Radioactive exchange between two surfaces

Percentage Accurate: 86.3% → 99.9%

Time: 9.8s

Alternatives: 6

Speedup: 13.7×

• 60.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.3%

   \[{x}^{4} - {y}^{4} \]
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 93.1%

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

   1. *-commutative N/A

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

   2. difference-of-squares N/A

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

   3. associate-*r* N/A

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

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

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

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

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

   6. +-commutative N/A

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

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

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

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

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

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

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

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

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

   11. --lowering--.f64 99.9%

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

6. Applied egg-rr 99.9%

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

Alternative 2: 91.7% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - y\right) \cdot \left(y \cdot \left(y \cdot y\right)\right)\\ t_1 := \left(y \cdot y\right) \cdot \left(x \cdot x - y \cdot y\right)\\ \mathbf{if}\;y \leq -4.5 \cdot 10^{+177}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{-50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-101}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+116}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- x y) (* y (* y y)))) (t_1 (* (* y y) (- (* x x) (* y y)))))
   (if (<= y -4.5e+177)
     t_0
     (if (<= y -4.8e-50)
       t_1
       (if (<= y 2.3e-101)
         (* x (* x (* x x)))
         (if (<= y 8.2e+116) t_1 t_0))))))

C

double code(double x, double y) {
	double t_0 = (x - y) * (y * (y * y));
	double t_1 = (y * y) * ((x * x) - (y * y));
	double tmp;
	if (y <= -4.5e+177) {
		tmp = t_0;
	} else if (y <= -4.8e-50) {
		tmp = t_1;
	} else if (y <= 2.3e-101) {
		tmp = x * (x * (x * x));
	} else if (y <= 8.2e+116) {
		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 - y) * (y * (y * y))
    t_1 = (y * y) * ((x * x) - (y * y))
    if (y <= (-4.5d+177)) then
        tmp = t_0
    else if (y <= (-4.8d-50)) then
        tmp = t_1
    else if (y <= 2.3d-101) then
        tmp = x * (x * (x * x))
    else if (y <= 8.2d+116) 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 - y) * (y * (y * y));
	double t_1 = (y * y) * ((x * x) - (y * y));
	double tmp;
	if (y <= -4.5e+177) {
		tmp = t_0;
	} else if (y <= -4.8e-50) {
		tmp = t_1;
	} else if (y <= 2.3e-101) {
		tmp = x * (x * (x * x));
	} else if (y <= 8.2e+116) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) * (y * (y * y))
	t_1 = (y * y) * ((x * x) - (y * y))
	tmp = 0
	if y <= -4.5e+177:
		tmp = t_0
	elif y <= -4.8e-50:
		tmp = t_1
	elif y <= 2.3e-101:
		tmp = x * (x * (x * x))
	elif y <= 8.2e+116:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) * Float64(y * Float64(y * y)))
	t_1 = Float64(Float64(y * y) * Float64(Float64(x * x) - Float64(y * y)))
	tmp = 0.0
	if (y <= -4.5e+177)
		tmp = t_0;
	elseif (y <= -4.8e-50)
		tmp = t_1;
	elseif (y <= 2.3e-101)
		tmp = Float64(x * Float64(x * Float64(x * x)));
	elseif (y <= 8.2e+116)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) * (y * (y * y));
	t_1 = (y * y) * ((x * x) - (y * y));
	tmp = 0.0;
	if (y <= -4.5e+177)
		tmp = t_0;
	elseif (y <= -4.8e-50)
		tmp = t_1;
	elseif (y <= 2.3e-101)
		tmp = x * (x * (x * x));
	elseif (y <= 8.2e+116)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y * y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.5e+177], t$95$0, If[LessEqual[y, -4.8e-50], t$95$1, If[LessEqual[y, 2.3e-101], N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.2e+116], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.4999999999999997e177 or 8.1999999999999996e116 < y

   1. Initial program 58.2%

      \[{x}^{4} - {y}^{4} \]
   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 69.1%

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

      1. *-commutative N/A

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

      2. difference-of-squares N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. +-commutative N/A

