Linear.Quaternion:$c/ from linear-1.19.1.3, E

Percentage Accurate: 99.9% → 99.9%

Time: 5.8s

Alternatives: 4

Speedup: 1.0×

• 44.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y \]
2. Step-by-step derivation

   1. +-commutative 99.9%

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

   2. sqr-neg 99.9%

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

   3. +-commutative 99.9%

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

   4. sqr-neg 99.9%

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

   5. +-commutative 99.9%

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

   6. fma-define 100.0%

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

   7. sqr-neg 100.0%

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

   8. sqr-neg 100.0%

      \[\leadsto \mathsf{fma}\left(y, y, \left(x \cdot x + \color{blue}{y \cdot y}\right) + y \cdot y\right) \]

   9. associate-+l+ 100.0%

      \[\leadsto \mathsf{fma}\left(y, y, \color{blue}{x \cdot x + \left(y \cdot y + y \cdot y\right)}\right) \]

   10. fma-define 100.0%

       \[\leadsto \mathsf{fma}\left(y, y, \color{blue}{\mathsf{fma}\left(x, x, y \cdot y + y \cdot y\right)}\right) \]

   11. count-2 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(y, y, \mathsf{fma}\left(x, x, 2 \cdot \left(y \cdot y\right)\right)\right) \]
6. Add Preprocessing

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y \]
2. Add Preprocessing

3. Final simplification 99.9%

   \[\leadsto y \cdot y + \left(y \cdot y + \left(y \cdot y + x \cdot x\right)\right) \]
4. Add Preprocessing

Alternative 3: 58.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ y \cdot \left(y \cdot 3\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* y (* y 3.0)))

C

double code(double x, double y) {
	return y * (y * 3.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y * (y * 3.0d0)
end function

Java

public static double code(double x, double y) {
	return y * (y * 3.0);
}

Python

def code(x, y):
	return y * (y * 3.0)

Julia

function code(x, y)
	return Float64(y * Float64(y * 3.0))
end

MATLAB

function tmp = code(x, y)
	tmp = y * (y * 3.0);
end

Wolfram

code[x_, y_] := N[(y * N[(y * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. add-cbrt-cube 73.1%

      \[\leadsto \left(\color{blue}{\sqrt[3]{\left(\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\right) \cdot \left(x \cdot x + y \cdot y\right)}} + y \cdot y\right) + y \cdot y \]

   2. pow1/3 72.0%

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

   3. pow3 72.0%

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

   4. add-sqr-sqrt 72.0%

      \[\leadsto \left({\left({\color{blue}{\left(\sqrt{x \cdot x + y \cdot y} \cdot \sqrt{x \cdot x + y \cdot y}\right)}}^{3}\right)}^{0.3333333333333333} + y \cdot y\right) + y \cdot y \]

   5. pow2 72.0%

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

   6. hypot-define 72.0%

      \[\leadsto \left({\left({\left({\color{blue}{\left(\mathsf{hypot}\left(x, y\right)\right)}}^{2}\right)}^{3}\right)}^{0.3333333333333333} + y \cdot y\right) + y \cdot y \]

4. Applied egg-rr 72.0%

   \[\leadsto \left(\color{blue}{{\left({\left({\left(\mathsf{hypot}\left(x, y\right)\right)}^{2}\right)}^{3}\right)}^{0.3333333333333333}} + y \cdot y\right) + y \cdot y \]

5. Taylor expanded in x around 0 52.7%

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

   1. pow2 52.7%

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

   2. metadata-eval 52.7%

      \[\leadsto \left({\left({y}^{6}\right)}^{0.3333333333333333} + y \cdot y\right) + {y}^{\color{blue}{\left(6 \cdot 0.3333333333333333\right)}} \]

   3. pow-pow 52.5%

      \[\leadsto \left({\left({y}^{6}\right)}^{0.3333333333333333} + y \cdot y\right) + \color{blue}{{\left({y}^{6}\right)}^{0.3333333333333333}} \]

   4. +-commutative 52.5%

      \[\leadsto \color{blue}{{\left({y}^{6}\right)}^{0.3333333333333333} + \left({\left({y}^{6}\right)}^{0.3333333333333333} + y \cdot y\right)} \]

   5. pow2 52.5%

      \[\leadsto {\left({y}^{6}\right)}^{0.3333333333333333} + \left({\left({y}^{6}\right)}^{0.3333333333333333} + \color{blue}{{y}^{2}}\right) \]

   6. metadata-eval 52.5%

      \[\leadsto {\left({y}^{6}\right)}^{0.3333333333333333} + \left({\left({y}^{6}\right)}^{0.3333333333333333} + {y}^{\color{blue}{\left(6 \cdot 0.3333333333333333\right)}}\right) \]

