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

Percentage Accurate: 99.9% → 100.0%

Time: 7.7s

Alternatives: 8

Speedup: 1.8×

• 43.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 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(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: 88.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-89}:\\ \;\;\;\;\mathsf{fma}\left(y, y + y, y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2 + y, y, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 2e-89) (fma y (+ y y) (* y y)) (fma (+ 2.0 y) y (* x x))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 2e-89) {
		tmp = fma(y, (y + y), (y * y));
	} else {
		tmp = fma((2.0 + y), y, (x * x));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 2e-89)
		tmp = fma(y, Float64(y + y), Float64(y * y));
	else
		tmp = fma(Float64(2.0 + y), y, Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e-89], N[(y * N[(y + y), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + y), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.00000000000000008e-89

   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 84.3

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

   7. Applied rewrites 84.3%

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

   if 2.00000000000000008e-89 < (*.f64 x x)

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. 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. flip-+ N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 17.4%

      \[\leadsto \color{blue}{\frac{{x}^{4} - \mathsf{fma}\left(y, y, 2 \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(y, y, 2 \cdot \left(y \cdot y\right)\right)}{x \cdot x - \mathsf{fma}\left(y, y, 2 \cdot \left(y \cdot y\right)\right)}} \]

   6. Applied rewrites 17.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(3 \cdot y, y, x \cdot x\right) \cdot \mathsf{fma}\left(x, x, -3 \cdot \left(y \cdot y\right)\right)}}{x \cdot x - \mathsf{fma}\left(y, y, 2 \cdot \left(y \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. difference-of-squares-rev N/A

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

      16. lift--.f64 N/A

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

      17. lift-fma.f64 N/A

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

      18. lift-*.f64 N/A

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

   8. Applied rewrites 94.1%

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

Alternative 3: 82.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{-29}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y + y, y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 1e-29) (* x x) (fma y (+ y y) (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1e-29) {
		tmp = x * x;
	} else {
		tmp = fma(y, (y + y), (y * y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 1e-29)
		tmp = Float64(x * x);
	else
		tmp = fma(y, Float64(y + y), Float64(y * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 1e-29], N[(x * x), $MachinePrecision], N[(y * N[(y + y), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 9.99999999999999943e-30

   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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto {x}^{2} \cdot \color{blue}{\left(1 + 3 \cdot \frac{{y}^{2}}{{x}^{2}}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 93.3%

      \[\leadsto \mathsf{fma}\left(3 \cdot \frac{y}{x}, \frac{y}{x}, 1\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   9. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 89.2

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

   11. Applied rewrites 89.2%

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

   if 9.99999999999999943e-30 < (*.f64 y y)

   1. Initial program 99.9%

      \[\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y \]
   2. Add Preprocessing
   3. 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 82.8

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

   7. Applied rewrites 82.8%

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

Alternative 4: 82.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{-29}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 1e-29) (* x x) (* 3.0 (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1e-29) {
		tmp = x * x;
	} else {
		tmp = 3.0 * (y * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * y) <= 1d-29) then
        tmp = x * x
    else
        tmp = 3.0d0 * (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1e-29) {
		tmp = x * x;
	} else {
		tmp = 3.0 * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 1e-29:
		tmp = x * x
	else:
		tmp = 3.0 * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 1e-29)
		tmp = Float64(x * x);
	else
		tmp = Float64(3.0 * Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 1e-29)
		tmp = x * x;
	else
		tmp = 3.0 * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 1e-29], N[(x * x), $MachinePrecision], N[(3.0 * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 9.99999999999999943e-30

   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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto {x}^{2} \cdot \color{blue}{\left(1 + 3 \cdot \frac{{y}^{2}}{{x}^{2}}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 93.3%

      \[\leadsto \mathsf{fma}\left(3 \cdot \frac{y}{x}, \frac{y}{x}, 1\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   9. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 89.2

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

   11. Applied rewrites 89.2%

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

   if 9.99999999999999943e-30 < (*.f64 y y)

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 82.7

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

   5. Applied rewrites 82.7%

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

Alternative 5: 75.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 4 \cdot 10^{+303}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y y) 4e+303) (* x x) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 4e+303) {
		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) <= 4d+303) 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) <= 4e+303) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 4e+303:
		tmp = x * x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 4e+303)
		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) <= 4e+303)
		tmp = x * x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 4e+303], N[(x * x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 4e303

   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. Taylor expanded in x around inf

      \[\leadsto {x}^{2} \cdot \color{blue}{\left(1 + 3 \cdot \frac{{y}^{2}}{{x}^{2}}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 86.1%

      \[\leadsto \mathsf{fma}\left(3 \cdot \frac{y}{x}, \frac{y}{x}, 1\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   9. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 73.2

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

   11. Applied rewrites 73.2%

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

   if 4e303 < (*.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. 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 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 100.0%

      \[\leadsto \left|\mathsf{fma}\left(y, y, 0\right)\right| \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

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

Alternative 6: 99.9% accurate, 1.8× speedup

Math

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

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: 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. 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. Taylor expanded in x around inf

   \[\leadsto {x}^{2} \cdot \color{blue}{\left(1 + 3 \cdot \frac{{y}^{2}}{{x}^{2}}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 83.4%

   \[\leadsto \mathsf{fma}\left(3 \cdot \frac{y}{x}, \frac{y}{x}, 1\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

9. Taylor expanded in x around inf

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

    1. unpow2 N/A

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

    2. lower-*.f64 61.1

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

11. Applied rewrites 61.1%

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

Alternative 8: 3.6% accurate, 7.5× speedup

Math

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

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. 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 55.0

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

5. Applied rewrites 55.0%

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

7. Applied rewrites 30.2%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 3.9%

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

12. Applied rewrites 3.9%

    \[\leadsto y + y \]
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 2024327 
(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)))