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

Percentage Accurate: 99.9% → 99.9%

Time: 5.1s

Alternatives: 8

Speedup: 1.8×

• 44.1% 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, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

4. Applied rewrites 100.0%

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

Alternative 2: 89.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5.5 \cdot 10^{-186}:\\ \;\;\;\;\mathsf{fma}\left(y, y + y, y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 + y, y, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 5.5e-186) (fma y (+ y y) (* y y)) (fma (+ 2.0 y) y (* x x))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 5.5e-186) {
		tmp = fma(y, (y + y), (y * y));
	} else {
		tmp = fma((2.0 + y), y, (x * x));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 5.5e-186)
		tmp = fma(y, Float64(y + y), Float64(y * y));
	else
		tmp = fma(Float64(2.0 + y), y, Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 5.5e-186], N[(y * N[(y + y), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + y), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5.5000000000000001e-186

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(y, y + y, \color{blue}{{y}^{2}}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 93.4

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

   7. Applied rewrites 93.4%

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

   if 5.5000000000000001e-186 < (*.f64 x x)

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(y, y + y, \color{blue}{{y}^{2}}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 40.9

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

   7. Applied rewrites 40.9%

      \[\leadsto \mathsf{fma}\left(y, y + y, \color{blue}{y \cdot y}\right) \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

         \[\leadsto y \cdot \color{blue}{\frac{y \cdot y - y \cdot y}{y - y}} + y \cdot y \]

      4. +-inverses N/A

         \[\leadsto y \cdot \frac{y \cdot y - y \cdot y}{\color{blue}{0}} + y \cdot y \]

      5. +-inverses N/A

         \[\leadsto y \cdot \frac{y \cdot y - y \cdot y}{\color{blue}{y \cdot y - y \cdot y}} + y \cdot y \]

      6. associate-*r/ N/A

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

      7. +-inverses N/A

         \[\leadsto \frac{y \cdot \color{blue}{0}}{y \cdot y - y \cdot y} + y \cdot y \]

      8. +-inverses N/A

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

      9. distribute-lft-out-- N/A

         \[\leadsto \frac{\color{blue}{y \cdot y - y \cdot y}}{y \cdot y - y \cdot y} + y \cdot y \]

      10. +-inverses N/A

          \[\leadsto \frac{y \cdot y - y \cdot y}{\color{blue}{0}} + y \cdot y \]

      11. +-inverses N/A

          \[\leadsto \frac{y \cdot y - y \cdot y}{\color{blue}{y - y}} + y \cdot y \]

      12. flip-+ N/A

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

      13. count-2 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 34.1

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

   9. Applied rewrites 34.1%

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

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. associate-+r+ N/A

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

       3. unpow2 N/A

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

       4. distribute-rgt-in N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-+.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 91.1

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

   12. Applied rewrites 91.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(2 + y, y, x \cdot x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 89.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5.5 \cdot 10^{-186}:\\ \;\;\;\;3 \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 + y, y, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 5.5e-186) (* 3.0 (* y y)) (fma (+ 2.0 y) y (* x x))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 5.5e-186) {
		tmp = 3.0 * (y * y);
	} else {
		tmp = fma((2.0 + y), y, (x * x));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 5.5e-186)
		tmp = Float64(3.0 * Float64(y * y));
	else
		tmp = fma(Float64(2.0 + y), y, Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 5.5e-186], N[(3.0 * N[(y * y), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + y), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5.5000000000000001e-186

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 93.2

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

   5. Applied rewrites 93.2%

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

   if 5.5000000000000001e-186 < (*.f64 x x)

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(y, y + y, \color{blue}{{y}^{2}}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 40.9

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

   7. Applied rewrites 40.9%

      \[\leadsto \mathsf{fma}\left(y, y + y, \color{blue}{y \cdot y}\right) \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

         \[\leadsto y \cdot \color{blue}{\frac{y \cdot y - y \cdot y}{y - y}} + y \cdot y \]

      4. +-inverses N/A

         \[\leadsto y \cdot \frac{y \cdot y - y \cdot y}{\color{blue}{0}} + y \cdot y \]

      5. +-inverses N/A

         \[\leadsto y \cdot \frac{y \cdot y - y \cdot y}{\color{blue}{y \cdot y - y \cdot y}} + y \cdot y \]

      6. associate-*r/ N/A

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

      7. +-inverses N/A

         \[\leadsto \frac{y \cdot \color{blue}{0}}{y \cdot y - y \cdot y} + y \cdot y \]

      8. +-inverses N/A

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

      9. distribute-lft-out-- N/A

         \[\leadsto \frac{\color{blue}{y \cdot y - y \cdot y}}{y \cdot y - y \cdot y} + y \cdot y \]

      10. +-inverses N/A

          \[\leadsto \frac{y \cdot y - y \cdot y}{\color{blue}{0}} + y \cdot y \]

      11. +-inverses N/A

          \[\leadsto \frac{y \cdot y - y \cdot y}{\color{blue}{y - y}} + y \cdot y \]

