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

Percentage Accurate: 99.9% → 100.0%

Time: 8.7s

Alternatives: 8

Speedup: 1.8×

• 43.1% 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: 100.0% 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(Float64(y + y), y, fma(y, y, Float64(x * x)))
end

Wolfram

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

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

   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. lower-+.f64 100.0

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. lower-fma.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 90.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.7 \cdot 10^{-193}:\\ \;\;\;\;\mathsf{fma}\left(y + y, y, y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y + y, y, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 1.7e-193) (fma (+ y y) y (* y y)) (fma (+ y y) y (* x x))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 1.7e-193) {
		tmp = fma((y + y), y, (y * y));
	} else {
		tmp = fma((y + y), y, (x * x));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 1.7e-193)
		tmp = fma(Float64(y + y), y, Float64(y * y));
	else
		tmp = fma(Float64(y + y), y, Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 1.7e-193], N[(N[(y + y), $MachinePrecision] * y + N[(y * y), $MachinePrecision]), $MachinePrecision], N[(N[(y + y), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.7000000000000001e-193

   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

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

      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. lower-+.f64 99.9

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(y + y, y, \color{blue}{{y}^{2}}\right) \]
   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 91.1

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

   7. Applied rewrites 91.1%

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

   if 1.7000000000000001e-193 < (*.f64 x 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. +-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. lower-+.f64 100.0

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 90.6

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

   7. Applied rewrites 90.6%

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

Alternative 3: 90.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.7 \cdot 10^{-193}:\\ \;\;\;\;\left(3 \cdot y\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y + y, y, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 1.7e-193) (* (* 3.0 y) y) (fma (+ y y) y (* x x))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 1.7e-193) {
		tmp = (3.0 * y) * y;
	} else {
		tmp = fma((y + y), y, (x * x));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 1.7e-193)
		tmp = Float64(Float64(3.0 * y) * y);
	else
		tmp = fma(Float64(y + y), y, Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 1.7e-193], N[(N[(3.0 * y), $MachinePrecision] * y), $MachinePrecision], N[(N[(y + y), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.7000000000000001e-193

   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 y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 91.0

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

   5. Applied rewrites 91.0%

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

   7. Applied rewrites 91.1%

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

   if 1.7000000000000001e-193 < (*.f64 x 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. +-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. lower-+.f64 100.0

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 90.6

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

   7. Applied rewrites 90.6%

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

Alternative 4: 90.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-193}:\\ \;\;\;\;\left(3 \cdot y\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 2e-193) (* (* 3.0 y) y) (fma y y (* x x))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 2e-193) {
		tmp = (3.0 * y) * y;
	} else {
		tmp = fma(y, y, (x * x));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 2e-193)
		tmp = Float64(Float64(3.0 * y) * y);
	else
		tmp = fma(y, y, Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e-193], N[(N[(3.0 * y), $MachinePrecision] * y), $MachinePrecision], N[(y * y + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.0000000000000001e-193

   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 y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 91.0

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

   5. Applied rewrites 91.0%

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

   7. Applied rewrites 91.1%

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

   if 2.0000000000000001e-193 < (*.f64 x 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. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 90.4

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

   5. Applied rewrites 90.4%

      \[\leadsto \color{blue}{x \cdot x} + y \cdot y \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 90.4

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

   7. Applied rewrites 90.4%

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

Alternative 5: 82.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 3.9 \cdot 10^{-7}:\\ \;\;\;\;\left(3 \cdot y\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 3.9e-7) (* (* 3.0 y) y) (* x x)))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 3.9e-7) {
		tmp = (3.0 * y) * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x * x) <= 3.9d-7) then
        tmp = (3.0d0 * y) * y
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x * x) <= 3.9e-7) {
		tmp = (3.0 * y) * y;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 3.9e-7:
		tmp = (3.0 * y) * y
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 3.9e-7)
		tmp = Float64(Float64(3.0 * y) * y);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * x) <= 3.9e-7)
		tmp = (3.0 * y) * y;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 3.9e-7], N[(N[(3.0 * y), $MachinePrecision] * y), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 3.90000000000000025e-7

   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 y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 84.2

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

   5. Applied rewrites 84.2%

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

   7. Applied rewrites 84.3%

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

   if 3.90000000000000025e-7 < (*.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. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 86.3

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

   5. Applied rewrites 86.3%

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

Alternative 6: 74.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 1.12 \cdot 10^{+308}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y y) 1.12e+308) (* x x) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1.12e+308) {
		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) <= 1.12d+308) 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) <= 1.12e+308) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 1.12e+308:
		tmp = x * x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 1.12e+308)
		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) <= 1.12e+308)
		tmp = x * x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 1.12e+308], N[(x * x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1.12e308

   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 y around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 69.8

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

   5. Applied rewrites 69.8%

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

   if 1.12e308 < (*.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

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

      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. lower-+.f64 100.0

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around inf

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. flip-+ N/A

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

      6. +-inverses N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. +-inverses N/A

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

      11. metadata-eval N/A

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

      12. flip-- N/A

          \[\leadsto y \cdot y + y \cdot \color{blue}{\left(0 - 0\right)} \]

      13. metadata-eval N/A

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

      14. +-inverses N/A

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

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

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

      16. +-inverses N/A

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

      17. lower-+.f64 100.0

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

   9. Applied rewrites 100.0%

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

   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 7: 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 8: 36.8% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(y * y), $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. lower-+.f64 100.0

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. lower-fma.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in y around inf

   \[\leadsto \mathsf{fma}\left(y + y, y, \color{blue}{{y}^{2}}\right) \]
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 54.1

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

7. Applied rewrites 54.1%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lift-+.f64 N/A

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

   5. flip-+ N/A

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

   6. +-inverses N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. +-inverses N/A

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

   11. metadata-eval N/A

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

   12. flip-- N/A

       \[\leadsto y \cdot y + y \cdot \color{blue}{\left(0 - 0\right)} \]

   13. metadata-eval N/A

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

   14. +-inverses N/A

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

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

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

   16. +-inverses N/A

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

   17. lower-+.f64 34.9

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

9. Applied rewrites 34.9%

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

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 34.9

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

12. Applied rewrites 34.9%

    \[\leadsto \color{blue}{y \cdot y} \]
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 2024235 
(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)))