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

Percentage Accurate: 99.9% → 100.0%

Time: 9.1s

Alternatives: 9

Speedup: 1.8×

• 43.8% 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: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate-+l+ N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-out N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. +-lowering-+.f64 N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. *-lowering-*.f64 99.9

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

4. Applied egg-rr 99.9%

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

Alternative 2: 89.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-304}:\\ \;\;\;\;\mathsf{fma}\left(y + y, y, y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y + y, y, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 2e-304) (fma (+ y y) y (* y y)) (fma (+ y y) y (* x x))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 2e-304) {
		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) <= 2e-304)
		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], 2e-304], 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) < 1.99999999999999994e-304

   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. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. +-lowering-+.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   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. *-lowering-*.f64 99.0

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

   7. Simplified 99.0%

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

   if 1.99999999999999994e-304 < (*.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. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. +-lowering-+.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 89.1

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

   7. Simplified 89.1%

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

Alternative 3: 89.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-304}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y + y, y, x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 2e-304) (* y (* y 3.0)) (fma (+ y y) y (* x x))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 2e-304) {
		tmp = y * (y * 3.0);
	} else {
		tmp = fma((y + y), y, (x * x));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 2e-304)
		tmp = Float64(y * Float64(y * 3.0));
	else
		tmp = fma(Float64(y + y), y, Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e-304], N[(y * N[(y * 3.0), $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) < 1.99999999999999994e-304

   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. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 98.9

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

   5. Simplified 98.9%

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

   if 1.99999999999999994e-304 < (*.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. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. +-lowering-+.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 89.1

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

   7. Simplified 89.1%

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

Alternative 4: 89.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-304}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 2e-304) (* y (* y 3.0)) (fma x x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 2e-304) {
		tmp = y * (y * 3.0);
	} else {
		tmp = fma(x, x, (y * y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 2e-304)
		tmp = Float64(y * Float64(y * 3.0));
	else
		tmp = fma(x, x, Float64(y * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e-304], N[(y * N[(y * 3.0), $MachinePrecision]), $MachinePrecision], N[(x * x + N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.99999999999999994e-304

   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. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 98.9

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

   5. Simplified 98.9%

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

   if 1.99999999999999994e-304 < (*.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. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. +-lowering-+.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 99.9

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

   4. Applied egg-rr 99.9%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. flip-+ N/A

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

      4. +-inverses N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. +-inverses N/A

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

      9. metadata-eval N/A

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

      10. flip-- N/A

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

      11. metadata-eval N/A

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

      12. +-inverses N/A

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

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

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

      14. +-inverses N/A

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

      15. associate-+r+ N/A

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

      16. +-rgt-identity N/A

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

      17. accelerator-lowering-fma.f64 N/A

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

      18. *-lowering-*.f64 88.8

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

   6. Applied egg-rr 88.8%

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

Alternative 5: 80.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 4 \cdot 10^{+21}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 4e+21) (* x x) (* (* y y) 3.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 4e+21) {
		tmp = x * x;
	} else {
		tmp = (y * y) * 3.0;
	}
	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+21) then
        tmp = x * x
    else
        tmp = (y * y) * 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 4e+21) {
		tmp = x * x;
	} else {
		tmp = (y * y) * 3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 4e+21:
		tmp = x * x
	else:
		tmp = (y * y) * 3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 4e+21)
		tmp = Float64(x * x);
	else
		tmp = Float64(Float64(y * y) * 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 4e+21)
		tmp = x * x;
	else
		tmp = (y * y) * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 4e+21], N[(x * x), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 4e21

   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 inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 89.4

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

   5. Simplified 89.4%

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

   if 4e21 < (*.f64 y 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. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-out N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. +-lowering-+.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 84.2

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

   7. Simplified 84.2%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 4 \cdot 10^{+21}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot 3\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{+277}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y, y + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 2e+277) (* x x) (fma y y (+ y y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 2e+277) {
		tmp = x * x;
	} else {
		tmp = fma(y, y, (y + y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 2e+277)
		tmp = Float64(x * x);
	else
		tmp = fma(y, y, Float64(y + y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 2e+277], N[(x * x), $MachinePrecision], N[(y * y + N[(y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 2.00000000000000001e277

   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 inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 70.5

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

   5. Simplified 70.5%

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

   if 2.00000000000000001e277 < (*.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. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 98.7

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

   5. Simplified 98.7%

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-lowering-*.f64 98.7

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

   7. Applied egg-rr 98.7%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. distribute-lft1-in N/A

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

      5. count-2 N/A

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

      6. associate-+l+ N/A

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

      7. distribute-rgt-out N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. flip-+ N/A

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

      11. +-inverses N/A

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

      12. +-inverses N/A

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

      13. associate-*r/ N/A

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

      14. +-inverses N/A

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

      15. +-inverses N/A

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

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

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

      17. +-inverses N/A

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

      18. +-inverses N/A

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

      19. flip-+ N/A

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

      20. +-lowering-+.f64 91.9

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

   9. Applied egg-rr 91.9%

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

Alternative 7: 74.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 3.15 \cdot 10^{+277}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y y) 3.15e+277) (* x x) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 3.15e+277) {
		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) <= 3.15d+277) 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) <= 3.15e+277) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 3.15e+277:
		tmp = x * x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 3.15e+277)
		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) <= 3.15e+277)
		tmp = x * x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 3.15e+277], N[(x * x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 3.1500000000000002e277

   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 inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 70.5

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

   5. Simplified 70.5%

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

   if 3.1500000000000002e277 < (*.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 inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 93.1

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

   5. Simplified 93.1%

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

   6. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 91.9

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

   8. Simplified 91.9%

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

Alternative 8: 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. associate-+l+ N/A

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

   2. associate-+l+ N/A

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

   3. +-commutative N/A

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

   4. count-2 N/A

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

   5. distribute-lft1-in N/A

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

   6. metadata-eval N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 99.9

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

4. Applied egg-rr 99.9%

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

Alternative 9: 56.2% 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 inf

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 58.3

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

5. Simplified 58.3%

   \[\leadsto \color{blue}{x \cdot x} \]
6. 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 2024204 
(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)))