From Rump in a 1983 paper

Percentage Accurate: 18.8% → 100.0%

Time: 16.5s

Alternatives: 5

Speedup: 1.0×

Specification

\[x = 10864 \land y = 18817\]

Math

\[\begin{array}{l} \\ \left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \left(y \cdot y\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ (- (* 9.0 (pow x 4.0)) (pow y 4.0)) (* 2.0 (* y y))))

C

double code(double x, double y) {
	return ((9.0 * pow(x, 4.0)) - pow(y, 4.0)) + (2.0 * (y * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((9.0d0 * (x ** 4.0d0)) - (y ** 4.0d0)) + (2.0d0 * (y * y))
end function

Java

public static double code(double x, double y) {
	return ((9.0 * Math.pow(x, 4.0)) - Math.pow(y, 4.0)) + (2.0 * (y * y));
}

Python

def code(x, y):
	return ((9.0 * math.pow(x, 4.0)) - math.pow(y, 4.0)) + (2.0 * (y * y))

Julia

function code(x, y)
	return Float64(Float64(Float64(9.0 * (x ^ 4.0)) - (y ^ 4.0)) + Float64(2.0 * Float64(y * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = ((9.0 * (x ^ 4.0)) - (y ^ 4.0)) + (2.0 * (y * y));
end

Wolfram

code[x_, y_] := N[(N[(N[(9.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] - N[Power[y, 4.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 18.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \left(y \cdot y\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ (- (* 9.0 (pow x 4.0)) (pow y 4.0)) (* 2.0 (* y y))))

C

double code(double x, double y) {
	return ((9.0 * pow(x, 4.0)) - pow(y, 4.0)) + (2.0 * (y * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((9.0d0 * (x ** 4.0d0)) - (y ** 4.0d0)) + (2.0d0 * (y * y))
end function

Java

public static double code(double x, double y) {
	return ((9.0 * Math.pow(x, 4.0)) - Math.pow(y, 4.0)) + (2.0 * (y * y));
}

Python

def code(x, y):
	return ((9.0 * math.pow(x, 4.0)) - math.pow(y, 4.0)) + (2.0 * (y * y))

Julia

function code(x, y)
	return Float64(Float64(Float64(9.0 * (x ^ 4.0)) - (y ^ 4.0)) + Float64(2.0 * Float64(y * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = ((9.0 * (x ^ 4.0)) - (y ^ 4.0)) + (2.0 * (y * y));
end

Wolfram

code[x_, y_] := N[(N[(N[(9.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] - N[Power[y, 4.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left({y}^{2}, -{y}^{2}, 9 \cdot {x}^{4}\right) + 2 \cdot \left(y \cdot y\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ (fma (pow y 2.0) (- (pow y 2.0)) (* 9.0 (pow x 4.0))) (* 2.0 (* y y))))

C

double code(double x, double y) {
	return fma(pow(y, 2.0), -pow(y, 2.0), (9.0 * pow(x, 4.0))) + (2.0 * (y * y));
}

Julia

function code(x, y)
	return Float64(fma((y ^ 2.0), Float64(-(y ^ 2.0)), Float64(9.0 * (x ^ 4.0))) + Float64(2.0 * Float64(y * y)))
end

Wolfram

code[x_, y_] := N[(N[(N[Power[y, 2.0], $MachinePrecision] * (-N[Power[y, 2.0], $MachinePrecision]) + N[(9.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.8%

   \[\left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \left(y \cdot y\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg 18.8%

      \[\leadsto \color{blue}{\left(9 \cdot {x}^{4} + \left(-{y}^{4}\right)\right)} + 2 \cdot \left(y \cdot y\right) \]

   2. +-commutative 18.8%

      \[\leadsto \color{blue}{\left(\left(-{y}^{4}\right) + 9 \cdot {x}^{4}\right)} + 2 \cdot \left(y \cdot y\right) \]

   3. metadata-eval 18.8%

      \[\leadsto \left(\left(-{y}^{\color{blue}{\left(2 + 2\right)}}\right) + 9 \cdot {x}^{4}\right) + 2 \cdot \left(y \cdot y\right) \]

   4. pow-prod-up 18.8%

      \[\leadsto \left(\left(-\color{blue}{{y}^{2} \cdot {y}^{2}}\right) + 9 \cdot {x}^{4}\right) + 2 \cdot \left(y \cdot y\right) \]

   5. pow2 18.8%

      \[\leadsto \left(\left(-\color{blue}{\left(y \cdot y\right)} \cdot {y}^{2}\right) + 9 \cdot {x}^{4}\right) + 2 \cdot \left(y \cdot y\right) \]

