Rump's expression from Stadtherr's award speech

Percentage Accurate: 1.4% → 18.9%

Time: 13.5s

Alternatives: 3

Speedup: 87.8×

Specification

\[x = 77617 \land y = 33096\]

Math

\[\begin{array}{l} \\ -0.8273960599468214 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -0.8273960599468214)

C

double code(double x, double y) {
	return -0.8273960599468214;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -0.8273960599468214d0
end function

Java

public static double code(double x, double y) {
	return -0.8273960599468214;
}

Python

def code(x, y):
	return -0.8273960599468214

Julia

function code(x, y)
	return -0.8273960599468214
end

MATLAB

function tmp = code(x, y)
	tmp = -0.8273960599468214;
end

Wolfram

code[x_, y_] := -0.8273960599468214

Initial Program: 1.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(333.75 \cdot {y}^{6} + \left(x \cdot x\right) \cdot \left(\left(\left(\left(\left(\left(11 \cdot x\right) \cdot x\right) \cdot y\right) \cdot y - {y}^{6}\right) - 121 \cdot {y}^{4}\right) - 2\right)\right) + 5.5 \cdot {y}^{8}\right) + \frac{x}{2 \cdot y} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+
  (+
   (+
    (* 333.75 (pow y 6.0))
    (*
     (* x x)
     (-
      (- (- (* (* (* (* 11.0 x) x) y) y) (pow y 6.0)) (* 121.0 (pow y 4.0)))
      2.0)))
   (* 5.5 (pow y 8.0)))
  (/ x (* 2.0 y))))

C

double code(double x, double y) {
	return (((333.75 * pow(y, 6.0)) + ((x * x) * (((((((11.0 * x) * x) * y) * y) - pow(y, 6.0)) - (121.0 * pow(y, 4.0))) - 2.0))) + (5.5 * pow(y, 8.0))) + (x / (2.0 * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (((333.75d0 * (y ** 6.0d0)) + ((x * x) * (((((((11.0d0 * x) * x) * y) * y) - (y ** 6.0d0)) - (121.0d0 * (y ** 4.0d0))) - 2.0d0))) + (5.5d0 * (y ** 8.0d0))) + (x / (2.0d0 * y))
end function

Java

public static double code(double x, double y) {
	return (((333.75 * Math.pow(y, 6.0)) + ((x * x) * (((((((11.0 * x) * x) * y) * y) - Math.pow(y, 6.0)) - (121.0 * Math.pow(y, 4.0))) - 2.0))) + (5.5 * Math.pow(y, 8.0))) + (x / (2.0 * y));
}

Python

def code(x, y):
	return (((333.75 * math.pow(y, 6.0)) + ((x * x) * (((((((11.0 * x) * x) * y) * y) - math.pow(y, 6.0)) - (121.0 * math.pow(y, 4.0))) - 2.0))) + (5.5 * math.pow(y, 8.0))) + (x / (2.0 * y))