      12. flip-+ N/A

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

      13. count-2 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 34.1

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

   9. Applied rewrites 34.1%

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

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. associate-+r+ N/A

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

       3. unpow2 N/A

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

       4. distribute-rgt-in N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-+.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 91.1

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

   12. Applied rewrites 91.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(2 + y, y, x \cdot x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 82.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 50000000000:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 50000000000.0) (* x x) (* 3.0 (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 50000000000.0) {
		tmp = x * x;
	} else {
		tmp = 3.0 * (y * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * y) <= 50000000000.0d0) then
        tmp = x * x
    else
        tmp = 3.0d0 * (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 50000000000.0) {
		tmp = x * x;
	} else {
		tmp = 3.0 * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 50000000000.0:
		tmp = x * x
	else:
		tmp = 3.0 * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 50000000000.0)
		tmp = Float64(x * x);
	else
		tmp = Float64(3.0 * Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 50000000000.0)
		tmp = x * x;
	else
		tmp = 3.0 * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 50000000000.0], N[(x * x), $MachinePrecision], N[(3.0 * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 5e10

   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. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(y, y + y, \color{blue}{{y}^{2}}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 29.7

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

   7. Applied rewrites 29.7%

      \[\leadsto \mathsf{fma}\left(y, y + y, \color{blue}{y \cdot y}\right) \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

         \[\leadsto y \cdot \color{blue}{\frac{y \cdot y - y \cdot y}{y - y}} + y \cdot y \]

      4. +-inverses N/A

         \[\leadsto y \cdot \frac{y \cdot y - y \cdot y}{\color{blue}{0}} + y \cdot y \]

      5. +-inverses N/A

         \[\leadsto y \cdot \frac{y \cdot y - y \cdot y}{\color{blue}{y \cdot y - y \cdot y}} + y \cdot y \]

      6. associate-*r/ N/A

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

      7. +-inverses N/A

         \[\leadsto \frac{y \cdot \color{blue}{0}}{y \cdot y - y \cdot y} + y \cdot y \]

      8. +-inverses N/A

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

      9. distribute-lft-out-- N/A

         \[\leadsto \frac{\color{blue}{y \cdot y - y \cdot y}}{y \cdot y - y \cdot y} + y \cdot y \]

      10. +-inverses N/A

          \[\leadsto \frac{y \cdot y - y \cdot y}{\color{blue}{0}} + y \cdot y \]

      11. +-inverses N/A

          \[\leadsto \frac{y \cdot y - y \cdot y}{\color{blue}{y - y}} + y \cdot y \]

      12. flip-+ N/A

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

      13. count-2 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 3.9

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

   9. Applied rewrites 3.9%

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

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{{x}^{2}} \]
   11. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \color{blue}{x \cdot x} \]

       2. lower-*.f64 81.2

          \[\leadsto \color{blue}{x \cdot x} \]

   12. Applied rewrites 81.2%

       \[\leadsto \color{blue}{x \cdot x} \]

   if 5e10 < (*.f64 y y)

   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

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 85.9

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

   5. Applied rewrites 85.9%

      \[\leadsto \color{blue}{3 \cdot \left(y \cdot y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 75.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{+299}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(2 + y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 1e+299) (* x x) (* (+ 2.0 y) y)))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1e+299) {
		tmp = x * x;
	} else {
		tmp = (2.0 + y) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * y) <= 1d+299) then
        tmp = x * x
    else
        tmp = (2.0d0 + y) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1e+299) {
		tmp = x * x;
	} else {
		tmp = (2.0 + y) * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 1e+299:
		tmp = x * x
	else:
		tmp = (2.0 + y) * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 1e+299)
		tmp = Float64(x * x);
	else
		tmp = Float64(Float64(2.0 + y) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 1e+299)
		tmp = x * x;
	else
		tmp = (2.0 + y) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 1e+299], N[(x * x), $MachinePrecision], N[(N[(2.0 + y), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1.0000000000000001e299

   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. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(y, y + y, \color{blue}{{y}^{2}}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 41.7

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

   7. Applied rewrites 41.7%

      \[\leadsto \mathsf{fma}\left(y, y + y, \color{blue}{y \cdot y}\right) \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

         \[\leadsto y \cdot \color{blue}{\frac{y \cdot y - y \cdot y}{y - y}} + y \cdot y \]

      4. +-inverses N/A

         \[\leadsto y \cdot \frac{y \cdot y - y \cdot y}{\color{blue}{0}} + y \cdot y \]

      5. +-inverses N/A

         \[\leadsto y \cdot \frac{y \cdot y - y \cdot y}{\color{blue}{y \cdot y - y \cdot y}} + y \cdot y \]

      6. associate-*r/ N/A

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

      7. +-inverses N/A

         \[\leadsto \frac{y \cdot \color{blue}{0}}{y \cdot y - y \cdot y} + y \cdot y \]

      8. +-inverses N/A

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

      9. distribute-lft-out-- N/A

         \[\leadsto \frac{\color{blue}{y \cdot y - y \cdot y}}{y \cdot y - y \cdot y} + y \cdot y \]