   6. pow2 18.8%

      \[\leadsto \left(\left(-\left(y \cdot y\right) \cdot \color{blue}{\left(y \cdot y\right)}\right) + 9 \cdot {x}^{4}\right) + 2 \cdot \left(y \cdot y\right) \]

   7. distribute-rgt-neg-in 18.8%

      \[\leadsto \left(\color{blue}{\left(y \cdot y\right) \cdot \left(-y \cdot y\right)} + 9 \cdot {x}^{4}\right) + 2 \cdot \left(y \cdot y\right) \]

   8. fma-define 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot y, -y \cdot y, 9 \cdot {x}^{4}\right)} + 2 \cdot \left(y \cdot y\right) \]

   9. pow2 100.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{{y}^{2}}, -y \cdot y, 9 \cdot {x}^{4}\right) + 2 \cdot \left(y \cdot y\right) \]

   10. pow2 100.0%

       \[\leadsto \mathsf{fma}\left({y}^{2}, -\color{blue}{{y}^{2}}, 9 \cdot {x}^{4}\right) + 2 \cdot \left(y \cdot y\right) \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{2}, -{y}^{2}, 9 \cdot {x}^{4}\right)} + 2 \cdot \left(y \cdot y\right) \]
5. Add Preprocessing

Alternative 2: 18.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \left(9 \cdot {x}^{4} - {y}^{4}\right) + {\left({y}^{6} \cdot 8\right)}^{0.3333333333333333} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+
  (- (* 9.0 (pow x 4.0)) (pow y 4.0))
  (pow (* (pow y 6.0) 8.0) 0.3333333333333333)))

C

double code(double x, double y) {
	return ((9.0 * pow(x, 4.0)) - pow(y, 4.0)) + pow((pow(y, 6.0) * 8.0), 0.3333333333333333);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((9.0d0 * (x ** 4.0d0)) - (y ** 4.0d0)) + (((y ** 6.0d0) * 8.0d0) ** 0.3333333333333333d0)
end function

Java

public static double code(double x, double y) {
	return ((9.0 * Math.pow(x, 4.0)) - Math.pow(y, 4.0)) + Math.pow((Math.pow(y, 6.0) * 8.0), 0.3333333333333333);
}

Python

def code(x, y):
	return ((9.0 * math.pow(x, 4.0)) - math.pow(y, 4.0)) + math.pow((math.pow(y, 6.0) * 8.0), 0.3333333333333333)

Julia

function code(x, y)
	return Float64(Float64(Float64(9.0 * (x ^ 4.0)) - (y ^ 4.0)) + (Float64((y ^ 6.0) * 8.0) ^ 0.3333333333333333))
end

MATLAB

function tmp = code(x, y)
	tmp = ((9.0 * (x ^ 4.0)) - (y ^ 4.0)) + (((y ^ 6.0) * 8.0) ^ 0.3333333333333333);
end

Wolfram

code[x_, y_] := N[(N[(N[(9.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] - N[Power[y, 4.0], $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[Power[y, 6.0], $MachinePrecision] * 8.0), $MachinePrecision], 0.3333333333333333], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.8%

   \[\left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \left(y \cdot y\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-cbrt-cube 18.7%

      \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \color{blue}{\sqrt[3]{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)}} \]

   2. pow1/3 18.8%

      \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \color{blue}{{\left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)\right)}^{0.3333333333333333}} \]

   3. pow2 18.8%

      \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot {\left(\left(\color{blue}{{y}^{2}} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)\right)}^{0.3333333333333333} \]

   4. pow2 18.8%

      \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot {\left(\left({y}^{2} \cdot \color{blue}{{y}^{2}}\right) \cdot \left(y \cdot y\right)\right)}^{0.3333333333333333} \]

   5. pow-prod-up 18.8%

      \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot {\left(\color{blue}{{y}^{\left(2 + 2\right)}} \cdot \left(y \cdot y\right)\right)}^{0.3333333333333333} \]

   6. metadata-eval 18.8%

      \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot {\left({y}^{\color{blue}{4}} \cdot \left(y \cdot y\right)\right)}^{0.3333333333333333} \]

   7. pow2 18.8%

      \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot {\left({y}^{4} \cdot \color{blue}{{y}^{2}}\right)}^{0.3333333333333333} \]

   8. pow-prod-up 18.8%

      \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot {\color{blue}{\left({y}^{\left(4 + 2\right)}\right)}}^{0.3333333333333333} \]

   9. metadata-eval 18.8%

      \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot {\left({y}^{\color{blue}{6}}\right)}^{0.3333333333333333} \]

4. Applied egg-rr 18.8%

   \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \color{blue}{{\left({y}^{6}\right)}^{0.3333333333333333}} \]
5. Step-by-step derivation