Julia

function code(x, y)
	return Float64(Float64(Float64(Float64(333.75 * (y ^ 6.0)) + Float64(Float64(x * x) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(11.0 * x) * x) * y) * y) - (y ^ 6.0)) - Float64(121.0 * (y ^ 4.0))) - 2.0))) + Float64(5.5 * (y ^ 8.0))) + Float64(x / Float64(2.0 * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = (((333.75 * (y ^ 6.0)) + ((x * x) * (((((((11.0 * x) * x) * y) * y) - (y ^ 6.0)) - (121.0 * (y ^ 4.0))) - 2.0))) + (5.5 * (y ^ 8.0))) + (x / (2.0 * y));
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(333.75 * N[Power[y, 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(11.0 * x), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] - N[Power[y, 6.0], $MachinePrecision]), $MachinePrecision] - N[(121.0 * N[Power[y, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(5.5 * N[Power[y, 8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / N[(2.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 18.9% accurate, 5.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot -2\right) + -0.5\\ t_1 := y \cdot \left(y \cdot y\right)\\ \frac{\frac{0.125 + \left(-8 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot t\_1}{\frac{y}{x}}}{0.015625 + \left(\left(x \cdot -2\right) \cdot \left(y \cdot t\_0\right)\right) \cdot \left(\left(x \cdot \left(y \cdot \left(y \cdot \left(x \cdot 4\right)\right)\right)\right) \cdot \left(t\_0 \cdot t\_0\right)\right)} \cdot \left(0.0625 + x \cdot \left(y \cdot -0.25 + \left(x \cdot t\_1\right) \cdot \left(x \cdot 8\right)\right)\right) \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (* y (* x -2.0)) -0.5)) (t_1 (* y (* y y))))
   (*
    (/
     (/ (+ 0.125 (* (* -8.0 (* x (* x x))) t_1)) (/ y x))
     (+
      0.015625
      (*
       (* (* x -2.0) (* y t_0))
       (* (* x (* y (* y (* x 4.0)))) (* t_0 t_0)))))
    (+ 0.0625 (* x (+ (* y -0.25) (* (* x t_1) (* x 8.0))))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (y * (x * -2.0)) + -0.5;
	double t_1 = y * (y * y);
	return (((0.125 + ((-8.0 * (x * (x * x))) * t_1)) / (y / x)) / (0.015625 + (((x * -2.0) * (y * t_0)) * ((x * (y * (y * (x * 4.0)))) * (t_0 * t_0))))) * (0.0625 + (x * ((y * -0.25) + ((x * t_1) * (x * 8.0)))));
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    t_0 = (y * (x * (-2.0d0))) + (-0.5d0)
    t_1 = y * (y * y)
    code = (((0.125d0 + (((-8.0d0) * (x * (x * x))) * t_1)) / (y / x)) / (0.015625d0 + (((x * (-2.0d0)) * (y * t_0)) * ((x * (y * (y * (x * 4.0d0)))) * (t_0 * t_0))))) * (0.0625d0 + (x * ((y * (-0.25d0)) + ((x * t_1) * (x * 8.0d0)))))
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = (y * (x * -2.0)) + -0.5;
	double t_1 = y * (y * y);
	return (((0.125 + ((-8.0 * (x * (x * x))) * t_1)) / (y / x)) / (0.015625 + (((x * -2.0) * (y * t_0)) * ((x * (y * (y * (x * 4.0)))) * (t_0 * t_0))))) * (0.0625 + (x * ((y * -0.25) + ((x * t_1) * (x * 8.0)))));
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (y * (x * -2.0)) + -0.5
	t_1 = y * (y * y)
	return (((0.125 + ((-8.0 * (x * (x * x))) * t_1)) / (y / x)) / (0.015625 + (((x * -2.0) * (y * t_0)) * ((x * (y * (y * (x * 4.0)))) * (t_0 * t_0))))) * (0.0625 + (x * ((y * -0.25) + ((x * t_1) * (x * 8.0)))))

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(y * Float64(x * -2.0)) + -0.5)
	t_1 = Float64(y * Float64(y * y))
	return Float64(Float64(Float64(Float64(0.125 + Float64(Float64(-8.0 * Float64(x * Float64(x * x))) * t_1)) / Float64(y / x)) / Float64(0.015625 + Float64(Float64(Float64(x * -2.0) * Float64(y * t_0)) * Float64(Float64(x * Float64(y * Float64(y * Float64(x * 4.0)))) * Float64(t_0 * t_0))))) * Float64(0.0625 + Float64(x * Float64(Float64(y * -0.25) + Float64(Float64(x * t_1) * Float64(x * 8.0))))))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	t_0 = (y * (x * -2.0)) + -0.5;
	t_1 = y * (y * y);
	tmp = (((0.125 + ((-8.0 * (x * (x * x))) * t_1)) / (y / x)) / (0.015625 + (((x * -2.0) * (y * t_0)) * ((x * (y * (y * (x * 4.0)))) * (t_0 * t_0))))) * (0.0625 + (x * ((y * -0.25) + ((x * t_1) * (x * 8.0)))));
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(y * N[(x * -2.0), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(0.125 + N[(N[(-8.0 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision] / N[(0.015625 + N[(N[(N[(x * -2.0), $MachinePrecision] * N[(y * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(x * N[(y * N[(y * N[(x * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.0625 + N[(x * N[(N[(y * -0.25), $MachinePrecision] + N[(N[(x * t$95$1), $MachinePrecision] * N[(x * 8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 9.2%

