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

Percentage Accurate: 99.9% → 100.0%

Time: 5.4s

Alternatives: 8

Speedup: 1.7×

• 43.4% 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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(y * y + N[(x * x + N[(y * N[(y + y), $MachinePrecision]), $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. Step-by-step derivation

   1. +-commutative 99.9%

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

   2. fma-def 100.0%

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

   3. associate-+l+ 100.0%

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

   4. fma-def 100.0%

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

   5. distribute-lft-out 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. +-commutative 99.9%

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

   2. +-commutative 99.9%

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

   3. associate-+r+ 99.9%

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

   4. distribute-lft-in 99.9%

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

   5. *-commutative 99.9%

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

   6. fma-def 99.9%

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

3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

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

Alternative 3: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[(y * 3.0), $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. Taylor expanded in x around 0 99.9%

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

4. Simplified 99.9%

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

   1. associate-*r* 99.9%

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

   2. fma-def 99.9%

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

   3. *-commutative 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 4: 81.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 1.95 \cdot 10^{-31} \lor \neg \left(y \cdot y \leq 3.6 \cdot 10^{+37}\right) \land y \cdot y \leq 2.9 \cdot 10^{+71}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* y y) 1.95e-31)
         (and (not (<= (* y y) 3.6e+37)) (<= (* y y) 2.9e+71)))
   (* x x)
   (* (* y y) 3.0)))

C

double code(double x, double y) {
	double tmp;
	if (((y * y) <= 1.95e-31) || (!((y * y) <= 3.6e+37) && ((y * y) <= 2.9e+71))) {
		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) <= 1.95d-31) .or. (.not. ((y * y) <= 3.6d+37)) .and. ((y * y) <= 2.9d+71)) 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) <= 1.95e-31) || (!((y * y) <= 3.6e+37) && ((y * y) <= 2.9e+71))) {
		tmp = x * x;
	} else {
		tmp = (y * y) * 3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((y * y) <= 1.95e-31) or (not ((y * y) <= 3.6e+37) and ((y * y) <= 2.9e+71)):
		tmp = x * x
	else:
		tmp = (y * y) * 3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(y * y) <= 1.95e-31) || (!(Float64(y * y) <= 3.6e+37) && (Float64(y * y) <= 2.9e+71)))
		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) <= 1.95e-31) || (~(((y * y) <= 3.6e+37)) && ((y * y) <= 2.9e+71)))
		tmp = x * x;
	else
		tmp = (y * y) * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(y * y), $MachinePrecision], 1.95e-31], And[N[Not[LessEqual[N[(y * y), $MachinePrecision], 3.6e+37]], $MachinePrecision], LessEqual[N[(y * y), $MachinePrecision], 2.9e+71]]], N[(x * x), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1.9500000000000001e-31 or 3.59999999999999998e37 < (*.f64 y y) < 2.90000000000000007e71

   1. Initial program 99.9%

      \[\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y \]

   2. Taylor expanded in x around inf 86.2%

      \[\leadsto \color{blue}{{x}^{2}} \]

   4. Simplified 86.2%

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

   if 1.9500000000000001e-31 < (*.f64 y y) < 3.59999999999999998e37 or 2.90000000000000007e71 < (*.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. Taylor expanded in x around 0 85.5%

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

   4. Simplified 85.5%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 1.95 \cdot 10^{-31} \lor \neg \left(y \cdot y \leq 3.6 \cdot 10^{+37}\right) \land y \cdot y \leq 2.9 \cdot 10^{+71}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot 3\\ \end{array} \]