   1. pow-pow 18.8%

      \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \color{blue}{{y}^{\left(6 \cdot 0.3333333333333333\right)}} \]

   2. metadata-eval 18.8%

      \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot {y}^{\color{blue}{2}} \]

   3. add-cbrt-cube 18.7%

      \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + \color{blue}{\sqrt[3]{\left(\left(2 \cdot {y}^{2}\right) \cdot \left(2 \cdot {y}^{2}\right)\right) \cdot \left(2 \cdot {y}^{2}\right)}} \]

   4. pow1/3 18.8%

      \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + \color{blue}{{\left(\left(\left(2 \cdot {y}^{2}\right) \cdot \left(2 \cdot {y}^{2}\right)\right) \cdot \left(2 \cdot {y}^{2}\right)\right)}^{0.3333333333333333}} \]

   5. pow2 18.8%

      \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + {\left(\left(\left(2 \cdot \color{blue}{\left(y \cdot y\right)}\right) \cdot \left(2 \cdot {y}^{2}\right)\right) \cdot \left(2 \cdot {y}^{2}\right)\right)}^{0.3333333333333333} \]

   6. pow2 18.8%

      \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + {\left(\left(\left(2 \cdot \left(y \cdot y\right)\right) \cdot \left(2 \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \cdot \left(2 \cdot {y}^{2}\right)\right)}^{0.3333333333333333} \]

   7. pow2 18.8%

      \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + {\left(\left(\left(2 \cdot \left(y \cdot y\right)\right) \cdot \left(2 \cdot \left(y \cdot y\right)\right)\right) \cdot \left(2 \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)}^{0.3333333333333333} \]

   8. pow3 18.8%

      \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + {\color{blue}{\left({\left(2 \cdot \left(y \cdot y\right)\right)}^{3}\right)}}^{0.3333333333333333} \]

   9. pow2 18.8%

      \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + {\left({\left(2 \cdot \color{blue}{{y}^{2}}\right)}^{3}\right)}^{0.3333333333333333} \]

   10. *-commutative 18.8%

       \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + {\left({\color{blue}{\left({y}^{2} \cdot 2\right)}}^{3}\right)}^{0.3333333333333333} \]

   11. unpow-prod-down 18.8%

       \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + {\color{blue}{\left({\left({y}^{2}\right)}^{3} \cdot {2}^{3}\right)}}^{0.3333333333333333} \]

   12. pow2 18.8%

       \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + {\left({\color{blue}{\left(y \cdot y\right)}}^{3} \cdot {2}^{3}\right)}^{0.3333333333333333} \]

   13. pow-prod-down 18.8%

       \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + {\left(\color{blue}{\left({y}^{3} \cdot {y}^{3}\right)} \cdot {2}^{3}\right)}^{0.3333333333333333} \]

   14. pow-sqr 18.8%

       \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + {\left(\color{blue}{{y}^{\left(2 \cdot 3\right)}} \cdot {2}^{3}\right)}^{0.3333333333333333} \]

   15. metadata-eval 18.8%

       \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + {\left({y}^{\color{blue}{6}} \cdot {2}^{3}\right)}^{0.3333333333333333} \]

   16. metadata-eval 18.8%

       \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + {\left({y}^{6} \cdot \color{blue}{8}\right)}^{0.3333333333333333} \]

6. Applied egg-rr 18.8%

   \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + \color{blue}{{\left({y}^{6} \cdot 8\right)}^{0.3333333333333333}} \]
7. Add Preprocessing

Alternative 3: 18.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot {\left({y}^{6}\right)}^{0.3333333333333333} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+
  (- (* 9.0 (pow x 4.0)) (pow y 4.0))
  (* 2.0 (pow (pow y 6.0) 0.3333333333333333))))

C

double code(double x, double y) {
	return ((9.0 * pow(x, 4.0)) - pow(y, 4.0)) + (2.0 * pow(pow(y, 6.0), 0.3333333333333333));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((9.0d0 * (x ** 4.0d0)) - (y ** 4.0d0)) + (2.0d0 * ((y ** 6.0d0) ** 0.3333333333333333d0))
end function

Java

public static double code(double x, double y) {
	return ((9.0 * Math.pow(x, 4.0)) - Math.pow(y, 4.0)) + (2.0 * Math.pow(Math.pow(y, 6.0), 0.3333333333333333));
}

Python

def code(x, y):
	return ((9.0 * math.pow(x, 4.0)) - math.pow(y, 4.0)) + (2.0 * math.pow(math.pow(y, 6.0), 0.3333333333333333))

Julia

function code(x, y)
	return Float64(Float64(Float64(9.0 * (x ^ 4.0)) - (y ^ 4.0)) + Float64(2.0 * ((y ^ 6.0) ^ 0.3333333333333333)))
end

