Rump's expression from Stadtherr's award speech

Percentage Accurate: 1.4% → 11.0%

Time: 12.2s

Alternatives: 4

Speedup: 44.0×

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: 11.0% accurate, 17.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \mathsf{fma}\left(x, x \cdot -2, \frac{x}{y} \cdot 1.5\right) \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (fma x (* x -2.0) (* (/ x y) 1.5)))

C

assert(x < y);
double code(double x, double y) {
	return fma(x, (x * -2.0), ((x / y) * 1.5));
}

Julia

x, y = sort([x, y])
function code(x, y)
	return fma(x, Float64(x * -2.0), Float64(Float64(x / y) * 1.5))
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] + N[(N[(x / y), $MachinePrecision] * 1.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.2%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{-2 \cdot {x}^{2}} + \frac{x}{2 \cdot y} \]
4. Step-by-step derivation

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

      \[\leadsto \color{blue}{-2 \cdot {x}^{2}} + \frac{x}{2 \cdot y} \]

   2. unpow2 N/A

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

   3. *-lowering-*.f64 10.8

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

5. Simplified 10.8%

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

7. Applied egg-rr 10.8%

   \[\leadsto \color{blue}{\frac{x \cdot \mathsf{fma}\left(x, -2, \frac{0.5}{y}\right)}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot -8, \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.125}{y \cdot \left(y \cdot y\right)}\right)} \cdot \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{0.5}{y}\right)}{\frac{1}{\mathsf{fma}\left(-2 \cdot \left(x \cdot x\right), x \cdot \mathsf{fma}\left(x, -2, \frac{0.5}{y}\right), \frac{\left(x \cdot x\right) \cdot 0.25}{y \cdot y}\right)}}} \]

8. Taylor expanded in x around inf

   \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right)}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot -8, \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{8}}{y \cdot \left(y \cdot y\right)}\right)} \cdot \color{blue}{\left({x}^{6} \cdot \left(4 \cdot \frac{1}{x \cdot y} - 8\right)\right)} \]
9. Step-by-step derivation

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

      \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right)}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot -8, \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{8}}{y \cdot \left(y \cdot y\right)}\right)} \cdot \color{blue}{\left({x}^{6} \cdot \left(4 \cdot \frac{1}{x \cdot y} - 8\right)\right)} \]

   2. metadata-eval N/A

      \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right)}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot -8, \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{8}}{y \cdot \left(y \cdot y\right)}\right)} \cdot \left({x}^{\color{blue}{\left(2 \cdot 3\right)}} \cdot \left(4 \cdot \frac{1}{x \cdot y} - 8\right)\right) \]

   3. pow-sqr N/A

      \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right)}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot -8, \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{8}}{y \cdot \left(y \cdot y\right)}\right)} \cdot \left(\color{blue}{\left({x}^{3} \cdot {x}^{3}\right)} \cdot \left(4 \cdot \frac{1}{x \cdot y} - 8\right)\right) \]

   4. cube-mult N/A

      \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right)}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot -8, \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{8}}{y \cdot \left(y \cdot y\right)}\right)} \cdot \left(\left(\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot {x}^{3}\right) \cdot \left(4 \cdot \frac{1}{x \cdot y} - 8\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right)}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot -8, \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{8}}{y \cdot \left(y \cdot y\right)}\right)} \cdot \left(\left(\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot {x}^{3}\right) \cdot \left(4 \cdot \frac{1}{x \cdot y} - 8\right)\right) \]

   6. cube-mult N/A

      \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right)}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot -8, \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{8}}{y \cdot \left(y \cdot y\right)}\right)} \cdot \left(\left(\left(x \cdot {x}^{2}\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}\right) \cdot \left(4 \cdot \frac{1}{x \cdot y} - 8\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right)}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot -8, \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{8}}{y \cdot \left(y \cdot y\right)}\right)} \cdot \left(\left(\left(x \cdot {x}^{2}\right) \cdot \left(x \cdot \color{blue}{{x}^{2}}\right)\right) \cdot \left(4 \cdot \frac{1}{x \cdot y} - 8\right)\right) \]

   8. swap-sqr N/A

      \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right)}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot -8, \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{8}}{y \cdot \left(y \cdot y\right)}\right)} \cdot \left(\color{blue}{\left(\left(x \cdot x\right) \cdot \left({x}^{2} \cdot {x}^{2}\right)\right)} \cdot \left(4 \cdot \frac{1}{x \cdot y} - 8\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right)}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot -8, \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{8}}{y \cdot \left(y \cdot y\right)}\right)} \cdot \left(\left(\color{blue}{{x}^{2}} \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot \left(4 \cdot \frac{1}{x \cdot y} - 8\right)\right) \]

