Rump's expression from Stadtherr's award speech

Percentage Accurate: 9.2% → 20.2%

Time: 8.1s

Alternatives: 3

Speedup: 17.3×

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: 9.2% 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: 20.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \frac{x}{y} - x \cdot \left(x \cdot -2\right)\\ \frac{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, -2 \cdot {x}^{5}, 0.5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right), x \cdot \left(\left(x \cdot x\right) \cdot 0.125\right)\right)}{y \cdot \left(y \cdot y\right)}}{t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (* 0.5 (/ x y)) (* x (* x -2.0)))))
   (/
    (/
     (fma
      y
      (fma y (* -2.0 (pow x 5.0)) (* 0.5 (* x (* x (* x x)))))
      (* x (* (* x x) 0.125)))
     (* y (* y y)))
    (* t_0 t_0))))

C

double code(double x, double y) {
	double t_0 = (0.5 * (x / y)) - (x * (x * -2.0));
	return (fma(y, fma(y, (-2.0 * pow(x, 5.0)), (0.5 * (x * (x * (x * x))))), (x * ((x * x) * 0.125))) / (y * (y * y))) / (t_0 * t_0);
}

Julia

function code(x, y)
	t_0 = Float64(Float64(0.5 * Float64(x / y)) - Float64(x * Float64(x * -2.0)))
	return Float64(Float64(fma(y, fma(y, Float64(-2.0 * (x ^ 5.0)), Float64(0.5 * Float64(x * Float64(x * Float64(x * x))))), Float64(x * Float64(Float64(x * x) * 0.125))) / Float64(y * Float64(y * y))) / Float64(t_0 * t_0))
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision] - N[(x * N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(y * N[(y * N[(-2.0 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(x * x), $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$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. lower-*.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. lower-*.f64 10.8

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

5. Applied rewrites 10.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. associate-*l/ N/A

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

   9. lift-/.f64 N/A

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

   10. lower-fma.f64 10.8

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

7. Applied rewrites 10.8%

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

   1. lift-fma.f64 N/A

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

   2. flip-+ N/A

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

9. Applied rewrites 10.8%

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

10. Taylor expanded in y around 0

    \[\leadsto \frac{\color{blue}{\frac{\frac{1}{8} \cdot {x}^{3} + y \cdot \left(-2 \cdot \left({x}^{5} \cdot y\right) + \frac{1}{2} \cdot {x}^{4}\right)}{{y}^{3}}}}{\left(\frac{x}{y} \cdot \frac{1}{2} - x \cdot \left(x \cdot -2\right)\right) \cdot \left(\frac{x}{y} \cdot \frac{1}{2} - x \cdot \left(x \cdot -2\right)\right)} \]
11. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \frac{\color{blue}{\frac{\frac{1}{8} \cdot {x}^{3} + y \cdot \left(-2 \cdot \left({x}^{5} \cdot y\right) + \frac{1}{2} \cdot {x}^{4}\right)}{{y}^{3}}}}{\left(\frac{x}{y} \cdot \frac{1}{2} - x \cdot \left(x \cdot -2\right)\right) \cdot \left(\frac{x}{y} \cdot \frac{1}{2} - x \cdot \left(x \cdot -2\right)\right)} \]

12. Applied rewrites 20.2%

    \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, -2 \cdot {x}^{5}, 0.5 \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot x\right)\right), x \cdot \left(\left(x \cdot x\right) \cdot 0.125\right)\right)}{y \cdot \left(y \cdot y\right)}}}{\left(\frac{x}{y} \cdot 0.5 - x \cdot \left(x \cdot -2\right)\right) \cdot \left(\frac{x}{y} \cdot 0.5 - x \cdot \left(x \cdot -2\right)\right)} \]

13. Final simplification 20.2%

    \[\leadsto \frac{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, -2 \cdot {x}^{5}, 0.5 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right), x \cdot \left(\left(x \cdot x\right) \cdot 0.125\right)\right)}{y \cdot \left(y \cdot y\right)}}{\left(0.5 \cdot \frac{x}{y} - x \cdot \left(x \cdot -2\right)\right) \cdot \left(0.5 \cdot \frac{x}{y} - x \cdot \left(x \cdot -2\right)\right)} \]
14. Add Preprocessing

Alternative 2: 10.8% accurate, 17.3× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (fma (/ x y) 0.5 (* x (* x -2.0))))

C

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

Julia

function code(x, y)
	return fma(Float64(x / y), 0.5, Float64(x * Float64(x * -2.0)))
end

Wolfram

code[x_, y_] := N[(N[(x / y), $MachinePrecision] * 0.5 + N[(x * N[(x * -2.0), $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. lower-*.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. lower-*.f64 10.8

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

5. Applied rewrites 10.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. associate-*l/ N/A

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

   9. lift-/.f64 N/A

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

   10. lower-fma.f64 10.8

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

7. Applied rewrites 10.8%

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

Alternative 3: 1.6% accurate, 28.5× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{x}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* 0.5 (/ x y)))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.5d0 * (x / y)
end function

Java

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

Python

def code(x, y):
	return 0.5 * (x / y)

Julia

function code(x, y)
	return Float64(0.5 * Float64(x / y))
end

MATLAB

function tmp = code(x, y)
	tmp = 0.5 * (x / y);
end

Wolfram

code[x_, y_] := N[(0.5 * N[(x / y), $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

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x}{y}} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x}{y}} \]

   2. lower-/.f64 1.6

      \[\leadsto 0.5 \cdot \color{blue}{\frac{x}{y}} \]

5. Applied rewrites 1.6%

   \[\leadsto \color{blue}{0.5 \cdot \frac{x}{y}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024222 
(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))))