Rump's expression from Stadtherr's award speech

Percentage Accurate: 1.4% → 11.0%

Time: 12.1s

Alternatives: 2

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: 11.0% accurate, 4.0× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	return pow(x, 2.0) * ((0.5 / (x * y)) + -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 ** 2.0d0) * ((0.5d0 / (x * y)) + (-2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. log1p-expm1-u 3.1%

      \[\leadsto \left(\left(333.75 \cdot {y}^{6} + \left(x \cdot x\right) \cdot \left(\left(\left(\left(\left(\left(11 \cdot x\right) \cdot x\right) \cdot y\right) \cdot y - {y}^{6}\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(121 \cdot {y}^{4}\right)\right)}\right) - 2\right)\right) + 5.5 \cdot {y}^{8}\right) + \frac{x}{2 \cdot y} \]

4. Applied egg-rr 3.1%

   \[\leadsto \left(\left(333.75 \cdot {y}^{6} + \left(x \cdot x\right) \cdot \left(\left(\left(\left(\left(\left(11 \cdot x\right) \cdot x\right) \cdot y\right) \cdot y - {y}^{6}\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(121 \cdot {y}^{4}\right)\right)}\right) - 2\right)\right) + 5.5 \cdot {y}^{8}\right) + \frac{x}{2 \cdot y} \]

5. Taylor expanded in y around 0 1.4%

   \[\leadsto \left(\left(333.75 \cdot {y}^{6} + \left(x \cdot x\right) \cdot \color{blue}{-2}\right) + 5.5 \cdot {y}^{8}\right) + \frac{x}{2 \cdot y} \]
6. Step-by-step derivation

   1. log1p-expm1-u 3.1%

      \[\leadsto \left(\left(333.75 \cdot {y}^{6} + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot x\right)\right)} \cdot -2\right) + 5.5 \cdot {y}^{8}\right) + \frac{x}{2 \cdot y} \]

   2. pow2 3.1%

      \[\leadsto \left(\left(333.75 \cdot {y}^{6} + \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{{x}^{2}}\right)\right) \cdot -2\right) + 5.5 \cdot {y}^{8}\right) + \frac{x}{2 \cdot y} \]

7. Applied egg-rr 3.1%

   \[\leadsto \left(\left(333.75 \cdot {y}^{6} + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({x}^{2}\right)\right)} \cdot -2\right) + 5.5 \cdot {y}^{8}\right) + \frac{x}{2 \cdot y} \]

8. Taylor expanded in x around inf 10.8%

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.5 \cdot \frac{1}{x \cdot y} - 2\right)} \]
9. Step-by-step derivation

   1. sub-neg 10.8%

      \[\leadsto {x}^{2} \cdot \color{blue}{\left(0.5 \cdot \frac{1}{x \cdot y} + \left(-2\right)\right)} \]

   2. associate-*r/ 10.8%

      \[\leadsto {x}^{2} \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{x \cdot y}} + \left(-2\right)\right) \]

   3. metadata-eval 10.8%

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

   4. metadata-eval 10.8%

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

10. Simplified 10.8%

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

Alternative 2: 1.6% accurate, 87.8× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	return x / (2.0 * 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 / (2.0d0 * y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. log1p-expm1-u 3.1%

      \[\leadsto \left(\left(333.75 \cdot {y}^{6} + \left(x \cdot x\right) \cdot \left(\left(\left(\left(\left(\left(11 \cdot x\right) \cdot x\right) \cdot y\right) \cdot y - {y}^{6}\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(121 \cdot {y}^{4}\right)\right)}\right) - 2\right)\right) + 5.5 \cdot {y}^{8}\right) + \frac{x}{2 \cdot y} \]

4. Applied egg-rr 3.1%

   \[\leadsto \left(\left(333.75 \cdot {y}^{6} + \left(x \cdot x\right) \cdot \left(\left(\left(\left(\left(\left(11 \cdot x\right) \cdot x\right) \cdot y\right) \cdot y - {y}^{6}\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(121 \cdot {y}^{4}\right)\right)}\right) - 2\right)\right) + 5.5 \cdot {y}^{8}\right) + \frac{x}{2 \cdot y} \]

5. Taylor expanded in y around inf 1.5%

   \[\leadsto \left(\left(333.75 \cdot {y}^{6} + \color{blue}{-1 \cdot \left({x}^{2} \cdot {y}^{6}\right)}\right) + 5.5 \cdot {y}^{8}\right) + \frac{x}{2 \cdot y} \]
6. Step-by-step derivation

   1. mul-1-neg 1.5%

      \[\leadsto \left(\left(333.75 \cdot {y}^{6} + \color{blue}{\left(-{x}^{2} \cdot {y}^{6}\right)}\right) + 5.5 \cdot {y}^{8}\right) + \frac{x}{2 \cdot y} \]

   2. *-commutative 1.5%

      \[\leadsto \left(\left(333.75 \cdot {y}^{6} + \left(-\color{blue}{{y}^{6} \cdot {x}^{2}}\right)\right) + 5.5 \cdot {y}^{8}\right) + \frac{x}{2 \cdot y} \]

   3. distribute-lft-neg-in 1.5%

      \[\leadsto \left(\left(333.75 \cdot {y}^{6} + \color{blue}{\left(-{y}^{6}\right) \cdot {x}^{2}}\right) + 5.5 \cdot {y}^{8}\right) + \frac{x}{2 \cdot y} \]

7. Simplified 1.5%

   \[\leadsto \left(\left(333.75 \cdot {y}^{6} + \color{blue}{\left(-{y}^{6}\right) \cdot {x}^{2}}\right) + 5.5 \cdot {y}^{8}\right) + \frac{x}{2 \cdot y} \]

8. Taylor expanded in y around inf 1.4%

   \[\leadsto \color{blue}{5.5 \cdot {y}^{8}} + \frac{x}{2 \cdot y} \]

9. Taylor expanded in y around 0 1.6%

   \[\leadsto \color{blue}{0.5 \cdot \frac{x}{y}} \]
10. Step-by-step derivation

    1. metadata-eval 1.6%

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

    2. times-frac 1.6%

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

    3. neg-mul-1 1.6%

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

    4. *-commutative 1.6%

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

    5. distribute-neg-frac 1.6%

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

    6. associate-/r* 1.6%

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

    7. distribute-neg-frac2 1.6%

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

    8. metadata-eval 1.6%

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

    9. associate-/r* 1.6%

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

11. Simplified 1.6%

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

12. Final simplification 1.6%

    \[\leadsto \frac{x}{2 \cdot y} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024144 
(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))))