   \[\left(\left(333.75 \cdot {y}^{6} + \left(x \cdot x\right) \cdot \left(\left(\left(\left(\left(\left(11 \cdot x\right) \cdot x\right) \cdot y\right) \cdot y - {y}^{6}\right) - 121 \cdot {y}^{4}\right) - 2\right)\right) + 5.5 \cdot {y}^{8}\right) + \frac{x}{2 \cdot y} \]

3. Simplified 3.1%

   \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot \left(y \cdot \left(y \cdot \left(x \cdot \left(x \cdot 11\right)\right)\right)\right) + \left({y}^{6} \cdot \left(333.75 + \left(0 - x \cdot x\right)\right) + \left(x \cdot x\right) \cdot \left(-2 - 121 \cdot {y}^{4}\right)\right)\right) + \left(5.5 \cdot {y}^{8} + \frac{x}{y \cdot 2}\right)} \]
4. Add Preprocessing

5. Taylor expanded in y around 0

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

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

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

   2. associate-*r* N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{1}{2} + \left(\left(x \cdot x\right) \cdot y\right) \cdot -2\right), y\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{1}{2} + \left(x \cdot \left(x \cdot y\right)\right) \cdot -2\right), y\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{1}{2} + x \cdot \left(\left(x \cdot y\right) \cdot -2\right)\right), y\right) \]

   10. distribute-lft-out N/A

       \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(\frac{1}{2} + \left(x \cdot y\right) \cdot -2\right)\right), y\right) \]

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

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

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

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

   13. *-commutative N/A

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

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

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

   15. *-lowering-*.f64 10.8%

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

7. Simplified 10.8%

   \[\leadsto \color{blue}{\frac{x \cdot \left(0.5 + \left(y \cdot x\right) \cdot -2\right)}{y}} \]
8. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

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

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

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

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

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

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

   7. associate-*l* N/A

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

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

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

   9. *-lowering-*.f64 10.8%

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

9. Applied egg-rr 10.8%

   \[\leadsto \color{blue}{\frac{x}{y} \cdot \left(0.5 + y \cdot \left(x \cdot -2\right)\right)} \]

11. Applied egg-rr 10.8%

    \[\leadsto \color{blue}{\frac{\frac{0.125 + \left(-8 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(y \cdot \left(y \cdot y\right)\right)}{\frac{y}{x}}}{0.015625 + \left(\left(x \cdot -2\right) \cdot \left(y \cdot \left(\left(x \cdot -2\right) \cdot y + -0.5\right)\right)\right) \cdot \left(\left(x \cdot \left(y \cdot \left(\left(x \cdot 4\right) \cdot y\right)\right)\right) \cdot \left(\left(\left(x \cdot -2\right) \cdot y + -0.5\right) \cdot \left(\left(x \cdot -2\right) \cdot y + -0.5\right)\right)\right)} \cdot \left(0.0625 + \left(\left(x \cdot -2\right) \cdot \left(y \cdot \left(\left(x \cdot -2\right) \cdot y + -0.5\right)\right)\right) \cdot \left(\left(x \cdot -2\right) \cdot \left(y \cdot \left(\left(x \cdot -2\right) \cdot y + -0.5\right)\right) - 0.25\right)\right)} \]

12. Taylor expanded in x around 0

    \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right)\right), \mathsf{/.f64}\left(y, x\right)\right), \mathsf{+.f64}\left(\frac{1}{64}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, -2\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, -2\right), y\right), \frac{-1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, 4\right), y\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, -2\right), y\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, -2\right), y\right), \frac{-1}{2}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{16}, \color{blue}{\left(x \cdot \left(\frac{-1}{4} \cdot y + x \cdot \left(-2 \cdot \left(y \cdot \left(\frac{-1}{2} \cdot y + \frac{1}{2} \cdot y\right)\right) + 8 \cdot \left(x \cdot {y}^{3}\right)\right)\right)\right)}\right)\right) \]

