}^{2}\right) + {\color{blue}{x}}^{2}\right) \]

      8. associate-+r+ N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(-1 \cdot {y}^{2} + \color{blue}{\left({y}^{2} + {x}^{2}\right)}\right) \]

      9. +-commutative N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(-1 \cdot {y}^{2} + \left({x}^{2} + \color{blue}{{y}^{2}}\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(-1 \cdot {y}^{2} + \left({x}^{2} + {y}^{2}\right)\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto x \cdot \left(x \cdot \left(-1 \cdot {y}^{2} + \left({y}^{2} + \color{blue}{{x}^{2}}\right)\right)\right) \]

      12. associate-+r+ N/A

          \[\leadsto x \cdot \left(x \cdot \left(\left(-1 \cdot {y}^{2} + {y}^{2}\right) + \color{blue}{{x}^{2}}\right)\right) \]

      13. distribute-lft1-in N/A

          \[\leadsto x \cdot \left(x \cdot \left(\left(-1 + 1\right) \cdot {y}^{2} + {\color{blue}{x}}^{2}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto x \cdot \left(x \cdot \left(0 \cdot {y}^{2} + {x}^{2}\right)\right) \]

      15. mul0-lft N/A

          \[\leadsto x \cdot \left(x \cdot \left(0 + {\color{blue}{x}}^{2}\right)\right) \]

      16. +-lft-identity N/A

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

      17. unpow2 N/A

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

      18. cube-mult N/A

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

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

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

      20. cube-mult N/A

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

      21. unpow2 N/A

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

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

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

      23. unpow2 N/A

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

   7. Simplified 94.1%

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

   if 1.35000000000000003e154 < y

   1. Initial program 53.3%

      \[{x}^{4} - {y}^{4} \]
   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. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. fmm-def N/A

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

      7. metadata-eval N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \mathsf{fma}\left(x, x, \mathsf{neg}\left({y}^{2}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \mathsf{fma}\left(x, x, \mathsf{neg}\left(y \cdot y\right)\right) \]

      9. *-commutative N/A

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

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

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

   4. Applied egg-rr 66.7%

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

      1. flip-- N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. remove-double-div N/A

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

      5. +-commutative N/A

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

      6. associate-/l/ N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 86.7%

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

   9. Simplified 86.7%

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

   10. Taylor expanded in y around inf

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

       1. unpow2 N/A

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

       2. *-lowering-*.f64 86.7%

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

   12. Simplified 86.7%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -950000000:\\ \;\;\;\;\frac{x \cdot x - y \cdot y}{\frac{1}{y \cdot y}}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-36}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{x \cdot x - y \cdot y}{\frac{1}{y \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \frac{1}{\frac{-1}{y \cdot y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.2% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot y\right) \cdot \left(x \cdot x - y \cdot y\right)\\ \mathbf{if}\;y \leq -850000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{-35}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \frac{1}{\frac{-1}{y \cdot y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y y) (- (* x x) (* y y)))))
   (if (<= y -850000000.0)
     t_0
     (if (<= y 1.02e-35)
       (* x (* x (* x x)))
       (if (<= y 1.35e+154) t_0 (* (* y y) (/ 1.0 (/ -1.0 (* y y)))))))))