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

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

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

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

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

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

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

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

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

      11. --lowering--.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({y}^{3}\right)}, \mathsf{\_.f64}\left(x, y\right)\right) \]
   8. Step-by-step derivation

      1. cube-mult N/A

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

      2. unpow2 N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 96.4%

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

   9. Simplified 96.4%

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

   if -4.4999999999999997e177 < y < -4.80000000000000004e-50 or 2.2999999999999999e-101 < y < 8.1999999999999996e116

   1. Initial program 87.5%

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

      1. sqr-pow N/A

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

      2. sqr-pow N/A

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

      3. difference-of-squares N/A

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

      4. 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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(y, y\right)\right), \color{blue}{\left({y}^{2}\right)}\right) \]
   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 -4.80000000000000004e-50 < y < 2.2999999999999999e-101

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. pow-lowering-pow.f64 98.6%

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

   5. Simplified 98.6%

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. cube-unmult N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot x \]

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

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

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

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

      6. *-lowering-*.f64 98.4%

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

   7. Applied egg-rr 98.4%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+177}:\\ \;\;\;\;\left(x - y\right) \cdot \left(y \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{-50}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(x \cdot x - y \cdot y\right)\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-101}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+116}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(x \cdot x - y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \left(y \cdot \left(y \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.5% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - y\right) \cdot \left(y \cdot \left(y \cdot y\right)\right)\\ \mathbf{if}\;y \leq -7.5 \cdot 10^{+62}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 180000:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (- x y) (* y (* y y)))))
   (if (<= y -7.5e+62) t_0 (if (<= y 180000.0) (* x (* x (* x x))) t_0))))

C

double code(double x, double y) {
	double t_0 = (x - y) * (y * (y * y));
	double tmp;
	if (y <= -7.5e+62) {
		tmp = t_0;
	} else if (y <= 180000.0) {
		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 = (x - y) * (y * (y * y))
    if (y <= (-7.5d+62)) then
        tmp = t_0
    else if (y <= 180000.0d0) 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 = (x - y) * (y * (y * y));
	double tmp;
	if (y <= -7.5e+62) {
		tmp = t_0;
	} else if (y <= 180000.0) {
		tmp = x * (x * (x * x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) * (y * (y * y))
	tmp = 0
	if y <= -7.5e+62:
		tmp = t_0
	elif y <= 180000.0:
		tmp = x * (x * (x * x))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) * Float64(y * Float64(y * y)))
	tmp = 0.0
	if (y <= -7.5e+62)
		tmp = t_0;
	elseif (y <= 180000.0)
		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 = (x - y) * (y * (y * y));
	tmp = 0.0;
	if (y <= -7.5e+62)
		tmp = t_0;
	elseif (y <= 180000.0)
		tmp = x * (x * (x * x));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.5e+62], t$95$0, If[LessEqual[y, 180000.0], N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.49999999999999998e62 or 1.8e5 < y

   1. Initial program 65.7%

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

      1. sqr-pow N/A

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

      2. sqr-pow N/A

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

      3. difference-of-squares N/A

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

      4. 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 83.3%

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

      1. *-commutative N/A

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

      2. difference-of-squares N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. +-commutative N/A

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

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

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

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

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

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

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

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

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

      11. --lowering--.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({y}^{3}\right)}, \mathsf{\_.f64}\left(x, y\right)\right) \]
   8. Step-by-step derivation

      1. cube-mult N/A

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

      2. unpow2 N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 90.2%

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

   9. Simplified 90.2%

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

   if -7.49999999999999998e62 < y < 1.8e5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. pow-lowering-pow.f64 86.6%

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

   5. Simplified 86.6%

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. cube-unmult N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot x \]

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

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

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

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

      6. *-lowering-*.f64 86.5%

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

   7. Applied egg-rr 86.5%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+62}:\\ \;\;\;\;\left(x - y\right) \cdot \left(y \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;y \leq 180000:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \left(y \cdot \left(y \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.2% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-62}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+19}:\\ \;\;\;\;y \cdot \left(y \cdot \left(y \cdot \left(0 - y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* x (* x x)))))
   (if (<= x -2.5e-62)
     t_0
     (if (<= x 1.5e+19) (* y (* y (* y (- 0.0 y)))) t_0))))