      10. +-inverses N/A

          \[\leadsto \frac{y \cdot y - y \cdot y}{\color{blue}{0}} + y \cdot y \]

      11. +-inverses N/A

          \[\leadsto \frac{y \cdot y - y \cdot y}{\color{blue}{y - y}} + y \cdot y \]

      12. flip-+ N/A

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

      13. count-2 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 7.0

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

   9. Applied rewrites 7.0%

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

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{{x}^{2}} \]
   11. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \color{blue}{x \cdot x} \]

       2. lower-*.f64 66.8

          \[\leadsto \color{blue}{x \cdot x} \]

   12. Applied rewrites 66.8%

       \[\leadsto \color{blue}{x \cdot x} \]

   if 1.0000000000000001e299 < (*.f64 y y)

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(y, y + y, \color{blue}{{y}^{2}}\right) \]
   6. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(y, y + y, \color{blue}{y \cdot y}\right) \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

         \[\leadsto y \cdot \color{blue}{\frac{y \cdot y - y \cdot y}{y - y}} + y \cdot y \]

      4. +-inverses N/A

         \[\leadsto y \cdot \frac{y \cdot y - y \cdot y}{\color{blue}{0}} + y \cdot y \]

      5. +-inverses N/A

         \[\leadsto y \cdot \frac{y \cdot y - y \cdot y}{\color{blue}{y \cdot y - y \cdot y}} + y \cdot y \]

      6. associate-*r/ N/A

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

      7. +-inverses N/A

         \[\leadsto \frac{y \cdot \color{blue}{0}}{y \cdot y - y \cdot y} + y \cdot y \]

      8. +-inverses N/A

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

      9. distribute-lft-out-- N/A

         \[\leadsto \frac{\color{blue}{y \cdot y - y \cdot y}}{y \cdot y - y \cdot y} + y \cdot y \]

      10. +-inverses N/A

          \[\leadsto \frac{y \cdot y - y \cdot y}{\color{blue}{0}} + y \cdot y \]

      11. +-inverses N/A

          \[\leadsto \frac{y \cdot y - y \cdot y}{\color{blue}{y - y}} + y \cdot y \]

      12. flip-+ N/A

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

      13. count-2 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 96.7

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

   9. Applied rewrites 96.7%

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

   10. Taylor expanded in x around 0

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

       1. unpow2 N/A

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

       2. distribute-rgt-in N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-+.f64 96.7

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

   12. Applied rewrites 96.7%

       \[\leadsto \color{blue}{\left(2 + y\right) \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. distribute-lft1-in N/A

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

   4. metadata-eval N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 99.9

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

5. Applied rewrites 99.9%

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

7. Applied rewrites 99.9%

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

Alternative 7: 99.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. distribute-lft1-in N/A

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

   4. metadata-eval N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 99.9

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

5. Applied rewrites 99.9%

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

Alternative 8: 57.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ x \cdot x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(x * x), $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. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(y, y + y, \color{blue}{{y}^{2}}\right) \]
6. Step-by-step derivation

   1. unpow2 N/A

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

   2. lower-*.f64 58.5

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

7. Applied rewrites 58.5%

   \[\leadsto \mathsf{fma}\left(y, y + y, \color{blue}{y \cdot y}\right) \]
8. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. flip-+ N/A

      \[\leadsto y \cdot \color{blue}{\frac{y \cdot y - y \cdot y}{y - y}} + y \cdot y \]

   4. +-inverses N/A

      \[\leadsto y \cdot \frac{y \cdot y - y \cdot y}{\color{blue}{0}} + y \cdot y \]

   5. +-inverses N/A

      \[\leadsto y \cdot \frac{y \cdot y - y \cdot y}{\color{blue}{y \cdot y - y \cdot y}} + y \cdot y \]

   6. associate-*r/ N/A

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

   7. +-inverses N/A

      \[\leadsto \frac{y \cdot \color{blue}{0}}{y \cdot y - y \cdot y} + y \cdot y \]

   8. +-inverses N/A

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

   9. distribute-lft-out-- N/A

      \[\leadsto \frac{\color{blue}{y \cdot y - y \cdot y}}{y \cdot y - y \cdot y} + y \cdot y \]

   10. +-inverses N/A

       \[\leadsto \frac{y \cdot y - y \cdot y}{\color{blue}{0}} + y \cdot y \]

   11. +-inverses N/A

       \[\leadsto \frac{y \cdot y - y \cdot y}{\color{blue}{y - y}} + y \cdot y \]

   12. flip-+ N/A

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

   13. count-2 N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 32.9

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

9. Applied rewrites 32.9%

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

10. Taylor expanded in x around inf

    \[\leadsto \color{blue}{{x}^{2}} \]
11. Step-by-step derivation

    1. unpow2 N/A

       \[\leadsto \color{blue}{x \cdot x} \]

    2. lower-*.f64 54.6

       \[\leadsto \color{blue}{x \cdot x} \]

12. Applied rewrites 54.6%

    \[\leadsto \color{blue}{x \cdot x} \]
13. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.5× 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 2024320 
(FPCore (x y)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, E"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* x x) (* y (+ y (+ y y)))))

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