Rump's expression from Stadtherr's award speech

Percentage Accurate: 9.2% → 7.9%

Time: 18.1s

Alternatives: 3

Speedup: N/A×

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: 7.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{{y}^{7} \cdot \left(333.75 - {x}^{2}\right) + 0.5 \cdot x}{y} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (+ (* (pow y 7.0) (- 333.75 (pow x 2.0))) (* 0.5 x)) y))

C

double code(double x, double y) {
	return ((pow(y, 7.0) * (333.75 - pow(x, 2.0))) + (0.5 * x)) / y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (((y ** 7.0d0) * (333.75d0 - (x ** 2.0d0))) + (0.5d0 * x)) / y
end function

Java

public static double code(double x, double y) {
	return ((Math.pow(y, 7.0) * (333.75 - Math.pow(x, 2.0))) + (0.5 * x)) / y;
}

Python

def code(x, y):
	return ((math.pow(y, 7.0) * (333.75 - math.pow(x, 2.0))) + (0.5 * x)) / y

Julia

function code(x, y)
	return Float64(Float64(Float64((y ^ 7.0) * Float64(333.75 - (x ^ 2.0))) + Float64(0.5 * x)) / y)
end

MATLAB

function tmp = code(x, y)
	tmp = (((y ^ 7.0) * (333.75 - (x ^ 2.0))) + (0.5 * x)) / y;
end

Wolfram

code[x_, y_] := N[(N[(N[(N[Power[y, 7.0], $MachinePrecision] * N[(333.75 - N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * x), $MachinePrecision]), $MachinePrecision] / y), $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-rgt-neg-in 1.5%

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

7. Simplified 1.5%

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

   1. pow2 1.5%

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

   2. distribute-rgt-neg-out 1.5%

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

   3. distribute-lft-neg-in 1.5%

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

   4. add-sqr-sqrt 1.5%

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

   5. sqrt-unprod 1.5%

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

   6. sqr-neg 1.5%

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

   7. sqrt-unprod 0.0%

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

   8. add-sqr-sqrt 1.4%

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

   9. cancel-sign-sub-inv 1.4%

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

   10. pow2 1.4%

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

   11. log1p-expm1-u 0.7%

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

   12. *-commutative 0.7%

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

   13. pow2 0.7%

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

   14. distribute-lft-out-- 0.7%

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

9. Applied egg-rr 3.1%

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

10. Taylor expanded in y around 0 7.9%

    \[\leadsto \color{blue}{\frac{0.5 \cdot x + {y}^{7} \cdot \left(333.75 - {x}^{2}\right)}{y}} \]

11. Final simplification 7.9%

    \[\leadsto \frac{{y}^{7} \cdot \left(333.75 - {x}^{2}\right) + 0.5 \cdot x}{y} \]
12. Add Preprocessing

Alternative 2: 7.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot x - {y}^{7} \cdot {x}^{2}}{y} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (- (* 0.5 x) (* (pow y 7.0) (pow x 2.0))) y))

C

double code(double x, double y) {
	return ((0.5 * x) - (pow(y, 7.0) * pow(x, 2.0))) / y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((0.5d0 * x) - ((y ** 7.0d0) * (x ** 2.0d0))) / y
end function

Java

public static double code(double x, double y) {
	return ((0.5 * x) - (Math.pow(y, 7.0) * Math.pow(x, 2.0))) / y;
}

Python

def code(x, y):
	return ((0.5 * x) - (math.pow(y, 7.0) * math.pow(x, 2.0))) / y

Julia

function code(x, y)
	return Float64(Float64(Float64(0.5 * x) - Float64((y ^ 7.0) * (x ^ 2.0))) / y)
end

MATLAB

function tmp = code(x, y)
	tmp = ((0.5 * x) - ((y ^ 7.0) * (x ^ 2.0))) / y;
end

Wolfram

code[x_, y_] := N[(N[(N[(0.5 * x), $MachinePrecision] - N[(N[Power[y, 7.0], $MachinePrecision] * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $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-rgt-neg-in 1.5%

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

7. Simplified 1.5%

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

8. Taylor expanded in x around inf 8.6%

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

   1. mul-1-neg 8.6%

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

   2. distribute-rgt-neg-in 8.6%

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

10. Simplified 8.6%

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

11. Taylor expanded in y around 0 7.9%

    \[\leadsto \color{blue}{\frac{-1 \cdot \left({x}^{2} \cdot {y}^{7}\right) + 0.5 \cdot x}{y}} \]
12. Step-by-step derivation

    1. +-commutative 7.9%

       \[\leadsto \frac{\color{blue}{0.5 \cdot x + -1 \cdot \left({x}^{2} \cdot {y}^{7}\right)}}{y} \]

    2. *-commutative 7.9%

       \[\leadsto \frac{\color{blue}{x \cdot 0.5} + -1 \cdot \left({x}^{2} \cdot {y}^{7}\right)}{y} \]

    3. mul-1-neg 7.9%

       \[\leadsto \frac{x \cdot 0.5 + \color{blue}{\left(-{x}^{2} \cdot {y}^{7}\right)}}{y} \]

    4. unsub-neg 7.9%

       \[\leadsto \frac{\color{blue}{x \cdot 0.5 - {x}^{2} \cdot {y}^{7}}}{y} \]

13. Simplified 7.9%

    \[\leadsto \color{blue}{\frac{x \cdot 0.5 - {x}^{2} \cdot {y}^{7}}{y}} \]

14. Final simplification 7.9%

    \[\leadsto \frac{0.5 \cdot x - {y}^{7} \cdot {x}^{2}}{y} \]
15. Add Preprocessing

Alternative 3: 1.6% accurate, 87.8× speedup

Math

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

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(Float64(0.5 * x) / y)
end

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(0.5 * x), $MachinePrecision] / y), $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-rgt-neg-in 1.5%

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

7. Simplified 1.5%

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

   1. pow2 1.5%

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

   2. distribute-rgt-neg-out 1.5%

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

   3. distribute-lft-neg-in 1.5%

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

   4. add-sqr-sqrt 1.5%

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

   5. sqrt-unprod 1.5%

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

   6. sqr-neg 1.5%

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

   7. sqrt-unprod 0.0%

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

   8. add-sqr-sqrt 1.4%

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

   9. cancel-sign-sub-inv 1.4%

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

   10. pow2 1.4%

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

   11. log1p-expm1-u 0.7%

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

   12. *-commutative 0.7%

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

   13. pow2 0.7%

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

   14. distribute-lft-out-- 0.7%

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

9. Applied egg-rr 3.1%

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

10. Taylor expanded in y around 0 1.6%

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

    1. associate-*r/ 1.6%

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

    2. *-commutative 1.6%

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

12. Simplified 1.6%

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

13. Final simplification 1.6%

    \[\leadsto \frac{0.5 \cdot x}{y} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024111 
(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))))