14. Simplified 17.8%

    \[\leadsto \frac{\frac{0.125 + \left(-8 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(y \cdot \left(y \cdot y\right)\right)}{\frac{y}{x}}}{0.015625 + \left(\left(x \cdot -2\right) \cdot \left(y \cdot \left(\left(x \cdot -2\right) \cdot y + -0.5\right)\right)\right) \cdot \left(\left(x \cdot \left(y \cdot \left(\left(x \cdot 4\right) \cdot y\right)\right)\right) \cdot \left(\left(\left(x \cdot -2\right) \cdot y + -0.5\right) \cdot \left(\left(x \cdot -2\right) \cdot y + -0.5\right)\right)\right)} \cdot \left(0.0625 + \color{blue}{x \cdot \left(\left(y \cdot -0.25 + 0\right) + \left(x \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot \left(x \cdot 8\right)\right)}\right) \]

15. Final simplification 17.8%

    \[\leadsto \frac{\frac{0.125 + \left(-8 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(y \cdot \left(y \cdot y\right)\right)}{\frac{y}{x}}}{0.015625 + \left(\left(x \cdot -2\right) \cdot \left(y \cdot \left(y \cdot \left(x \cdot -2\right) + -0.5\right)\right)\right) \cdot \left(\left(x \cdot \left(y \cdot \left(y \cdot \left(x \cdot 4\right)\right)\right)\right) \cdot \left(\left(y \cdot \left(x \cdot -2\right) + -0.5\right) \cdot \left(y \cdot \left(x \cdot -2\right) + -0.5\right)\right)\right)} \cdot \left(0.0625 + x \cdot \left(y \cdot -0.25 + \left(x \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot \left(x \cdot 8\right)\right)\right) \]
16. Add Preprocessing

Alternative 2: 11.0% accurate, 48.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ x \cdot \left(x \cdot -2 + \frac{0.5}{y}\right) \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (* x (+ (* x -2.0) (/ 0.5 y))))

C

assert(x < y);
double code(double x, double y) {
	return x * ((x * -2.0) + (0.5 / y));
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * ((x * (-2.0d0)) + (0.5d0 / y))
end function

Java

assert x < y;
public static double code(double x, double y) {
	return x * ((x * -2.0) + (0.5 / y));
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return x * ((x * -2.0) + (0.5 / y))

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(x * Float64(Float64(x * -2.0) + Float64(0.5 / y)))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = x * ((x * -2.0) + (0.5 / y));
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(x * N[(N[(x * -2.0), $MachinePrecision] + N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.2%

   \[\left(\left(333.75 \cdot {y}^{6} + \left(x \cdot x\right) \cdot \left(\left(\left(\left(\left(\left(11 \cdot x\right) \cdot x\right) \cdot y\right) \cdot y - {y}^{6}\right) - 121 \cdot {y}^{4}\right) - 2\right)\right) + 5.5 \cdot {y}^{8}\right) + \frac{x}{2 \cdot y} \]

3. Simplified 3.1%

   \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot \left(y \cdot \left(y \cdot \left(x \cdot \left(x \cdot 11\right)\right)\right)\right) + \left({y}^{6} \cdot \left(333.75 + \left(0 - x \cdot x\right)\right) + \left(x \cdot x\right) \cdot \left(-2 - 121 \cdot {y}^{4}\right)\right)\right) + \left(5.5 \cdot {y}^{8} + \frac{x}{y \cdot 2}\right)} \]
4. Add Preprocessing

5. Taylor expanded in y around 0

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

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

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

   2. associate-*r* N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{1}{2} + \left(\left(x \cdot x\right) \cdot y\right) \cdot -2\right), y\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{1}{2} + \left(x \cdot \left(x \cdot y\right)\right) \cdot -2\right), y\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{1}{2} + x \cdot \left(\left(x \cdot y\right) \cdot -2\right)\right), y\right) \]

