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

Percentage Accurate: 99.9% → 99.9%

Time: 7.8s

Alternatives: 8

Speedup: 1.8×

• 42.8% 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: 73.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.75 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(y, y + y, y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \mathsf{fma}\left(y, y, y + y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.75e-51) (fma y (+ y y) (* y y)) (fma x x (fma y y (+ y y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.75e-51) {
		tmp = fma(y, (y + y), (y * y));
	} else {
		tmp = fma(x, x, fma(y, y, (y + y)));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.75e-51)
		tmp = fma(y, Float64(y + y), Float64(y * y));
	else
		tmp = fma(x, x, fma(y, y, Float64(y + y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.75e-51], N[(y * N[(y + y), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(y * y + N[(y + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.7499999999999999e-51

   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 99.9%

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

   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 63.2

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

   7. Applied rewrites 63.2%

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

   if 1.7499999999999999e-51 < x

   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. lift-+.f64 N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2 N/A

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

      11. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. flip-+ N/A

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

      7. +-inverses N/A

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

      8. +-inverses N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. +-inverses N/A

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

      15. +-inverses N/A

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

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

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. +-inverses N/A

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

      20. +-inverses N/A

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

      21. flip-+ N/A

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

      22. lift-+.f64 85.6

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

   6. Applied rewrites 85.6%

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

Alternative 3: 69.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-20}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y + y, y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 8.5e-20) (* x x) (fma y (+ y y) (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8.5e-20) {
		tmp = x * x;
	} else {
		tmp = fma(y, (y + y), (y * y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8.5e-20)
		tmp = Float64(x * x);
	else
		tmp = fma(y, Float64(y + y), Float64(y * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8.5e-20], N[(x * x), $MachinePrecision], N[(y * N[(y + y), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.5000000000000005e-20

   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. lift-+.f64 N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2 N/A

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

      11. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

   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 68.2

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

   9. Applied rewrites 68.2%

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

   if 8.5000000000000005e-20 < y

   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 99.9%

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

   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 85.1

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

   7. Applied rewrites 85.1%

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

Alternative 4: 69.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-20}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 8.5e-20) (* x x) (* 3.0 (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8.5e-20) {
		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 <= 8.5d-20) 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 <= 8.5e-20) {
		tmp = x * x;
	} else {
		tmp = 3.0 * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8.5e-20:
		tmp = x * x
	else:
		tmp = 3.0 * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8.5e-20)
		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 <= 8.5e-20)
		tmp = x * x;
	else
		tmp = 3.0 * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8.5e-20], N[(x * x), $MachinePrecision], N[(3.0 * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.5000000000000005e-20

   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. lift-+.f64 N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2 N/A

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

      11. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

   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 68.2

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

   9. Applied rewrites 68.2%

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

   if 8.5000000000000005e-20 < y

   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 85.0

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

   5. Applied rewrites 85.0%

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

Alternative 5: 99.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   5. associate-+l+ N/A

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

   6. lift-*.f64 N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. count-2 N/A

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

   11. lower-*.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-rgt1-in N/A

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

   5. metadata-eval N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. lift-*.f64 N/A

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

   9. lower-*.f64 99.9

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

6. Applied rewrites 99.9%

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

Alternative 6: 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 7: 66.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.65 \cdot 10^{+151}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 3.65e+151) (* x x) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.65e+151) {
		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 <= 3.65d+151) 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 <= 3.65e+151) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.65e+151:
		tmp = x * x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.65e+151)
		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 <= 3.65e+151)
		tmp = x * x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.65e+151], N[(x * x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.65e151

   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. lift-+.f64 N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2 N/A

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

      11. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

   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 63.3

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

   9. Applied rewrites 63.3%

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

   if 3.65e151 < 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. lift-+.f64 N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2 N/A

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

      11. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. flip-+ N/A

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

      7. +-inverses N/A

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

      8. +-inverses N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. +-inverses N/A

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

      15. +-inverses N/A

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

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

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. +-inverses N/A

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

      20. +-inverses N/A

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

      21. flip-+ N/A

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

      22. lift-+.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot y + {y}^{2}} \]
   8. 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 100.0

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in y around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 100.0

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

   12. Applied rewrites 100.0%

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

Alternative 8: 57.3% 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. lift-+.f64 N/A

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

   5. associate-+l+ N/A

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

   6. lift-*.f64 N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. count-2 N/A

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

   11. lower-*.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-rgt1-in N/A

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

   5. metadata-eval N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. lift-*.f64 N/A

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

   9. lower-*.f64 99.9

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

6. Applied rewrites 99.9%

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

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 57.6

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

9. Applied rewrites 57.6%

   \[\leadsto \color{blue}{x \cdot x} \]
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 2024337 
(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)))