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

Percentage Accurate: 99.9% → 99.9%

Time: 5.3s

Alternatives: 6

Speedup: 1.8×

• 43.6% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   5. count-2 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. count-2 N/A

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

   9. lower-fma.f64 N/A

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

   10. count-2 N/A

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

   11. lower-*.f64 100.0

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lower-fma.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. count-2 N/A

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

   5. distribute-lft-out N/A

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

   6. lift-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-fma.f64 N/A

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

   12. lift-*.f64 100.0

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

6. Applied rewrites 100.0%

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

Alternative 2: 82.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{-7}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y + y, y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 1e-7) (* x x) (fma y (+ y y) (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1e-7) {
		tmp = x * x;
	} else {
		tmp = fma(y, (y + y), (y * y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 1e-7)
		tmp = Float64(x * x);
	else
		tmp = fma(y, Float64(y + y), Float64(y * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 1e-7], N[(x * x), $MachinePrecision], N[(y * N[(y + y), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 9.9999999999999995e-8

   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

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

      5. count-2 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. count-2 N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2 N/A

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

      11. lower-*.f64 100.0

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. count-2 N/A

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

      5. distribute-lft-out N/A

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

      6. lift-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 88.9

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

   9. Applied rewrites 88.9%

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

   if 9.9999999999999995e-8 < (*.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 85.7

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

   5. Applied rewrites 85.7%

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

   7. Applied rewrites 85.8%

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

Alternative 3: 82.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{-7}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 1e-7) (* x x) (* 3.0 (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1e-7) {
		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) <= 1d-7) 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) <= 1e-7) {
		tmp = x * x;
	} else {
		tmp = 3.0 * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 1e-7:
		tmp = x * x
	else:
		tmp = 3.0 * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 1e-7)
		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) <= 1e-7)
		tmp = x * x;
	else
		tmp = 3.0 * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 1e-7], N[(x * x), $MachinePrecision], N[(3.0 * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 9.9999999999999995e-8

   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

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

      5. count-2 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. count-2 N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2 N/A

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

      11. lower-*.f64 100.0

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. count-2 N/A

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

      5. distribute-lft-out N/A

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

      6. lift-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 88.9

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

   9. Applied rewrites 88.9%

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

   if 9.9999999999999995e-8 < (*.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 85.7

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

   5. Applied rewrites 85.7%

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

Alternative 4: 75.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2.05 \cdot 10^{+282}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y y) 2.05e+282) (* x x) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 2.05e+282) {
		tmp = x * x;
	} else {
		tmp = 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) <= 2.05d+282) then
        tmp = x * x
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 2.05e+282) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 2.05e+282:
		tmp = x * x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 2.05e+282)
		tmp = Float64(x * x);
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 2.05e+282)
		tmp = x * x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 2.05e+282], N[(x * x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 2.04999999999999997e282

   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

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

      5. count-2 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. count-2 N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2 N/A

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

      11. lower-*.f64 99.9

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. count-2 N/A

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

      5. distribute-lft-out N/A

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

      6. lift-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 71.9

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

   9. Applied rewrites 71.9%

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

   if 2.04999999999999997e282 < (*.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 99.4

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 87.2%

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

   9. Applied rewrites 87.2%

      \[\leadsto \frac{3 \cdot \left(\mathsf{fma}\left(y, y, 0\right) \cdot y\right)}{\color{blue}{y \cdot 2} - y} \]

   11. Applied rewrites 87.1%

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

Alternative 5: 99.9% accurate, 1.8× speedup

Math

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

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(3.0, Float64(y * y), Float64(x * x))
end

Wolfram

code[x_, y_] := N[(3.0 * N[(y * y), $MachinePrecision] + 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. 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. lift-+.f64 N/A

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

   4. associate-+l+ N/A

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

   5. associate-+l+ N/A

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

   6. +-commutative N/A

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

   7. count-2 N/A

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

   8. distribute-lft1-in N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 6: 56.5% 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

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

   5. count-2 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. count-2 N/A

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

   9. lower-fma.f64 N/A

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

   10. count-2 N/A

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

   11. lower-*.f64 100.0

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lower-fma.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. count-2 N/A

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

   5. distribute-lft-out N/A

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

   6. lift-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-fma.f64 N/A

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

   12. lift-*.f64 100.0

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. lower-*.f64 58.1

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

9. Applied rewrites 58.1%

   \[\leadsto \color{blue}{x \cdot x} \]
10. 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 2024294 
(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)))