C

double code(double x, double y) {
	double t_0 = (y * y) * ((x * x) - (y * y));
	double tmp;
	if (y <= -850000000.0) {
		tmp = t_0;
	} else if (y <= 1.02e-35) {
		tmp = x * (x * (x * x));
	} else if (y <= 1.35e+154) {
		tmp = t_0;
	} else {
		tmp = (y * y) * (1.0 / (-1.0 / (y * y)));
	}
	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) * ((x * x) - (y * y))
    if (y <= (-850000000.0d0)) then
        tmp = t_0
    else if (y <= 1.02d-35) then
        tmp = x * (x * (x * x))
    else if (y <= 1.35d+154) then
        tmp = t_0
    else
        tmp = (y * y) * (1.0d0 / ((-1.0d0) / (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * y) * ((x * x) - (y * y));
	double tmp;
	if (y <= -850000000.0) {
		tmp = t_0;
	} else if (y <= 1.02e-35) {
		tmp = x * (x * (x * x));
	} else if (y <= 1.35e+154) {
		tmp = t_0;
	} else {
		tmp = (y * y) * (1.0 / (-1.0 / (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * y) * ((x * x) - (y * y))
	tmp = 0
	if y <= -850000000.0:
		tmp = t_0
	elif y <= 1.02e-35:
		tmp = x * (x * (x * x))
	elif y <= 1.35e+154:
		tmp = t_0
	else:
		tmp = (y * y) * (1.0 / (-1.0 / (y * y)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * y) * Float64(Float64(x * x) - Float64(y * y)))
	tmp = 0.0
	if (y <= -850000000.0)
		tmp = t_0;
	elseif (y <= 1.02e-35)
		tmp = Float64(x * Float64(x * Float64(x * x)));
	elseif (y <= 1.35e+154)
		tmp = t_0;
	else
		tmp = Float64(Float64(y * y) * Float64(1.0 / Float64(-1.0 / Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * y) * ((x * x) - (y * y));
	tmp = 0.0;
	if (y <= -850000000.0)
		tmp = t_0;
	elseif (y <= 1.02e-35)
		tmp = x * (x * (x * x));
	elseif (y <= 1.35e+154)
		tmp = t_0;
	else
		tmp = (y * y) * (1.0 / (-1.0 / (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -850000000.0], t$95$0, If[LessEqual[y, 1.02e-35], N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e+154], t$95$0, N[(N[(y * y), $MachinePrecision] * N[(1.0 / N[(-1.0 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.5e8 or 1.01999999999999995e-35 < y < 1.35000000000000003e154

   1. Initial program 71.4%

      \[{x}^{4} - {y}^{4} \]
   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. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. fmm-def N/A

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

      7. metadata-eval N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \mathsf{fma}\left(x, x, \mathsf{neg}\left({y}^{2}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \mathsf{fma}\left(x, x, \mathsf{neg}\left(y \cdot y\right)\right) \]

      9. *-commutative N/A

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

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

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

   4. Applied egg-rr 92.6%

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

   5. Taylor expanded in y around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 86.6%

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

   7. Simplified 86.6%

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

   if -8.5e8 < y < 1.01999999999999995e-35

   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. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. fmm-def N/A

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

      7. metadata-eval N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \mathsf{fma}\left(x, x, \mathsf{neg}\left({y}^{2}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \mathsf{fma}\left(x, x, \mathsf{neg}\left(y \cdot y\right)\right) \]

      9. *-commutative N/A

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

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

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

   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. +-lft-identity N/A

         \[\leadsto {x}^{2} \cdot \left(0 + \color{blue}{{x}^{2}}\right) \]

      4. unpow2 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{0} + {x}^{2}\right) \]

      5. mul0-lft N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(0 \cdot {y}^{2} + {\color{blue}{x}}^{2}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(-1 + 1\right) \cdot {y}^{2} + {x}^{2}\right) \]

      7. distribute-lft1-in N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(-1 \cdot {y}^{2} + {y}^{2}\right) + {\color{blue}{x}}^{2}\right) \]

      8. associate-+r+ N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(-1 \cdot {y}^{2} + \color{blue}{\left({y}^{2} + {x}^{2}\right)}\right) \]

      9. +-commutative N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(-1 \cdot {y}^{2} + \left({x}^{2} + \color{blue}{{y}^{2}}\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(-1 \cdot {y}^{2} + \left({x}^{2} + {y}^{2}\right)\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto x \cdot \left(x \cdot \left(-1 \cdot {y}^{2} + \left({y}^{2} + \color{blue}{{x}^{2}}\right)\right)\right) \]

      12. associate-+r+ N/A

          \[\leadsto x \cdot \left(x \cdot \left(\left(-1 \cdot {y}^{2} + {y}^{2}\right) + \color{blue}{{x}^{2}}\right)\right) \]

      13. distribute-lft1-in N/A

          \[\leadsto x \cdot \left(x \cdot \left(\left(-1 + 1\right) \cdot {y}^{2} + {\color{blue}{x}}^{2}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto x \cdot \left(x \cdot \left(0 \cdot {y}^{2} + {x}^{2}\right)\right) \]