   10. pow-sqr N/A

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right)}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot -8, \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{8}}{y \cdot \left(y \cdot y\right)}\right)} \cdot \left(\left({x}^{2} \cdot \color{blue}{{x}^{\left(2 \cdot 2\right)}}\right) \cdot \left(4 \cdot \frac{1}{x \cdot y} - 8\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right)}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot -8, \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{8}}{y \cdot \left(y \cdot y\right)}\right)} \cdot \left(\left({x}^{2} \cdot {x}^{\color{blue}{4}}\right) \cdot \left(4 \cdot \frac{1}{x \cdot y} - 8\right)\right) \]

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

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right)}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot -8, \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{8}}{y \cdot \left(y \cdot y\right)}\right)} \cdot \left(\color{blue}{\left({x}^{2} \cdot {x}^{4}\right)} \cdot \left(4 \cdot \frac{1}{x \cdot y} - 8\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right)}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot -8, \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{8}}{y \cdot \left(y \cdot y\right)}\right)} \cdot \left(\left(\color{blue}{\left(x \cdot x\right)} \cdot {x}^{4}\right) \cdot \left(4 \cdot \frac{1}{x \cdot y} - 8\right)\right) \]

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

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right)}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot -8, \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{8}}{y \cdot \left(y \cdot y\right)}\right)} \cdot \left(\left(\color{blue}{\left(x \cdot x\right)} \cdot {x}^{4}\right) \cdot \left(4 \cdot \frac{1}{x \cdot y} - 8\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right)}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot -8, \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{8}}{y \cdot \left(y \cdot y\right)}\right)} \cdot \left(\left(\left(x \cdot x\right) \cdot {x}^{\color{blue}{\left(3 + 1\right)}}\right) \cdot \left(4 \cdot \frac{1}{x \cdot y} - 8\right)\right) \]

   16. pow-plus N/A

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right)}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot -8, \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{8}}{y \cdot \left(y \cdot y\right)}\right)} \cdot \left(\left(\left(x \cdot x\right) \cdot \color{blue}{\left({x}^{3} \cdot x\right)}\right) \cdot \left(4 \cdot \frac{1}{x \cdot y} - 8\right)\right) \]

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

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right)}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot -8, \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{8}}{y \cdot \left(y \cdot y\right)}\right)} \cdot \left(\left(\left(x \cdot x\right) \cdot \color{blue}{\left({x}^{3} \cdot x\right)}\right) \cdot \left(4 \cdot \frac{1}{x \cdot y} - 8\right)\right) \]

   18. cube-mult N/A

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right)}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot -8, \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{8}}{y \cdot \left(y \cdot y\right)}\right)} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot x\right)\right) \cdot \left(4 \cdot \frac{1}{x \cdot y} - 8\right)\right) \]

   19. unpow2 N/A

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right)}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot -8, \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{8}}{y \cdot \left(y \cdot y\right)}\right)} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot x\right)\right) \cdot \left(4 \cdot \frac{1}{x \cdot y} - 8\right)\right) \]

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

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right)}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot -8, \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{8}}{y \cdot \left(y \cdot y\right)}\right)} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot {x}^{2}\right)} \cdot x\right)\right) \cdot \left(4 \cdot \frac{1}{x \cdot y} - 8\right)\right) \]

   21. unpow2 N/A

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right)}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot -8, \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{8}}{y \cdot \left(y \cdot y\right)}\right)} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot x\right)\right) \cdot \left(4 \cdot \frac{1}{x \cdot y} - 8\right)\right) \]

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

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right)}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot -8, \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{8}}{y \cdot \left(y \cdot y\right)}\right)} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot x\right)\right) \cdot \left(4 \cdot \frac{1}{x \cdot y} - 8\right)\right) \]

   23. sub-neg N/A

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right)}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot -8, \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{8}}{y \cdot \left(y \cdot y\right)}\right)} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot x\right)\right) \cdot \color{blue}{\left(4 \cdot \frac{1}{x \cdot y} + \left(\mathsf{neg}\left(8\right)\right)\right)}\right) \]

   24. associate-*r/ N/A

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right)}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot -8, \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{8}}{y \cdot \left(y \cdot y\right)}\right)} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot x\right)\right) \cdot \left(\color{blue}{\frac{4 \cdot 1}{x \cdot y}} + \left(\mathsf{neg}\left(8\right)\right)\right)\right) \]

   25. metadata-eval N/A

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right)}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot -8, \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{8}}{y \cdot \left(y \cdot y\right)}\right)} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot x\right)\right) \cdot \left(\frac{\color{blue}{4}}{x \cdot y} + \left(\mathsf{neg}\left(8\right)\right)\right)\right) \]