   10. distribute-lft-out N/A

       \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(\frac{1}{2} + \left(x \cdot y\right) \cdot -2\right)\right), y\right) \]

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

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

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

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

   13. *-commutative N/A

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

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

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

   15. *-lowering-*.f64 10.8%

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

7. Simplified 10.8%

   \[\leadsto \color{blue}{\frac{x \cdot \left(0.5 + \left(y \cdot x\right) \cdot -2\right)}{y}} \]

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(-2 \cdot x + \frac{1}{2} \cdot \frac{1}{y}\right)} \]
9. Step-by-step derivation

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

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

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

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

   3. *-commutative N/A

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

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

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. /-lowering-/.f64 10.8%

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

10. Simplified 10.8%

    \[\leadsto \color{blue}{x \cdot \left(x \cdot -2 + \frac{0.5}{y}\right)} \]
11. Add Preprocessing

Alternative 3: 11.0% accurate, 87.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ x \cdot \left(x \cdot -2\right) \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (* x (* x -2.0)))

C

assert(x < y);
double code(double x, double y) {
	return x * (x * -2.0);
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (x * (-2.0d0))
end function

Java

assert x < y;
public static double code(double x, double y) {
	return x * (x * -2.0);
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return x * (x * -2.0)

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(x * Float64(x * -2.0))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = x * (x * -2.0);
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(x * N[(x * -2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.2%

   \[\left(\left(333.75 \cdot {y}^{6} + \left(x \cdot x\right) \cdot \left(\left(\left(\left(\left(\left(11 \cdot x\right) \cdot x\right) \cdot y\right) \cdot y - {y}^{6}\right) - 121 \cdot {y}^{4}\right) - 2\right)\right) + 5.5 \cdot {y}^{8}\right) + \frac{x}{2 \cdot y} \]

3. Simplified 3.1%

   \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot \left(y \cdot \left(y \cdot \left(x \cdot \left(x \cdot 11\right)\right)\right)\right) + \left({y}^{6} \cdot \left(333.75 + \left(0 - x \cdot x\right)\right) + \left(x \cdot x\right) \cdot \left(-2 - 121 \cdot {y}^{4}\right)\right)\right) + \left(5.5 \cdot {y}^{8} + \frac{x}{y \cdot 2}\right)} \]
4. Add Preprocessing

5. Taylor expanded in y around 0

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

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

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

   2. associate-*r* N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{1}{2} + \left(\left(x \cdot x\right) \cdot y\right) \cdot -2\right), y\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{1}{2} + \left(x \cdot \left(x \cdot y\right)\right) \cdot -2\right), y\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{1}{2} + x \cdot \left(\left(x \cdot y\right) \cdot -2\right)\right), y\right) \]

   10. distribute-lft-out N/A

       \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(\frac{1}{2} + \left(x \cdot y\right) \cdot -2\right)\right), y\right) \]

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

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

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

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

   13. *-commutative N/A

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

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

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

   15. *-lowering-*.f64 10.8%

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

7. Simplified 10.8%

   \[\leadsto \color{blue}{\frac{x \cdot \left(0.5 + \left(y \cdot x\right) \cdot -2\right)}{y}} \]

8. Taylor expanded in x around inf

   \[\leadsto \color{blue}{-2 \cdot {x}^{2}} \]
9. Step-by-step derivation

   1. unpow2 N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

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

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

   5. *-commutative N/A

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

   6. *-lowering-*.f64 10.8%

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

10. Simplified 10.8%

    \[\leadsto \color{blue}{x \cdot \left(x \cdot -2\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024192 
(FPCore (x y)
  :name "Rump's expression from Stadtherr's award speech"
  :precision binary64
  :pre (and (== x 77617.0) (== y 33096.0))
  (+ (+ (+ (* 333.75 (pow y 6.0)) (* (* x x) (- (- (- (* (* (* (* 11.0 x) x) y) y) (pow y 6.0)) (* 121.0 (pow y 4.0))) 2.0))) (* 5.5 (pow y 8.0))) (/ x (* 2.0 y))))