      15. mul0-lft N/A

          \[\leadsto x \cdot \left(x \cdot \left(0 + {\color{blue}{x}}^{2}\right)\right) \]

      16. +-lft-identity N/A

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

      17. unpow2 N/A

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

      18. cube-mult N/A

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

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

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

      20. cube-mult N/A

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

      21. unpow2 N/A

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

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

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

      23. unpow2 N/A

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

   7. Simplified 94.1%

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

   if 1.35000000000000003e154 < y

   1. Initial program 53.3%

      \[{x}^{4} - {y}^{4} \]
   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. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. fmm-def N/A

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

      7. metadata-eval N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \mathsf{fma}\left(x, x, \mathsf{neg}\left({y}^{2}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \mathsf{fma}\left(x, x, \mathsf{neg}\left(y \cdot y\right)\right) \]

      9. *-commutative N/A

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

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

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

   4. Applied egg-rr 66.7%

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

      1. flip-- N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. remove-double-div N/A

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

      5. +-commutative N/A

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

      6. associate-/l/ N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 86.7%

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

   9. Simplified 86.7%

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

   10. Taylor expanded in y around inf

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

       1. unpow2 N/A

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

       2. *-lowering-*.f64 86.7%

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

   12. Simplified 86.7%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -850000000:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(x \cdot x - y \cdot y\right)\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{-35}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(x \cdot x - y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \frac{1}{\frac{-1}{y \cdot y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.3% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot y\right) \cdot \left(x \cdot x - y \cdot y\right)\\ \mathbf{if}\;y \leq -1550000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{-39}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(0 - y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y y) (- (* x x) (* y y)))))
   (if (<= y -1550000000.0)
     t_0
     (if (<= y 4.1e-39)
       (* x (* x (* x x)))
       (if (<= y 1.3e+154) t_0 (* y (* y (- 0.0 (* y y)))))))))

C

double code(double x, double y) {
	double t_0 = (y * y) * ((x * x) - (y * y));
	double tmp;
	if (y <= -1550000000.0) {
		tmp = t_0;
	} else if (y <= 4.1e-39) {
		tmp = x * (x * (x * x));
	} else if (y <= 1.3e+154) {
		tmp = t_0;
	} else {
		tmp = y * (y * (0.0 - (y * y)));
	}
	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) * ((x * x) - (y * y))
    if (y <= (-1550000000.0d0)) then
        tmp = t_0
    else if (y <= 4.1d-39) then
        tmp = x * (x * (x * x))
    else if (y <= 1.3d+154) then
        tmp = t_0
    else
        tmp = y * (y * (0.0d0 - (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * y) * ((x * x) - (y * y));
	double tmp;
	if (y <= -1550000000.0) {
		tmp = t_0;
	} else if (y <= 4.1e-39) {
		tmp = x * (x * (x * x));
	} else if (y <= 1.3e+154) {
		tmp = t_0;
	} else {
		tmp = y * (y * (0.0 - (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * y) * ((x * x) - (y * y))
	tmp = 0
	if y <= -1550000000.0:
		tmp = t_0
	elif y <= 4.1e-39:
		tmp = x * (x * (x * x))
	elif y <= 1.3e+154:
		tmp = t_0
	else:
		tmp = y * (y * (0.0 - (y * y)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * y) * Float64(Float64(x * x) - Float64(y * y)))
	tmp = 0.0
	if (y <= -1550000000.0)
		tmp = t_0;
	elseif (y <= 4.1e-39)
		tmp = Float64(x * Float64(x * Float64(x * x)));
	elseif (y <= 1.3e+154)
		tmp = t_0;
	else
		tmp = Float64(y * Float64(y * Float64(0.0 - Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * y) * ((x * x) - (y * y));
	tmp = 0.0;
	if (y <= -1550000000.0)
		tmp = t_0;
	elseif (y <= 4.1e-39)
		tmp = x * (x * (x * x));
	elseif (y <= 1.3e+154)
		tmp = t_0;
	else
		tmp = y * (y * (0.0 - (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1550000000.0], t$95$0, If[LessEqual[y, 4.1e-39], N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e+154], t$95$0, N[(y * N[(y * N[(0.0 - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.55e9 or 4.1e-39 < y < 1.29999999999999994e154

