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

Percentage Accurate: 99.9% → 100.0%

Time: 5.2s

Alternatives: 8

Speedup: 1.8×

• 43.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 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: 89.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.3 \cdot 10^{-135}:\\ \;\;\;\;\mathsf{fma}\left(y + y, y, y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y + y, y, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 1.3e-135) (fma (+ y y) y (* y y)) (fma (+ y y) y (* x x))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 1.3e-135) {
		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) <= 1.3e-135)
		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], 1.3e-135], 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) < 1.30000000000000002e-135

   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 87.8

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

   7. Applied rewrites 87.8%

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

   if 1.30000000000000002e-135 < (*.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 90.8

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

   7. Applied rewrites 90.8%

      \[\leadsto \mathsf{fma}\left(y + y, y, \color{blue}{x \cdot x}\right) \]
3. Recombined 2 regimes into one program.
4. 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 2024230 
(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)))