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

Percentage Accurate: 99.9% → 99.9%

Time: 3.8s

Alternatives: 6

Speedup: 1.0×

• 43.9% 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: 99.9% 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, 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 3: 80.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 3.8 \cdot 10^{-52}:\\ \;\;\;\;\left(y \cdot y\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 3.8e-52) (* (* y y) 3.0) (* x x)))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 3.8e-52) {
		tmp = (y * y) * 3.0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x * x) <= 3.8d-52) then
        tmp = (y * y) * 3.0d0
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x * x) <= 3.8e-52) {
		tmp = (y * y) * 3.0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 3.8e-52:
		tmp = (y * y) * 3.0
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 3.8e-52)
		tmp = Float64(Float64(y * y) * 3.0);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * x) <= 3.8e-52)
		tmp = (y * y) * 3.0;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 3.8e-52], N[(N[(y * y), $MachinePrecision] * 3.0), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 3.8000000000000003e-52

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 89.4%

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

   4. Simplified 89.4%

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

   if 3.8000000000000003e-52 < (*.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. Taylor expanded in x around inf 80.4%

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

   4. Simplified 80.4%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 3.8 \cdot 10^{-52}:\\ \;\;\;\;\left(y \cdot y\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 4: 80.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 8.4 \cdot 10^{-54}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 8.4e-54) (* y (* y 3.0)) (* x x)))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 8.4e-54) {
		tmp = y * (y * 3.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x * x) <= 8.4d-54) then
        tmp = y * (y * 3.0d0)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x * x) <= 8.4e-54) {
		tmp = y * (y * 3.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 8.4e-54:
		tmp = y * (y * 3.0)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 8.4e-54)
		tmp = Float64(y * Float64(y * 3.0));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * x) <= 8.4e-54)
		tmp = y * (y * 3.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 8.4e-54], N[(y * N[(y * 3.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 8.4e-54

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 89.4%

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

   4. Simplified 89.4%

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

      1. add-sqr-sqrt 89.2%

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

      2. sqrt-unprod 66.8%

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

      3. swap-sqr 66.7%

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

      4. metadata-eval 66.7%

         \[\leadsto \sqrt{\color{blue}{9} \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)} \]

      5. pow2 66.7%

         \[\leadsto \sqrt{9 \cdot \left(\color{blue}{{y}^{2}} \cdot \left(y \cdot y\right)\right)} \]

      6. pow2 66.7%

         \[\leadsto \sqrt{9 \cdot \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)} \]

      7. pow-prod-up 66.8%

         \[\leadsto \sqrt{9 \cdot \color{blue}{{y}^{\left(2 + 2\right)}}} \]

      8. metadata-eval 66.8%

         \[\leadsto \sqrt{9 \cdot {y}^{\color{blue}{4}}} \]

   6. Applied egg-rr 66.8%

      \[\leadsto \color{blue}{\sqrt{9 \cdot {y}^{4}}} \]

   8. Simplified 66.8%

      \[\leadsto \color{blue}{\sqrt{{y}^{4} \cdot 9}} \]
   9. Step-by-step derivation

      1. *-commutative 66.8%

         \[\leadsto \sqrt{\color{blue}{9 \cdot {y}^{4}}} \]

      2. sqrt-prod 66.7%

         \[\leadsto \color{blue}{\sqrt{9} \cdot \sqrt{{y}^{4}}} \]

      3. metadata-eval 66.7%

         \[\leadsto \color{blue}{3} \cdot \sqrt{{y}^{4}} \]

      4. sqrt-pow1 89.4%

         \[\leadsto 3 \cdot \color{blue}{{y}^{\left(\frac{4}{2}\right)}} \]

      5. metadata-eval 89.4%

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

      6. pow2 89.4%

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

      7. associate-*r* 89.4%

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

      8. *-commutative 89.4%

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

   10. Applied egg-rr 89.4%

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

   if 8.4e-54 < (*.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. Taylor expanded in x around inf 80.4%

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

   4. Simplified 80.4%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 8.4 \cdot 10^{-54}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 5: 75.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 3.45 \cdot 10^{+304}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y y) 3.45e+304) (* x x) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 3.45e+304) {
		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) <= 3.45d+304) 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) <= 3.45e+304) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 3.45e+304:
		tmp = x * x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 3.45e+304)
		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) <= 3.45e+304)
		tmp = x * x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 3.45e+304], N[(x * x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 3.44999999999999999e304

   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 67.0%

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

   4. Simplified 67.0%

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

   if 3.44999999999999999e304 < (*.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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+r+ 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. *-commutative 100.0%

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

      6. add-cbrt-cube 98.7%

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

      7. cbrt-prod 100.0%

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

      8. associate-*r* 100.0%

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

      9. fma-def 100.0%

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

      10. cbrt-prod 100.0%

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

      11. pow2 100.0%

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

      12. add-sqr-sqrt 100.0%

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

      13. pow2 100.0%

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

      14. hypot-def 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in y around inf 98.8%

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

   6. Simplified 98.8%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 3.45 \cdot 10^{+304}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 6: 56.6% 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 56.4%

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

4. Simplified 56.4%

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

5. Final simplification 56.4%

   \[\leadsto x \cdot x \]

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 2023224 
(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)))