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

Percentage Accurate: 99.9% → 99.9%

Time: 7.4s

Alternatives: 6

Speedup: 1.0×

• 44.6% 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: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, y, \mathsf{fma}\left(x, x, 2 \cdot \left(y \cdot y\right)\right)\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma y y (fma x x (* 2.0 (* y y)))))

C

double code(double x, double y) {
	return fma(y, y, fma(x, x, (2.0 * (y * y))));
}

Julia

function code(x, y)
	return fma(y, y, fma(x, x, Float64(2.0 * Float64(y * y))))
end

Wolfram

code[x_, y_] := N[(y * y + N[(x * x + N[(2.0 * 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%

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

   2. sqr-neg 99.9%

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

   3. +-commutative 99.9%

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

   4. sqr-neg 99.9%

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

   5. +-commutative 99.9%

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

   6. fma-define 100.0%

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

   7. sqr-neg 100.0%

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

   8. sqr-neg 100.0%

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

   9. associate-+l+ 100.0%

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

   10. fma-define 100.0%

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

   11. count-2 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, y, \mathsf{fma}\left(x, x, 2 \cdot \left(y \cdot y\right)\right)\right)} \]
4. Add Preprocessing

5. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(y, y, \mathsf{fma}\left(x, x, 2 \cdot \left(y \cdot y\right)\right)\right) \]
6. Add Preprocessing

Alternative 2: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ y \cdot y + \mathsf{fma}\left(y \cdot 2, y, x \cdot x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (* y y) (fma (* y 2.0) y (* x x))))

C

double code(double x, double y) {
	return (y * y) + fma((y * 2.0), y, (x * x));
}

Julia

function code(x, y)
	return Float64(Float64(y * y) + fma(Float64(y * 2.0), y, Float64(x * x)))
end

Wolfram

code[x_, y_] := N[(N[(y * y), $MachinePrecision] + N[(N[(y * 2.0), $MachinePrecision] * 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. associate-+l+ 99.9%

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

   2. +-commutative 99.9%

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

   3. count-2 99.9%

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

   4. *-commutative 99.9%

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

   5. *-commutative 99.9%

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

   6. associate-*r* 99.9%

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

   7. fma-define 99.9%

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

   8. *-commutative 99.9%

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

   9. pow2 99.9%

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

4. Applied egg-rr 99.9%

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

   1. unpow2 80.4%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

   \[\leadsto y \cdot y + \mathsf{fma}\left(y \cdot 2, y, x \cdot x\right) \]
8. Add Preprocessing

Alternative 3: 77.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.8 \cdot 10^{-135}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(1 + 3 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.8e-135)
   (* y (* y 3.0))
   (* (* x x) (+ 1.0 (* 3.0 (* (/ y x) (/ y x)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.8e-135) {
		tmp = y * (y * 3.0);
	} else {
		tmp = (x * x) * (1.0 + (3.0 * ((y / x) * (y / 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 <= 1.8d-135) then
        tmp = y * (y * 3.0d0)
    else
        tmp = (x * x) * (1.0d0 + (3.0d0 * ((y / x) * (y / x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.8e-135) {
		tmp = y * (y * 3.0);
	} else {
		tmp = (x * x) * (1.0 + (3.0 * ((y / x) * (y / x))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.8e-135:
		tmp = y * (y * 3.0)
	else:
		tmp = (x * x) * (1.0 + (3.0 * ((y / x) * (y / x))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.8e-135)
		tmp = Float64(y * Float64(y * 3.0));
	else
		tmp = Float64(Float64(x * x) * Float64(1.0 + Float64(3.0 * Float64(Float64(y / x) * Float64(y / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.8e-135)
		tmp = y * (y * 3.0);
	else
		tmp = (x * x) * (1.0 + (3.0 * ((y / x) * (y / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.8e-135], N[(y * N[(y * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(1.0 + N[(3.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.79999999999999989e-135

   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. add-sqr-sqrt 99.7%

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

      2. pow2 99.7%

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

      3. +-commutative 99.7%

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

      4. add-sqr-sqrt 99.7%

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

      5. hypot-define 99.7%

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

      6. +-commutative 99.7%

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

      7. add-sqr-sqrt 99.7%

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

      8. hypot-define 99.7%

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

      9. hypot-define 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in y around inf 67.2%

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

      1. unpow-prod-down 67.1%

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

      2. pow2 67.1%

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

      3. pow2 67.1%

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

      4. rem-square-sqrt 67.4%

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

      5. associate-*l* 67.4%

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

   7. Applied egg-rr 67.4%

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

   if 1.79999999999999989e-135 < 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. Taylor expanded in x around inf 86.6%