Alternative 5: 81.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{-32}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;y \cdot y \leq 10^{+35}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \mathbf{elif}\;y \cdot y \leq 10^{+71}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 2e-32)
   (* x x)
   (if (<= (* y y) 1e+35)
     (* y (* y 3.0))
     (if (<= (* y y) 1e+71) (* x x) (* (* y y) 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 2e-32) {
		tmp = x * x;
	} else if ((y * y) <= 1e+35) {
		tmp = y * (y * 3.0);
	} else if ((y * y) <= 1e+71) {
		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) <= 2d-32) then
        tmp = x * x
    else if ((y * y) <= 1d+35) then
        tmp = y * (y * 3.0d0)
    else if ((y * y) <= 1d+71) 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) <= 2e-32) {
		tmp = x * x;
	} else if ((y * y) <= 1e+35) {
		tmp = y * (y * 3.0);
	} else if ((y * y) <= 1e+71) {
		tmp = x * x;
	} else {
		tmp = (y * y) * 3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 2e-32:
		tmp = x * x
	elif (y * y) <= 1e+35:
		tmp = y * (y * 3.0)
	elif (y * y) <= 1e+71:
		tmp = x * x
	else:
		tmp = (y * y) * 3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 2e-32)
		tmp = Float64(x * x);
	elseif (Float64(y * y) <= 1e+35)
		tmp = Float64(y * Float64(y * 3.0));
	elseif (Float64(y * y) <= 1e+71)
		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) <= 2e-32)
		tmp = x * x;
	elseif ((y * y) <= 1e+35)
		tmp = y * (y * 3.0);
	elseif ((y * y) <= 1e+71)
		tmp = x * x;
	else
		tmp = (y * y) * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 2e-32], N[(x * x), $MachinePrecision], If[LessEqual[N[(y * y), $MachinePrecision], 1e+35], N[(y * N[(y * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y * y), $MachinePrecision], 1e+71], N[(x * x), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * 3.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y y) < 2.00000000000000011e-32 or 9.9999999999999997e34 < (*.f64 y y) < 1e71

   1. Initial program 99.9%

      \[\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y \]

   2. Taylor expanded in x around inf 86.2%

      \[\leadsto \color{blue}{{x}^{2}} \]

   4. Simplified 86.2%

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

   if 2.00000000000000011e-32 < (*.f64 y y) < 9.9999999999999997e34

   1. Initial program 99.5%

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

      1. add-sqr-sqrt 99.0%

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

      2. fma-def 99.2%

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

      3. +-commutative 99.2%

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

      4. add-sqr-sqrt 99.2%

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

      5. hypot-def 99.5%

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

      6. hypot-def 99.5%

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

      7. +-commutative 99.5%

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

      8. add-sqr-sqrt 99.5%

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

      9. hypot-def 99.5%

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

      10. hypot-def 99.5%

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

   3. Applied egg-rr 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in y around inf 78.0%

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

      1. unpow2 78.0%

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

      2. rem-square-sqrt 78.7%

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

      3. count-2 78.7%

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

      4. unpow2 78.7%

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

      5. unpow2 78.7%

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

      6. distribute-lft-out 78.7%

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

   8. Simplified 78.7%

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

      1. distribute-lft-out 79.0%

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

      2. count-2 79.0%

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

      3. *-un-lft-identity 79.0%

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

      4. distribute-rgt-out 79.0%

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

      5. metadata-eval 79.0%

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

      6. *-commutative 79.0%

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

   10. Applied egg-rr 79.0%

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

   if 1e71 < (*.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. Taylor expanded in x around 0 86.1%

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

   4. Simplified 86.1%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{-32}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;y \cdot y \leq 10^{+35}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \mathbf{elif}\;y \cdot y \leq 10^{+71}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot 3\\ \end{array} \]

Alternative 6: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(y * y), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $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. Final simplification 99.9%

   \[\leadsto y \cdot y + \left(y \cdot y + \left(x \cdot x + y \cdot y\right)\right) \]

Alternative 7: 99.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * x) + ((y * y) * 3.0d0)
end function

Java

public static double code(double x, double y) {
	return (x * x) + ((y * y) * 3.0);
}

Python

def code(x, y):
	return (x * x) + ((y * y) * 3.0)

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = (x * x) + ((y * y) * 3.0);
end

Wolfram

code[x_, y_] := N[(N[(x * x), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] * 3.0), $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. Taylor expanded in x around 0 99.9%

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

4. Simplified 99.9%

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

5. Final simplification 99.9%

   \[\leadsto x \cdot x + \left(y \cdot y\right) \cdot 3 \]

Alternative 8: 57.8% 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. Taylor expanded in x around inf 61.7%

   \[\leadsto \color{blue}{{x}^{2}} \]

4. Simplified 61.7%

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

5. Final simplification 61.7%

   \[\leadsto x \cdot x \]

Developer target: 99.9% accurate, 1.4× 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 2023215 
(FPCore (x y)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, E"
  :precision binary64

  :herbie-target
  (+ (* x x) (* y (+ y (+ y y))))

  (+ (+ (+ (* x x) (* y y)) (* y y)) (* y y)))