10. Simplified 10.8%

    \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{0.5}{y}\right)}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot -8, \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.125}{y \cdot \left(y \cdot y\right)}\right)} \cdot \color{blue}{\left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot x\right)\right) \cdot \left(-8 + \frac{4}{x \cdot y}\right)\right)} \]

11. Taylor expanded in y around inf

    \[\leadsto \color{blue}{-2 \cdot {x}^{2} + \frac{3}{2} \cdot \frac{x}{y}} \]
12. Step-by-step derivation

    1. unpow2 N/A

       \[\leadsto -2 \cdot \color{blue}{\left(x \cdot x\right)} + \frac{3}{2} \cdot \frac{x}{y} \]

    2. associate-*r* N/A

       \[\leadsto \color{blue}{\left(-2 \cdot x\right) \cdot x} + \frac{3}{2} \cdot \frac{x}{y} \]

    3. *-commutative N/A

       \[\leadsto \color{blue}{x \cdot \left(-2 \cdot x\right)} + \frac{3}{2} \cdot \frac{x}{y} \]

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

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, -2 \cdot x, \frac{3}{2} \cdot \frac{x}{y}\right)} \]

    5. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -2}, \frac{3}{2} \cdot \frac{x}{y}\right) \]

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

       \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -2}, \frac{3}{2} \cdot \frac{x}{y}\right) \]

    7. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot -2, \color{blue}{\frac{x}{y} \cdot \frac{3}{2}}\right) \]

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

       \[\leadsto \mathsf{fma}\left(x, x \cdot -2, \color{blue}{\frac{x}{y} \cdot \frac{3}{2}}\right) \]

    9. /-lowering-/.f64 10.8

       \[\leadsto \mathsf{fma}\left(x, x \cdot -2, \color{blue}{\frac{x}{y}} \cdot 1.5\right) \]

13. Simplified 10.8%

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -2, \frac{x}{y} \cdot 1.5\right)} \]
14. Add Preprocessing

Alternative 2: 11.0% accurate, 21.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ x \cdot \mathsf{fma}\left(x, -2, \frac{1}{y}\right) \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (* x (fma x -2.0 (/ 1.0 y))))

C

assert(x < y);
double code(double x, double y) {
	return x * fma(x, -2.0, (1.0 / y));
}

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(x * fma(x, -2.0, Float64(1.0 / y)))
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 + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.2%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{-2 \cdot {x}^{2}} + \frac{x}{2 \cdot y} \]
4. Step-by-step derivation

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

      \[\leadsto \color{blue}{-2 \cdot {x}^{2}} + \frac{x}{2 \cdot y} \]

   2. unpow2 N/A

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

   3. *-lowering-*.f64 10.8

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

5. Simplified 10.8%

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

7. Applied egg-rr 10.8%

   \[\leadsto \color{blue}{\frac{x \cdot \mathsf{fma}\left(x, -2, \frac{0.5}{y}\right)}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot -8, \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.125}{y \cdot \left(y \cdot y\right)}\right)} \cdot \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{0.5}{y}\right)}{\frac{1}{\mathsf{fma}\left(-2 \cdot \left(x \cdot x\right), x \cdot \mathsf{fma}\left(x, -2, \frac{0.5}{y}\right), \frac{\left(x \cdot x\right) \cdot 0.25}{y \cdot y}\right)}}} \]

8. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{{x}^{4}}} \cdot \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right)}{\frac{1}{\mathsf{fma}\left(-2 \cdot \left(x \cdot x\right), x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right), \frac{\left(x \cdot x\right) \cdot \frac{1}{4}}{y \cdot y}\right)}} \]
9. Step-by-step derivation

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

      \[\leadsto \color{blue}{\frac{\frac{1}{4}}{{x}^{4}}} \cdot \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right)}{\frac{1}{\mathsf{fma}\left(-2 \cdot \left(x \cdot x\right), x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right), \frac{\left(x \cdot x\right) \cdot \frac{1}{4}}{y \cdot y}\right)}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{4}}{{x}^{\color{blue}{\left(3 + 1\right)}}} \cdot \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right)}{\frac{1}{\mathsf{fma}\left(-2 \cdot \left(x \cdot x\right), x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right), \frac{\left(x \cdot x\right) \cdot \frac{1}{4}}{y \cdot y}\right)}} \]

   3. pow-plus N/A

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{{x}^{3} \cdot x}} \cdot \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right)}{\frac{1}{\mathsf{fma}\left(-2 \cdot \left(x \cdot x\right), x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right), \frac{\left(x \cdot x\right) \cdot \frac{1}{4}}{y \cdot y}\right)}} \]