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

   5. Simplified 86.6%

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

      1. pow2 86.6%

         \[\leadsto {x}^{2} \cdot \left(1 + 3 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}}\right) \]

      2. unpow2 86.6%

         \[\leadsto {x}^{2} \cdot \left(1 + 3 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}}\right) \]

      3. times-frac 96.7%

         \[\leadsto {x}^{2} \cdot \left(1 + 3 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)}\right) \]

   7. Applied egg-rr 96.7%

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

      1. unpow2 96.7%

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

   9. Applied egg-rr 96.7%

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + 3 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.8 \cdot 10^{-135}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(1 + 3 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{-134}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(1 + 3 \cdot \frac{y}{x \cdot \frac{x}{y}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.3e-134)
   (* y (* y 3.0))
   (* (* x x) (+ 1.0 (* 3.0 (/ y (* x (/ x y))))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.3e-134) {
		tmp = y * (y * 3.0);
	} else {
		tmp = (x * x) * (1.0 + (3.0 * (y / (x * (x / y)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.3d-134) then
        tmp = y * (y * 3.0d0)
    else
        tmp = (x * x) * (1.0d0 + (3.0d0 * (y / (x * (x / y)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.3e-134) {
		tmp = y * (y * 3.0);
	} else {
		tmp = (x * x) * (1.0 + (3.0 * (y / (x * (x / y)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.3e-134:
		tmp = y * (y * 3.0)
	else:
		tmp = (x * x) * (1.0 + (3.0 * (y / (x * (x / y)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.3e-134)
		tmp = Float64(y * Float64(y * 3.0));
	else
		tmp = Float64(Float64(x * x) * Float64(1.0 + Float64(3.0 * Float64(y / Float64(x * Float64(x / y))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.3e-134)
		tmp = y * (y * 3.0);
	else
		tmp = (x * x) * (1.0 + (3.0 * (y / (x * (x / y)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.3e-134], N[(y * N[(y * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(1.0 + N[(3.0 * N[(y / N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.30000000000000011e-134

   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. add-sqr-sqrt 99.7%

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

      2. pow2 99.7%

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

      3. +-commutative 99.7%

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

      4. add-sqr-sqrt 99.7%

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

      5. hypot-define 99.7%

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

      6. +-commutative 99.7%

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

      7. add-sqr-sqrt 99.7%

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

      8. hypot-define 99.7%

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

      9. hypot-define 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in y around inf 67.2%

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

      1. unpow-prod-down 67.1%

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

      2. pow2 67.1%

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

      3. pow2 67.1%

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

      4. rem-square-sqrt 67.4%

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

      5. associate-*l* 67.4%

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

   7. Applied egg-rr 67.4%

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

   if 1.30000000000000011e-134 < 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. Taylor expanded in x around inf 86.6%

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

   5. Simplified 86.6%

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

      1. pow2 86.6%

         \[\leadsto {x}^{2} \cdot \left(1 + 3 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}}\right) \]

      2. unpow2 86.6%

         \[\leadsto {x}^{2} \cdot \left(1 + 3 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}}\right) \]

      3. times-frac 96.7%

         \[\leadsto {x}^{2} \cdot \left(1 + 3 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)}\right) \]

   7. Applied egg-rr 96.7%

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

      1. unpow2 96.7%

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

   9. Applied egg-rr 96.7%

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

       1. clear-num 96.7%

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

       2. frac-times 96.8%

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

       3. *-un-lft-identity 96.8%

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

   11. Applied egg-rr 96.8%

       \[\leadsto \left(x \cdot x\right) \cdot \left(1 + 3 \cdot \color{blue}{\frac{y}{\frac{x}{y} \cdot x}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{-134}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(1 + 3 \cdot \frac{y}{x \cdot \frac{x}{y}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Final simplification 99.9%

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

Alternative 6: 58.1% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(y * N[(y * 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. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 99.8%

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

   2. pow2 99.8%

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

   3. +-commutative 99.8%

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

   4. add-sqr-sqrt 99.7%

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

   5. hypot-define 99.8%

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

   6. +-commutative 99.8%

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

   7. add-sqr-sqrt 99.8%

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

   8. hypot-define 99.8%

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

   9. hypot-define 99.8%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in y around inf 58.7%

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

   1. unpow-prod-down 58.6%

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

   2. pow2 58.6%

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

   3. pow2 58.6%

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

   4. rem-square-sqrt 58.8%

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

   5. associate-*l* 58.8%

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

7. Applied egg-rr 58.8%

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

8. Final simplification 58.8%

   \[\leadsto y \cdot \left(y \cdot 3\right) \]
9. Add Preprocessing

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 2024060 
(FPCore (x y)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, E"
  :precision binary64

  :alt
  (+ (* x x) (* y (+ y (+ y y))))

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