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

Percentage Accurate: 99.9% → 99.9%

Time: 6.9s

Alternatives: 5

Speedup: 1.0×

• 44.4% 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 99.9%

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

   7. sqr-neg 99.9%

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

   8. sqr-neg 99.9%

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

   9. associate-+l+ 99.9%

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

   10. fma-define 99.9%

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

   11. count-2 99.9%

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

3. Simplified 99.9%

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

Alternative 2: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y \cdot 3, y, {x}^{2}\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma (* y 3.0) y (pow x 2.0)))

C

double code(double x, double y) {
	return fma((y * 3.0), y, pow(x, 2.0));
}

Julia

function code(x, y)
	return fma(Float64(y * 3.0), y, (x ^ 2.0))
end

Wolfram

code[x_, y_] := N[(N[(y * 3.0), $MachinePrecision] * y + N[Power[x, 2.0], $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. associate-+l+ 99.9%

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

   2. +-commutative 99.9%

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

   3. count-2 99.9%

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

   4. add-sqr-sqrt 99.8%

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

   5. fma-define 99.8%

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

   6. *-commutative 99.8%

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

   7. sqrt-prod 99.8%

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

   8. sqrt-prod 47.5%

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

   9. add-sqr-sqrt 72.7%

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

   10. *-commutative 72.7%

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

   11. sqrt-prod 72.6%

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

   12. sqrt-prod 47.5%

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

   13. add-sqr-sqrt 99.8%

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

   14. pow2 99.8%

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

4. Applied egg-rr 99.8%

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

6. Simplified 99.8%

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

   1. unpow-prod-down 99.7%

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

   2. pow2 99.7%

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

   3. rem-square-sqrt 99.9%

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

   4. pow2 99.9%

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

   5. associate-*l* 99.9%

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

   6. *-commutative 99.9%

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

8. Applied egg-rr 99.9%

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

   1. pow2 99.9%

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

   2. +-commutative 99.9%

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

   3. associate-+r+ 99.9%

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

   4. *-commutative 99.9%

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

   5. associate-*r* 99.9%

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

   6. pow2 99.9%

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

   7. distribute-rgt1-in 99.9%

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

   8. metadata-eval 99.9%

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

   9. metadata-eval 99.9%

      \[\leadsto \color{blue}{{3}^{1}} \cdot {y}^{2} + {x}^{2} \]

   10. metadata-eval 99.9%

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

   11. sqrt-pow2 99.6%

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

   12. pow2 99.6%

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

   13. associate-*r* 99.6%

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

   14. fma-define 99.6%

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

   15. sqrt-pow2 99.9%

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

   16. metadata-eval 99.9%

       \[\leadsto \mathsf{fma}\left({3}^{\color{blue}{1}} \cdot y, y, {x}^{2}\right) \]

   17. metadata-eval 99.9%

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

10. Applied egg-rr 99.9%

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

11. Final simplification 99.9%

    \[\leadsto \mathsf{fma}\left(y \cdot 3, y, {x}^{2}\right) \]
12. Add Preprocessing

Alternative 3: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(x * x + N[(N[(y * y), $MachinePrecision] * 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. Step-by-step derivation

   1. associate-+l+ 99.9%

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

   2. associate-+l+ 99.9%

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

   3. fma-define 99.9%

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

   4. *-lft-identity 99.9%

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{1 \cdot \left(y \cdot y\right)} + \left(y \cdot y + y \cdot y\right)\right) \]

   5. metadata-eval 99.9%

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

   6. count-2 99.9%

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

   7. distribute-rgt-out 99.9%

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

   8. metadata-eval 99.9%

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

   9. metadata-eval 99.9%

      \[\leadsto \mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot \color{blue}{3}\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot 3\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 4: 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 5: 57.2% 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. Step-by-step derivation

   1. associate-+l+ 99.9%

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

   2. associate-+l+ 99.9%

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

   3. fma-define 99.9%

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

   4. *-lft-identity 99.9%

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{1 \cdot \left(y \cdot y\right)} + \left(y \cdot y + y \cdot y\right)\right) \]

   5. metadata-eval 99.9%

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

   6. count-2 99.9%

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

   7. distribute-rgt-out 99.9%

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

   8. metadata-eval 99.9%

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

   9. metadata-eval 99.9%

      \[\leadsto \mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot \color{blue}{3}\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot 3\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 99.7%

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

   2. fma-undefine 99.7%

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

   3. add-sqr-sqrt 99.7%

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

   4. hypot-define 99.7%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(y \cdot y\right) \cdot 3}\right)} \cdot \sqrt{\mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot 3\right)} \]

   5. sqrt-prod 99.8%

      \[\leadsto \mathsf{hypot}\left(x, \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{3}}\right) \cdot \sqrt{\mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot 3\right)} \]

   6. sqrt-prod 47.5%

      \[\leadsto \mathsf{hypot}\left(x, \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{3}\right) \cdot \sqrt{\mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot 3\right)} \]

   7. add-sqr-sqrt 99.8%

      \[\leadsto \mathsf{hypot}\left(x, \color{blue}{y} \cdot \sqrt{3}\right) \cdot \sqrt{\mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot 3\right)} \]

   8. fma-undefine 99.8%

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

   9. add-sqr-sqrt 99.8%

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

   10. hypot-define 99.8%

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

   11. sqrt-prod 99.7%

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

   12. sqrt-prod 47.5%

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

   13. add-sqr-sqrt 99.7%

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

6. Applied egg-rr 99.7%

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

8. Simplified 99.7%

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

9. Taylor expanded in x around 0 57.1%

   \[\leadsto {\color{blue}{\left(y \cdot \sqrt{3}\right)}}^{2} \]
10. Step-by-step derivation

    1. *-commutative 57.1%

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

    2. unpow-prod-down 57.0%

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

    3. pow2 57.0%

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

    4. associate-*r* 57.0%

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

    5. sqrt-pow2 57.3%

       \[\leadsto \left(\color{blue}{{3}^{\left(\frac{2}{2}\right)}} \cdot y\right) \cdot y \]

    6. metadata-eval 57.3%

       \[\leadsto \left({3}^{\color{blue}{1}} \cdot y\right) \cdot y \]

    7. metadata-eval 57.3%

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

11. Applied egg-rr 57.3%

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

12. Final simplification 57.3%

    \[\leadsto y \cdot \left(y \cdot 3\right) \]
13. 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 2024088 
(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)))