   1. Initial program 71.4%

      \[{x}^{4} - {y}^{4} \]
   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. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. fmm-def N/A

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

      7. metadata-eval N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \mathsf{fma}\left(x, x, \mathsf{neg}\left({y}^{2}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \mathsf{fma}\left(x, x, \mathsf{neg}\left(y \cdot y\right)\right) \]

      9. *-commutative N/A

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

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

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

   4. Applied egg-rr 92.6%

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

   5. Taylor expanded in y around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 86.6%

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

   7. Simplified 86.6%

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

   if -1.55e9 < y < 4.1e-39

   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. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. fmm-def N/A

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

      7. metadata-eval N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \mathsf{fma}\left(x, x, \mathsf{neg}\left({y}^{2}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \mathsf{fma}\left(x, x, \mathsf{neg}\left(y \cdot y\right)\right) \]

      9. *-commutative N/A

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

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

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

   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. +-lft-identity N/A

         \[\leadsto {x}^{2} \cdot \left(0 + \color{blue}{{x}^{2}}\right) \]

      4. unpow2 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{0} + {x}^{2}\right) \]

      5. mul0-lft N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(0 \cdot {y}^{2} + {\color{blue}{x}}^{2}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(-1 + 1\right) \cdot {y}^{2} + {x}^{2}\right) \]

      7. distribute-lft1-in N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(-1 \cdot {y}^{2} + {y}^{2}\right) + {\color{blue}{x}}^{2}\right) \]

      8. associate-+r+ N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(-1 \cdot {y}^{2} + \color{blue}{\left({y}^{2} + {x}^{2}\right)}\right) \]

      9. +-commutative N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(-1 \cdot {y}^{2} + \left({x}^{2} + \color{blue}{{y}^{2}}\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(-1 \cdot {y}^{2} + \left({x}^{2} + {y}^{2}\right)\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto x \cdot \left(x \cdot \left(-1 \cdot {y}^{2} + \left({y}^{2} + \color{blue}{{x}^{2}}\right)\right)\right) \]

      12. associate-+r+ N/A

          \[\leadsto x \cdot \left(x \cdot \left(\left(-1 \cdot {y}^{2} + {y}^{2}\right) + \color{blue}{{x}^{2}}\right)\right) \]

      13. distribute-lft1-in N/A

          \[\leadsto x \cdot \left(x \cdot \left(\left(-1 + 1\right) \cdot {y}^{2} + {\color{blue}{x}}^{2}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto x \cdot \left(x \cdot \left(0 \cdot {y}^{2} + {x}^{2}\right)\right) \]

      15. mul0-lft N/A

          \[\leadsto x \cdot \left(x \cdot \left(0 + {\color{blue}{x}}^{2}\right)\right) \]

      16. +-lft-identity N/A

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

      17. unpow2 N/A

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

      18. cube-mult N/A

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

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

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

      20. cube-mult N/A

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

      21. unpow2 N/A

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

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

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

      23. unpow2 N/A

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

   7. Simplified 94.1%

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

   if 1.29999999999999994e154 < y

   1. Initial program 53.3%

      \[{x}^{4} - {y}^{4} \]
   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. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. fmm-def N/A

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

      7. metadata-eval N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \mathsf{fma}\left(x, x, \mathsf{neg}\left({y}^{2}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \mathsf{fma}\left(x, x, \mathsf{neg}\left(y \cdot y\right)\right) \]

      9. *-commutative N/A

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

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

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

   4. Applied egg-rr 66.7%

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

   5. Taylor expanded in y around inf

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

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

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

      2. unpow2 N/A

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

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

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

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

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 66.7%

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

   7. Simplified 66.7%

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

   8. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. mul-1-neg N/A

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

      10. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(0 - \color{blue}{{y}^{2}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(0, \color{blue}{\left({y}^{2}\right)}\right)\right)\right) \]

