Rump's expression from Stadtherr's award speech

Percentage Accurate: 1.4% → 11.1%

Time: 8.9s

Alternatives: 3

Speedup: 80.7×

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: 11.1% accurate, 24.2× speedup

Math

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

FPCore

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

C

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

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 * ((0.5d0 / y) - x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   \[\leadsto \color{blue}{-2 \cdot {x}^{10}} + \frac{x}{2 \cdot y} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{-2 \cdot {x}^{10}} + \frac{x}{2 \cdot y} \]

   2. lower-pow.f64 7.3

      \[\leadsto -2 \cdot \color{blue}{{x}^{10}} + \frac{x}{2 \cdot y} \]

5. Simplified 7.3%

   \[\leadsto \color{blue}{-2 \cdot {x}^{10}} + \frac{x}{2 \cdot y} \]

6. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 10.8

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

8. Simplified 10.8%

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

9. Taylor expanded in x around -inf

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

    1. mul-1-neg N/A

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

    2. lower-neg.f64 10.9

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

11. Simplified 10.9%

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

    1. lift-neg.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-/.f64 N/A

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

    5. +-commutative N/A

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

    6. lift-/.f64 N/A

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

    7. div-inv N/A

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

    8. lift-*.f64 N/A

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

    9. distribute-lft-out N/A

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

    10. lower-*.f64 N/A

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

    11. lift-*.f64 N/A

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

    12. associate-/r* N/A

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

    13. metadata-eval N/A

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

    14. lift-/.f64 N/A

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

    15. lift-neg.f64 N/A

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

    16. unsub-neg N/A

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

    17. lower--.f64 10.9

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

13. Applied egg-rr 10.9%

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

Alternative 2: 11.0% accurate, 44.0× speedup

Math

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

FPCore

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

C

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

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 = (-2.0d0) * (x * x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   \[\leadsto \color{blue}{-2 \cdot {x}^{10}} + \frac{x}{2 \cdot y} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{-2 \cdot {x}^{10}} + \frac{x}{2 \cdot y} \]

   2. lower-pow.f64 7.3

      \[\leadsto -2 \cdot \color{blue}{{x}^{10}} + \frac{x}{2 \cdot y} \]

5. Simplified 7.3%

   \[\leadsto \color{blue}{-2 \cdot {x}^{10}} + \frac{x}{2 \cdot y} \]

6. Taylor expanded in y around inf

   \[\leadsto \color{blue}{-2 \cdot {x}^{3}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{-2 \cdot {x}^{3}} \]

   2. cube-mult N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 9.9

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

8. Simplified 9.9%

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

9. Taylor expanded in x around 0

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

    1. lower-*.f64 N/A

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

    2. unpow2 N/A

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

    3. lower-*.f64 10.8

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

11. Simplified 10.8%

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

Alternative 3: 1.5% accurate, 80.7× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   \[\leadsto \color{blue}{-2 \cdot {x}^{10}} + \frac{x}{2 \cdot y} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{-2 \cdot {x}^{10}} + \frac{x}{2 \cdot y} \]

   2. lower-pow.f64 7.3

      \[\leadsto -2 \cdot \color{blue}{{x}^{10}} + \frac{x}{2 \cdot y} \]

5. Simplified 7.3%

   \[\leadsto \color{blue}{-2 \cdot {x}^{10}} + \frac{x}{2 \cdot y} \]

6. Taylor expanded in y around inf

   \[\leadsto \color{blue}{-2 \cdot {x}^{3}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{-2 \cdot {x}^{3}} \]

   2. cube-mult N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 9.9

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

8. Simplified 9.9%

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

9. Taylor expanded in x around inf

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

    1. unpow2 N/A

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

    2. lower-*.f64 1.5

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

11. Simplified 1.5%

    \[\leadsto \color{blue}{x \cdot x} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(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))))