Rump's expression from Stadtherr's award speech

Percentage Accurate: 9.2% → 20.2%

Time: 6.9s

Alternatives: 4

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, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{2 \cdot y}\\ t_1 := \left(x \cdot x\right) \cdot -2 - t\_0\\ \frac{\mathsf{fma}\left(t\_0, t\_0 \cdot t\_1, -2 \cdot \frac{{x}^{5}}{y}\right)}{t\_1 \cdot t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* 2.0 y))) (t_1 (- (* (* x x) -2.0) t_0)))
   (/ (fma t_0 (* t_0 t_1) (* -2.0 (/ (pow x 5.0) y))) (* t_1 t_1))))

C

double code(double x, double y) {
	double t_0 = x / (2.0 * y);
	double t_1 = ((x * x) * -2.0) - t_0;
	return fma(t_0, (t_0 * t_1), (-2.0 * (pow(x, 5.0) / y))) / (t_1 * t_1);
}

Julia

function code(x, y)
	t_0 = Float64(x / Float64(2.0 * y))
	t_1 = Float64(Float64(Float64(x * x) * -2.0) - t_0)
	return Float64(fma(t_0, Float64(t_0 * t_1), Float64(-2.0 * Float64((x ^ 5.0) / y))) / Float64(t_1 * t_1))
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(2.0 * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x * x), $MachinePrecision] * -2.0), $MachinePrecision] - t$95$0), $MachinePrecision]}, N[(N[(t$95$0 * N[(t$95$0 * t$95$1), $MachinePrecision] + N[(-2.0 * N[(N[Power[x, 5.0], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * t$95$1), $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. flip-+ N/A

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

7. Applied rewrites 10.8%

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

8. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{-2 \cdot \frac{{x}^{5}}{y}} - \left(\left(x \cdot x\right) \cdot -2 - \frac{x}{2 \cdot y}\right) \cdot {\left(\frac{x}{2 \cdot y}\right)}^{2}}{\left(\left(x \cdot x\right) \cdot -2 - \frac{x}{2 \cdot y}\right) \cdot \left(\left(x \cdot x\right) \cdot -2 - \frac{x}{2 \cdot y}\right)} \]
9. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-pow.f64 20.2

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

10. Applied rewrites 20.2%

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

12. Applied rewrites 20.2%

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

Alternative 2: 10.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot -2\\ t_1 := t\_0 - \frac{x}{2 \cdot y}\\ t_2 := t\_0 - \frac{x}{-2 \cdot y}\\ \frac{\mathsf{fma}\left(t\_2 \cdot t\_0, t\_0, \left(\frac{0.25}{y} \cdot \frac{x \cdot x}{y}\right) \cdot t\_2\right)}{t\_1 \cdot t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* x x) -2.0))
        (t_1 (- t_0 (/ x (* 2.0 y))))
        (t_2 (- t_0 (/ x (* -2.0 y)))))
   (/ (fma (* t_2 t_0) t_0 (* (* (/ 0.25 y) (/ (* x x) y)) t_2)) (* t_1 t_1))))

C

double code(double x, double y) {
	double t_0 = (x * x) * -2.0;
	double t_1 = t_0 - (x / (2.0 * y));
	double t_2 = t_0 - (x / (-2.0 * y));
	return fma((t_2 * t_0), t_0, (((0.25 / y) * ((x * x) / y)) * t_2)) / (t_1 * t_1);
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * x) * -2.0)
	t_1 = Float64(t_0 - Float64(x / Float64(2.0 * y)))
	t_2 = Float64(t_0 - Float64(x / Float64(-2.0 * y)))
	return Float64(fma(Float64(t_2 * t_0), t_0, Float64(Float64(Float64(0.25 / y) * Float64(Float64(x * x) / y)) * t_2)) / Float64(t_1 * t_1))
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[(x / N[(2.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - N[(x / N[(-2.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(t$95$2 * t$95$0), $MachinePrecision] * t$95$0 + N[(N[(N[(0.25 / y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * t$95$1), $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. flip-+ N/A

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

7. Applied rewrites 10.8%

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

9. Applied rewrites 10.8%

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

10. Taylor expanded in x around 0

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

    1. associate-*r/ N/A

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

    2. unpow2 N/A

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

    3. times-frac N/A

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

    4. lower-*.f64 N/A

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

    5. lower-/.f64 N/A

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

    6. lower-/.f64 N/A

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

    7. unpow2 N/A

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

    8. lower-*.f64 10.8

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

12. Applied rewrites 10.8%

    \[\leadsto \frac{\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot -2 - \frac{x}{-2 \cdot y}\right) \cdot \left(\left(x \cdot x\right) \cdot -2\right), \left(x \cdot x\right) \cdot -2, \color{blue}{\left(\frac{0.25}{y} \cdot \frac{x \cdot x}{y}\right)} \cdot \left(\left(x \cdot x\right) \cdot -2 - \frac{x}{-2 \cdot y}\right)\right)}{\left(\left(x \cdot x\right) \cdot -2 - \frac{x}{2 \cdot y}\right) \cdot \left(\left(x \cdot x\right) \cdot -2 - \frac{x}{2 \cdot y}\right)} \]
13. Add Preprocessing

Alternative 3: 10.8% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((-2.0d0) * (x * x)) + (x / (y + y))
end function

Java

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

Python

def code(x, y):
	return (-2.0 * (x * x)) + (x / (y + y))

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = (-2.0 * (x * x)) + (x / (y + y));
end

Wolfram

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

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

   2. count-2-rev N/A

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

   3. lower-+.f64 10.8

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

7. Applied rewrites 10.8%

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

Alternative 4: 1.6% accurate, 28.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. associate-*r/ N/A

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

   2. associate-*l/ N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-*.f64 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 1.6

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

5. Applied rewrites 1.6%

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

Reproduce

herbie shell --seed 2024327 
(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))))