C

double code(double x, double y) {
	double t_0 = x * (x * (x * x));
	double tmp;
	if (x <= -2.5e-62) {
		tmp = t_0;
	} else if (x <= 1.5e+19) {
		tmp = y * (y * (y * (0.0 - y)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x * (x * x))
    if (x <= (-2.5d-62)) then
        tmp = t_0
    else if (x <= 1.5d+19) then
        tmp = y * (y * (y * (0.0d0 - y)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (x * (x * x));
	double tmp;
	if (x <= -2.5e-62) {
		tmp = t_0;
	} else if (x <= 1.5e+19) {
		tmp = y * (y * (y * (0.0 - y)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (x * (x * x))
	tmp = 0
	if x <= -2.5e-62:
		tmp = t_0
	elif x <= 1.5e+19:
		tmp = y * (y * (y * (0.0 - y)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(x * Float64(x * x)))
	tmp = 0.0
	if (x <= -2.5e-62)
		tmp = t_0;
	elseif (x <= 1.5e+19)
		tmp = Float64(y * Float64(y * Float64(y * Float64(0.0 - y))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (x * (x * x));
	tmp = 0.0;
	if (x <= -2.5e-62)
		tmp = t_0;
	elseif (x <= 1.5e+19)
		tmp = y * (y * (y * (0.0 - y)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.5e-62], t$95$0, If[LessEqual[x, 1.5e+19], N[(y * N[(y * N[(y * N[(0.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5000000000000001e-62 or 1.5e19 < x

   1. Initial program 75.0%

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

   3. Taylor expanded in x around inf

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

      1. pow-lowering-pow.f64 80.5%

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

   5. Simplified 80.5%

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. cube-unmult N/A

         \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot x \]

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

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

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

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

      6. *-lowering-*.f64 80.4%

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

   7. Applied egg-rr 80.4%

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

   if -2.5000000000000001e-62 < x < 1.5e19

   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. Step-by-step derivation

      1. *-commutative N/A

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

      2. difference-of-squares N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. +-commutative N/A

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

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

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

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

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

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

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

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

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

      11. --lowering--.f64 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 90.3%

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

   9. Simplified 90.3%

      \[\leadsto \color{blue}{0 - \left(y \cdot y\right) \cdot \left(y \cdot y\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

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

       2. sub0-neg N/A

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

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

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

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

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

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

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

       6. *-lowering-*.f64 90.4%

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

   11. Applied egg-rr 90.4%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-62}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+19}:\\ \;\;\;\;y \cdot \left(y \cdot \left(y \cdot \left(0 - y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.4% 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 86.3%

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

3. Taylor expanded in x around inf

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

   1. pow-lowering-pow.f64 61.8%

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

5. Simplified 61.8%

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

   1. metadata-eval N/A

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

   2. pow-plus N/A

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

   3. cube-unmult N/A

      \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot x \]

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

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

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

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

   6. *-lowering-*.f64 61.7%

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

7. Applied egg-rr 61.7%

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

8. Final simplification 61.7%

   \[\leadsto x \cdot \left(x \cdot \left(x \cdot x\right)\right) \]
9. Add Preprocessing

Alternative 6: 57.3% accurate, 29.3× speedup

Math

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

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(Float64(x * x) * Float64(x * x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

3. Taylor expanded in x around inf

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

   1. pow-lowering-pow.f64 61.8%

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

5. Simplified 61.8%

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

   1. metadata-eval N/A

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

   2. pow-prod-up N/A

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

   3. pow2 N/A

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

   4. pow2 N/A

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

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

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

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

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

   7. *-lowering-*.f64 61.7%

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

7. Applied egg-rr 61.7%

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

Reproduce

herbie shell --seed 2024161 
(FPCore (x y)
  :name "Radioactive exchange between two surfaces"
  :precision binary64
  (- (pow x 4.0) (pow y 4.0)))