      12. unpow2 N/A

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

      13. *-lowering-*.f64 86.7%

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

   10. Simplified 86.7%

       \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(0 - y \cdot y\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1550000000:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(x \cdot x - y \cdot y\right)\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{-39}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+154}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(x \cdot x - y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(0 - y \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.0% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot \left(0 - y \cdot y\right)\right)\\ \mathbf{if}\;y \leq -500000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-30}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y (- 0.0 (* y y))))))
   (if (<= y -500000000000.0)
     t_0
     (if (<= y 1.15e-30) (* x (* x (* x x))) t_0))))

C

double code(double x, double y) {
	double t_0 = y * (y * (0.0 - (y * y)));
	double tmp;
	if (y <= -500000000000.0) {
		tmp = t_0;
	} else if (y <= 1.15e-30) {
		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 * (0.0d0 - (y * y)))
    if (y <= (-500000000000.0d0)) then
        tmp = t_0
    else if (y <= 1.15d-30) 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 * (0.0 - (y * y)));
	double tmp;
	if (y <= -500000000000.0) {
		tmp = t_0;
	} else if (y <= 1.15e-30) {
		tmp = x * (x * (x * x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * (0.0 - (y * y)))
	tmp = 0
	if y <= -500000000000.0:
		tmp = t_0
	elif y <= 1.15e-30:
		tmp = x * (x * (x * x))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * Float64(0.0 - Float64(y * y))))
	tmp = 0.0
	if (y <= -500000000000.0)
		tmp = t_0;
	elseif (y <= 1.15e-30)
		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 * (0.0 - (y * y)));
	tmp = 0.0;
	if (y <= -500000000000.0)
		tmp = t_0;
	elseif (y <= 1.15e-30)
		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[(y * N[(0.0 - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -500000000000.0], t$95$0, If[LessEqual[y, 1.15e-30], N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -5e11 or 1.14999999999999992e-30 < y

   1. Initial program 66.9%

      \[{x}^{4} - {y}^{4} \]
   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. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. fmm-def N/A

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

      7. metadata-eval N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \mathsf{fma}\left(x, x, \mathsf{neg}\left({y}^{2}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \mathsf{fma}\left(x, x, \mathsf{neg}\left(y \cdot y\right)\right) \]

      9. *-commutative N/A

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

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

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

   4. Applied egg-rr 86.4%

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

   5. Taylor expanded in y around inf

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

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

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

      2. unpow2 N/A

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

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

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

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

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 86.4%

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

   7. Simplified 86.4%

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

   8. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. mul-1-neg N/A

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

      10. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(0 - \color{blue}{{y}^{2}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(0, \color{blue}{\left({y}^{2}\right)}\right)\right)\right) \]

      12. unpow2 N/A

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

      13. *-lowering-*.f64 73.2%

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

   10. Simplified 73.2%

       \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(0 - y \cdot y\right)\right)} \]

   if -5e11 < y < 1.14999999999999992e-30

   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. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. fmm-def N/A

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

      7. metadata-eval N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \mathsf{fma}\left(x, x, \mathsf{neg}\left({y}^{2}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \mathsf{fma}\left(x, x, \mathsf{neg}\left(y \cdot y\right)\right) \]

      9. *-commutative N/A

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

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

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

   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. +-lft-identity N/A

         \[\leadsto {x}^{2} \cdot \left(0 + \color{blue}{{x}^{2}}\right) \]

      4. unpow2 N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{0} + {x}^{2}\right) \]

      5. mul0-lft N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(0 \cdot {y}^{2} + {\color{blue}{x}}^{2}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(-1 + 1\right) \cdot {y}^{2} + {x}^{2}\right) \]

      7. distribute-lft1-in N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(\left(-1 \cdot {y}^{2} + {y}^{2}\right) + {\color{blue}{x}}^{2}\right) \]

      8. associate-+r+ N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(-1 \cdot {y}^{2} + \color{blue}{\left({y}^{2} + {x}^{2}\right)}\right) \]

      9. +-commutative N/A

         \[\leadsto \left(x \cdot x\right) \cdot \left(-1 \cdot {y}^{2} + \left({x}^{2} + \color{blue}{{y}^{2}}\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(-1 \cdot {y}^{2} + \left({x}^{2} + {y}^{2}\right)\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto x \cdot \left(x \cdot \left(-1 \cdot {y}^{2} + \left({y}^{2} + \color{blue}{{x}^{2}}\right)\right)\right) \]

