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

Percentage Accurate: 99.9% → 99.9%

Time: 4.9s

Alternatives: 4

Speedup: 1.0×

• 44.7% 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, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[Power[y, 2.0], $MachinePrecision] * 3.0 + 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. Taylor expanded in x around 0 99.9%

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

   1. *-commutative 99.9%

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

   2. +-commutative 99.9%

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

   3. associate-+r+ 99.9%

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

   4. *-rgt-identity 99.9%

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

   5. distribute-lft-out 99.9%

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

   6. metadata-eval 99.9%

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

   7. fma-define 99.9%

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

5. Simplified 99.9%

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

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Final simplification 99.9%

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

Alternative 3: 58.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(y * y), $MachinePrecision] + N[(y * N[(y * 2.0), $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. Taylor expanded in x around 0 99.9%

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

   1. unpow2 99.9%

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

   2. associate-*r* 99.9%

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

   3. count-2 99.9%

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

   4. *-commutative 99.9%

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

   5. distribute-lft-in 99.9%

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

   6. unpow2 99.9%

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

   7. unpow2 99.9%

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

   8. associate-+r+ 99.9%

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

   9. +-commutative 99.9%

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

   10. unpow2 99.9%

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

   11. rem-square-sqrt 99.9%

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

   12. unpow2 99.9%

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

   13. unpow2 99.9%

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

   14. hypot-undefine 99.9%

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

   15. unpow2 99.9%

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

   16. unpow2 99.9%

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

   17. hypot-undefine 99.9%

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

   18. rem-square-sqrt 99.8%

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

5. Simplified 99.9%

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

6. Taylor expanded in y around inf 56.1%

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

   1. unpow-prod-down 56.0%

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

   2. pow2 56.0%

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

   3. pow2 56.0%

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

   4. rem-square-sqrt 56.2%

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

   5. associate-*l* 56.2%

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

8. Applied egg-rr 56.2%

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

9. Final simplification 56.2%

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

Alternative 4: 58.2% 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. Add Preprocessing

3. Taylor expanded in x around 0 99.9%

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

   1. unpow2 99.9%

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

   2. associate-*r* 99.9%

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

   3. count-2 99.9%

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

   4. *-commutative 99.9%

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

   5. distribute-lft-in 99.9%

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

   6. unpow2 99.9%

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

   7. unpow2 99.9%

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

   8. associate-+r+ 99.9%

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

   9. +-commutative 99.9%

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

   10. unpow2 99.9%

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

   11. rem-square-sqrt 99.9%

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

   12. unpow2 99.9%

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

   13. unpow2 99.9%

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

   14. hypot-undefine 99.9%

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

   15. unpow2 99.9%

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

   16. unpow2 99.9%

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

   17. hypot-undefine 99.9%

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

   18. rem-square-sqrt 99.8%

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

5. Simplified 99.9%

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

6. Taylor expanded in y around 0 99.9%

   \[\leadsto \color{blue}{3 \cdot {y}^{2} + {x}^{2}} \]
7. Step-by-step derivation

   1. +-commutative 99.9%

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

   2. unpow2 99.9%

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

   3. *-commutative 99.9%

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

   4. unpow2 99.9%

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

   5. rem-square-sqrt 99.7%

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

   6. swap-sqr 99.7%

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

   7. rem-square-sqrt 99.7%

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

   8. hypot-undefine 99.7%

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

   9. hypot-undefine 99.7%

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

   10. unpow2 99.7%

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

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 56.0%

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

    1. pow2 56.0%

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

    2. *-commutative 56.0%

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

    3. associate-*r* 56.0%

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

    4. sqrt-pow2 56.2%

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

    5. metadata-eval 56.2%

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

    6. metadata-eval 56.2%

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

11. Applied egg-rr 56.2%

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

12. Final simplification 56.2%

    \[\leadsto y \cdot \left(y \cdot 3\right) \]
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 2024096 
(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)))