MATLAB

function tmp = code(x, y)
	tmp = ((9.0 * (x ^ 4.0)) - (y ^ 4.0)) + (2.0 * ((y ^ 6.0) ^ 0.3333333333333333));
end

Wolfram

code[x_, y_] := N[(N[(N[(9.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] - N[Power[y, 4.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[Power[N[Power[y, 6.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.8%

   \[\left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \left(y \cdot y\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-cbrt-cube 18.7%

      \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \color{blue}{\sqrt[3]{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)}} \]

   2. pow1/3 18.8%

      \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \color{blue}{{\left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)\right)}^{0.3333333333333333}} \]

   3. pow2 18.8%

      \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot {\left(\left(\color{blue}{{y}^{2}} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)\right)}^{0.3333333333333333} \]

   4. pow2 18.8%

      \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot {\left(\left({y}^{2} \cdot \color{blue}{{y}^{2}}\right) \cdot \left(y \cdot y\right)\right)}^{0.3333333333333333} \]

   5. pow-prod-up 18.8%

      \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot {\left(\color{blue}{{y}^{\left(2 + 2\right)}} \cdot \left(y \cdot y\right)\right)}^{0.3333333333333333} \]

   6. metadata-eval 18.8%

      \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot {\left({y}^{\color{blue}{4}} \cdot \left(y \cdot y\right)\right)}^{0.3333333333333333} \]

   7. pow2 18.8%

      \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot {\left({y}^{4} \cdot \color{blue}{{y}^{2}}\right)}^{0.3333333333333333} \]

   8. pow-prod-up 18.8%

      \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot {\color{blue}{\left({y}^{\left(4 + 2\right)}\right)}}^{0.3333333333333333} \]

   9. metadata-eval 18.8%

      \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot {\left({y}^{\color{blue}{6}}\right)}^{0.3333333333333333} \]

4. Applied egg-rr 18.8%

   \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \color{blue}{{\left({y}^{6}\right)}^{0.3333333333333333}} \]
5. Add Preprocessing

Alternative 4: 18.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(y \cdot y\right) + \left(9 \cdot {x}^{4} - {y}^{4}\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ (* 2.0 (* y y)) (- (* 9.0 (pow x 4.0)) (pow y 4.0))))

C

double code(double x, double y) {
	return (2.0 * (y * y)) + ((9.0 * pow(x, 4.0)) - pow(y, 4.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 * (y * y)) + ((9.0d0 * (x ** 4.0d0)) - (y ** 4.0d0))
end function

Java

public static double code(double x, double y) {
	return (2.0 * (y * y)) + ((9.0 * Math.pow(x, 4.0)) - Math.pow(y, 4.0));
}

Python

def code(x, y):
	return (2.0 * (y * y)) + ((9.0 * math.pow(x, 4.0)) - math.pow(y, 4.0))

Julia

function code(x, y)
	return Float64(Float64(2.0 * Float64(y * y)) + Float64(Float64(9.0 * (x ^ 4.0)) - (y ^ 4.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = (2.0 * (y * y)) + ((9.0 * (x ^ 4.0)) - (y ^ 4.0));
end

Wolfram

code[x_, y_] := N[(N[(2.0 * N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(N[(9.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] - N[Power[y, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.8%

   \[\left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \left(y \cdot y\right) \]
2. Add Preprocessing

3. Final simplification 18.8%

   \[\leadsto 2 \cdot \left(y \cdot y\right) + \left(9 \cdot {x}^{4} - {y}^{4}\right) \]
4. Add Preprocessing

Alternative 5: 11.1% accurate, 42.6× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (* 2.0 (* y y)))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 2.0 * (y * y)

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(2.0 * N[(y * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.8%

   \[\left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \left(y \cdot y\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0 1.5%

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

   1. neg-mul-1 1.5%

      \[\leadsto \color{blue}{\left(-{y}^{4}\right)} + 2 \cdot \left(y \cdot y\right) \]

5. Simplified 1.5%

   \[\leadsto \color{blue}{\left(-{y}^{4}\right)} + 2 \cdot \left(y \cdot y\right) \]

6. Taylor expanded in y around 0 11.1%

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

   1. unpow2 11.1%

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

8. Applied egg-rr 11.1%

   \[\leadsto 2 \cdot \color{blue}{\left(y \cdot y\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024143 
(FPCore (x y)
  :name "From Rump in a 1983 paper"
  :precision binary64
  :pre (and (== x 10864.0) (== y 18817.0))
  (+ (- (* 9.0 (pow x 4.0)) (pow y 4.0)) (* 2.0 (* y y))))