      12. associate-+r+ N/A

          \[\leadsto x \cdot \left(x \cdot \left(\left(-1 \cdot {y}^{2} + {y}^{2}\right) + \color{blue}{{x}^{2}}\right)\right) \]

      13. distribute-lft1-in N/A

          \[\leadsto x \cdot \left(x \cdot \left(\left(-1 + 1\right) \cdot {y}^{2} + {\color{blue}{x}}^{2}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto x \cdot \left(x \cdot \left(0 \cdot {y}^{2} + {x}^{2}\right)\right) \]

      15. mul0-lft N/A

          \[\leadsto x \cdot \left(x \cdot \left(0 + {\color{blue}{x}}^{2}\right)\right) \]

      16. +-lft-identity N/A

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

      17. unpow2 N/A

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

      18. cube-mult N/A

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

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

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

      20. cube-mult N/A

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

      21. unpow2 N/A

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

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

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

      23. unpow2 N/A

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

   7. Simplified 94.1%

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

Alternative 7: 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 83.6%

   \[{x}^{4} - {y}^{4} \]
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. metadata-eval N/A

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

   5. unpow2 N/A

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

   6. fmm-def N/A

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

   7. metadata-eval N/A

      \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \mathsf{fma}\left(x, x, \mathsf{neg}\left({y}^{2}\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \mathsf{fma}\left(x, x, \mathsf{neg}\left(y \cdot y\right)\right) \]

   9. *-commutative N/A

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

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

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

4. Applied egg-rr 93.1%

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

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. +-lft-identity N/A

      \[\leadsto {x}^{2} \cdot \left(0 + \color{blue}{{x}^{2}}\right) \]

   4. unpow2 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{0} + {x}^{2}\right) \]

   5. mul0-lft N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(0 \cdot {y}^{2} + {\color{blue}{x}}^{2}\right) \]

   6. metadata-eval N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\left(-1 + 1\right) \cdot {y}^{2} + {x}^{2}\right) \]

   7. distribute-lft1-in N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\left(-1 \cdot {y}^{2} + {y}^{2}\right) + {\color{blue}{x}}^{2}\right) \]

   8. associate-+r+ N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(-1 \cdot {y}^{2} + \color{blue}{\left({y}^{2} + {x}^{2}\right)}\right) \]

   9. +-commutative N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(-1 \cdot {y}^{2} + \left({x}^{2} + \color{blue}{{y}^{2}}\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(-1 \cdot {y}^{2} + \left({x}^{2} + {y}^{2}\right)\right)\right)} \]

   11. +-commutative N/A

       \[\leadsto x \cdot \left(x \cdot \left(-1 \cdot {y}^{2} + \left({y}^{2} + \color{blue}{{x}^{2}}\right)\right)\right) \]

   12. associate-+r+ N/A

       \[\leadsto x \cdot \left(x \cdot \left(\left(-1 \cdot {y}^{2} + {y}^{2}\right) + \color{blue}{{x}^{2}}\right)\right) \]

   13. distribute-lft1-in N/A

       \[\leadsto x \cdot \left(x \cdot \left(\left(-1 + 1\right) \cdot {y}^{2} + {\color{blue}{x}}^{2}\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto x \cdot \left(x \cdot \left(0 \cdot {y}^{2} + {x}^{2}\right)\right) \]

   15. mul0-lft N/A

       \[\leadsto x \cdot \left(x \cdot \left(0 + {\color{blue}{x}}^{2}\right)\right) \]

   16. +-lft-identity N/A

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

   17. unpow2 N/A

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

   18. cube-mult N/A

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

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

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

   20. cube-mult N/A

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

   21. unpow2 N/A

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

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

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

   23. unpow2 N/A

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

7. Simplified 61.1%

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

Reproduce

herbie shell --seed 2024141 
(FPCore (x y)
  :name "Radioactive exchange between two surfaces"
  :precision binary64
  (- (pow x 4.0) (pow y 4.0)))