   7. pow-pow 51.8%

      \[\leadsto {\left({y}^{6}\right)}^{0.3333333333333333} + \left({\left({y}^{6}\right)}^{0.3333333333333333} + \color{blue}{{\left({y}^{6}\right)}^{0.3333333333333333}}\right) \]

   8. count-2 51.8%

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

   9. metadata-eval 51.8%

      \[\leadsto {\left({y}^{6}\right)}^{0.3333333333333333} + \color{blue}{{2}^{1}} \cdot {\left({y}^{6}\right)}^{0.3333333333333333} \]

   10. metadata-eval 51.8%

       \[\leadsto {\left({y}^{6}\right)}^{0.3333333333333333} + {2}^{\color{blue}{\left(\frac{2}{2}\right)}} \cdot {\left({y}^{6}\right)}^{0.3333333333333333} \]

   11. sqrt-pow2 51.8%

       \[\leadsto {\left({y}^{6}\right)}^{0.3333333333333333} + \color{blue}{{\left(\sqrt{2}\right)}^{2}} \cdot {\left({y}^{6}\right)}^{0.3333333333333333} \]

   12. distribute-rgt1-in 51.8%

       \[\leadsto \color{blue}{\left({\left(\sqrt{2}\right)}^{2} + 1\right) \cdot {\left({y}^{6}\right)}^{0.3333333333333333}} \]

   13. +-commutative 51.8%

       \[\leadsto \color{blue}{\left(1 + {\left(\sqrt{2}\right)}^{2}\right)} \cdot {\left({y}^{6}\right)}^{0.3333333333333333} \]

   14. pow-pow 58.6%

       \[\leadsto \left(1 + {\left(\sqrt{2}\right)}^{2}\right) \cdot \color{blue}{{y}^{\left(6 \cdot 0.3333333333333333\right)}} \]

   15. metadata-eval 58.6%

       \[\leadsto \left(1 + {\left(\sqrt{2}\right)}^{2}\right) \cdot {y}^{\color{blue}{2}} \]

   16. pow2 58.6%

       \[\leadsto \left(1 + {\left(\sqrt{2}\right)}^{2}\right) \cdot \color{blue}{\left(y \cdot y\right)} \]

   17. associate-*r* 58.7%

       \[\leadsto \color{blue}{\left(\left(1 + {\left(\sqrt{2}\right)}^{2}\right) \cdot y\right) \cdot y} \]

   18. sqrt-pow2 58.9%

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

   19. metadata-eval 58.9%

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

   20. metadata-eval 58.9%

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

   21. metadata-eval 58.9%

       \[\leadsto \left(\color{blue}{3} \cdot y\right) \cdot y \]

7. Applied egg-rr 58.9%

   \[\leadsto \color{blue}{\left(3 \cdot y\right) \cdot y} \]

8. Final simplification 58.9%

   \[\leadsto y \cdot \left(y \cdot 3\right) \]
9. Add Preprocessing

Alternative 4: 1.2% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return -2.0

Julia

function code(x, y)
	return -2.0
end

MATLAB

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

Wolfram

code[x_, y_] := -2.0

Derivation

1. Initial program 99.9%

   \[\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y \]
2. Step-by-step derivation

   1. +-commutative 99.9%

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

   2. sqr-neg 99.9%

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

   3. +-commutative 99.9%

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

   4. sqr-neg 99.9%

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

   5. +-commutative 99.9%

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

   6. fma-define 100.0%

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

   7. sqr-neg 100.0%

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

   8. sqr-neg 100.0%

      \[\leadsto \mathsf{fma}\left(y, y, \left(x \cdot x + \color{blue}{y \cdot y}\right) + y \cdot y\right) \]

   9. associate-+l+ 100.0%

      \[\leadsto \mathsf{fma}\left(y, y, \color{blue}{x \cdot x + \left(y \cdot y + y \cdot y\right)}\right) \]

   10. fma-define 100.0%

       \[\leadsto \mathsf{fma}\left(y, y, \color{blue}{\mathsf{fma}\left(x, x, y \cdot y + y \cdot y\right)}\right) \]

   11. count-2 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, y, \mathsf{fma}\left(x, x, 2 \cdot \left(y \cdot y\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 99.9%

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

   2. pow2 99.9%

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

   3. *-commutative 99.9%

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

   4. sqrt-prod 99.8%

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

   5. sqrt-prod 49.9%

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

   6. add-sqr-sqrt 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in y around inf 58.6%

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

   1. metadata-eval 58.6%

      \[\leadsto {y}^{\color{blue}{\left(6 \cdot 0.3333333333333333\right)}} \cdot \left(1 + {\left(\sqrt{2}\right)}^{2}\right) \]

   2. pow-pow 51.8%

      \[\leadsto \color{blue}{{\left({y}^{6}\right)}^{0.3333333333333333}} \cdot \left(1 + {\left(\sqrt{2}\right)}^{2}\right) \]