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

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{{x}^{3} \cdot x}} \cdot \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right)}{\frac{1}{\mathsf{fma}\left(-2 \cdot \left(x \cdot x\right), x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right), \frac{\left(x \cdot x\right) \cdot \frac{1}{4}}{y \cdot y}\right)}} \]

   5. cube-mult N/A

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot x} \cdot \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right)}{\frac{1}{\mathsf{fma}\left(-2 \cdot \left(x \cdot x\right), x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right), \frac{\left(x \cdot x\right) \cdot \frac{1}{4}}{y \cdot y}\right)}} \]

   6. unpow2 N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot x} \cdot \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right)}{\frac{1}{\mathsf{fma}\left(-2 \cdot \left(x \cdot x\right), x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right), \frac{\left(x \cdot x\right) \cdot \frac{1}{4}}{y \cdot y}\right)}} \]

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

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{\left(x \cdot {x}^{2}\right)} \cdot x} \cdot \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right)}{\frac{1}{\mathsf{fma}\left(-2 \cdot \left(x \cdot x\right), x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right), \frac{\left(x \cdot x\right) \cdot \frac{1}{4}}{y \cdot y}\right)}} \]

   8. unpow2 N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot x} \cdot \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right)}{\frac{1}{\mathsf{fma}\left(-2 \cdot \left(x \cdot x\right), x \cdot \mathsf{fma}\left(x, -2, \frac{\frac{1}{2}}{y}\right), \frac{\left(x \cdot x\right) \cdot \frac{1}{4}}{y \cdot y}\right)}} \]

   9. *-lowering-*.f64 10.8

      \[\leadsto \frac{0.25}{\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot x} \cdot \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{0.5}{y}\right)}{\frac{1}{\mathsf{fma}\left(-2 \cdot \left(x \cdot x\right), x \cdot \mathsf{fma}\left(x, -2, \frac{0.5}{y}\right), \frac{\left(x \cdot x\right) \cdot 0.25}{y \cdot y}\right)}} \]

10. Simplified 10.8%

    \[\leadsto \color{blue}{\frac{0.25}{\left(x \cdot \left(x \cdot x\right)\right) \cdot x}} \cdot \frac{x \cdot \mathsf{fma}\left(x, -2, \frac{0.5}{y}\right)}{\frac{1}{\mathsf{fma}\left(-2 \cdot \left(x \cdot x\right), x \cdot \mathsf{fma}\left(x, -2, \frac{0.5}{y}\right), \frac{\left(x \cdot x\right) \cdot 0.25}{y \cdot y}\right)}} \]

11. Taylor expanded in y around inf

    \[\leadsto \color{blue}{-2 \cdot {x}^{2} + \frac{x}{y}} \]
12. Step-by-step derivation

    1. unpow2 N/A

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

    2. associate-*r* N/A

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

    3. *-lft-identity N/A

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

    4. associate-*l/ N/A

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

    5. distribute-rgt-in N/A

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

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

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

    7. *-commutative N/A

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

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

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

    9. /-lowering-/.f64 10.8

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

13. Simplified 10.8%

    \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, -2, \frac{1}{y}\right)} \]
14. Add Preprocessing

Alternative 3: 11.0% accurate, 21.0× speedup

Math

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

FPCore

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

C

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

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(x * fma(x, -2.0, Float64(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[(x * -2.0 + N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.2%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{-2 \cdot {x}^{2}} + \frac{x}{2 \cdot y} \]
4. Step-by-step derivation

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

      \[\leadsto \color{blue}{-2 \cdot {x}^{2}} + \frac{x}{2 \cdot y} \]

   2. unpow2 N/A

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

   3. *-lowering-*.f64 10.8

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

5. Simplified 10.8%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. div-inv N/A

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

   4. distribute-lft-out N/A

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

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

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

   6. *-commutative N/A

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

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

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

   8. associate-/r* N/A

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

   9. metadata-eval N/A

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

   10. /-lowering-/.f64 10.8

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

7. Applied egg-rr 10.8%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, -2, \frac{0.5}{y}\right)} \]
8. Add Preprocessing

Alternative 4: 11.0% accurate, 44.0× 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%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{-2 \cdot {x}^{2}} + \frac{x}{2 \cdot y} \]
4. Step-by-step derivation

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

      \[\leadsto \color{blue}{-2 \cdot {x}^{2}} + \frac{x}{2 \cdot y} \]

   2. unpow2 N/A

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

   3. *-lowering-*.f64 10.8

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

5. Simplified 10.8%

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

6. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

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

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

   5. *-commutative N/A

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

   6. *-lowering-*.f64 10.8

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

8. Simplified 10.8%

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

Reproduce

herbie shell --seed 2024199 
(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))))