   3. distribute-rgt-in 51.8%

      \[\leadsto \color{blue}{1 \cdot {\left({y}^{6}\right)}^{0.3333333333333333} + {\left(\sqrt{2}\right)}^{2} \cdot {\left({y}^{6}\right)}^{0.3333333333333333}} \]

   4. *-un-lft-identity 51.8%

      \[\leadsto \color{blue}{{\left({y}^{6}\right)}^{0.3333333333333333}} + {\left(\sqrt{2}\right)}^{2} \cdot {\left({y}^{6}\right)}^{0.3333333333333333} \]

   5. sqrt-pow2 51.8%

      \[\leadsto {\left({y}^{6}\right)}^{0.3333333333333333} + \color{blue}{{2}^{\left(\frac{2}{2}\right)}} \cdot {\left({y}^{6}\right)}^{0.3333333333333333} \]

   6. metadata-eval 51.8%

      \[\leadsto {\left({y}^{6}\right)}^{0.3333333333333333} + {2}^{\color{blue}{1}} \cdot {\left({y}^{6}\right)}^{0.3333333333333333} \]

   7. metadata-eval 51.8%

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

   8. count-2 51.8%

      \[\leadsto {\left({y}^{6}\right)}^{0.3333333333333333} + \color{blue}{\left({\left({y}^{6}\right)}^{0.3333333333333333} + {\left({y}^{6}\right)}^{0.3333333333333333}\right)} \]

   9. pow-pow 52.5%

      \[\leadsto {\left({y}^{6}\right)}^{0.3333333333333333} + \left({\left({y}^{6}\right)}^{0.3333333333333333} + \color{blue}{{y}^{\left(6 \cdot 0.3333333333333333\right)}}\right) \]

   10. metadata-eval 52.5%

       \[\leadsto {\left({y}^{6}\right)}^{0.3333333333333333} + \left({\left({y}^{6}\right)}^{0.3333333333333333} + {y}^{\color{blue}{2}}\right) \]

   11. pow2 52.5%

       \[\leadsto {\left({y}^{6}\right)}^{0.3333333333333333} + \left({\left({y}^{6}\right)}^{0.3333333333333333} + \color{blue}{y \cdot y}\right) \]

   12. +-commutative 52.5%

       \[\leadsto \color{blue}{\left({\left({y}^{6}\right)}^{0.3333333333333333} + y \cdot y\right) + {\left({y}^{6}\right)}^{0.3333333333333333}} \]

   13. pow2 52.5%

       \[\leadsto \left({\left({y}^{6}\right)}^{0.3333333333333333} + \color{blue}{{y}^{2}}\right) + {\left({y}^{6}\right)}^{0.3333333333333333} \]

   14. metadata-eval 52.5%

       \[\leadsto \left({\left({y}^{6}\right)}^{0.3333333333333333} + {y}^{\color{blue}{\left(6 \cdot 0.3333333333333333\right)}}\right) + {\left({y}^{6}\right)}^{0.3333333333333333} \]

   15. pow-pow 51.8%

       \[\leadsto \left({\left({y}^{6}\right)}^{0.3333333333333333} + \color{blue}{{\left({y}^{6}\right)}^{0.3333333333333333}}\right) + {\left({y}^{6}\right)}^{0.3333333333333333} \]

   16. count-2 51.8%

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

   17. metadata-eval 51.8%

       \[\leadsto \color{blue}{{2}^{1}} \cdot {\left({y}^{6}\right)}^{0.3333333333333333} + {\left({y}^{6}\right)}^{0.3333333333333333} \]

   18. metadata-eval 51.8%

       \[\leadsto {2}^{\color{blue}{\left(\frac{2}{2}\right)}} \cdot {\left({y}^{6}\right)}^{0.3333333333333333} + {\left({y}^{6}\right)}^{0.3333333333333333} \]

   19. sqrt-pow2 51.8%

       \[\leadsto \color{blue}{{\left(\sqrt{2}\right)}^{2}} \cdot {\left({y}^{6}\right)}^{0.3333333333333333} + {\left({y}^{6}\right)}^{0.3333333333333333} \]

9. Applied egg-rr 58.9%

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

10. Taylor expanded in y around 0 58.8%

    \[\leadsto \color{blue}{3 \cdot {y}^{2}} \]

12. Simplified 1.2%

    \[\leadsto \color{blue}{-2} \]

13. Final simplification 1.2%

    \[\leadsto -2 \]
14. Add Preprocessing

Developer target: 99.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024071 
(FPCore (x y)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, E"
  :precision binary64

  :alt
  (+ (* x x) (* y (+ y (+ y y))))

  (+ (+ (+ (* x x) (* y y)) (* y y)) (* y y)))