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

Percentage Accurate: 99.9% → 99.9%

Time: 8.7s

Alternatives: 6

Speedup: 1.7×

• 44.4% 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: 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%

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

   2. fma-def 99.9%

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

   3. associate-+l+ 99.9%

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

   4. fma-def 99.9%

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

   5. count-2 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 90.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 7 \cdot 10^{-187}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \mathbf{elif}\;x \cdot x \leq 1.42 \cdot 10^{-149} \lor \neg \left(x \cdot x \leq 1.26 \cdot 10^{-83}\right):\\ \;\;\;\;y \cdot y + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 7e-187)
   (* y (* y 3.0))
   (if (or (<= (* x x) 1.42e-149) (not (<= (* x x) 1.26e-83)))
     (+ (* y y) (* x x))
     (* (* y y) 3.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 7e-187) {
		tmp = y * (y * 3.0);
	} else if (((x * x) <= 1.42e-149) || !((x * x) <= 1.26e-83)) {
		tmp = (y * y) + (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 ((x * x) <= 7d-187) then
        tmp = y * (y * 3.0d0)
    else if (((x * x) <= 1.42d-149) .or. (.not. ((x * x) <= 1.26d-83))) then
        tmp = (y * y) + (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 ((x * x) <= 7e-187) {
		tmp = y * (y * 3.0);
	} else if (((x * x) <= 1.42e-149) || !((x * x) <= 1.26e-83)) {
		tmp = (y * y) + (x * x);
	} else {
		tmp = (y * y) * 3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 7e-187:
		tmp = y * (y * 3.0)
	elif ((x * x) <= 1.42e-149) or not ((x * x) <= 1.26e-83):
		tmp = (y * y) + (x * x)
	else:
		tmp = (y * y) * 3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 7e-187)
		tmp = Float64(y * Float64(y * 3.0));
	elseif ((Float64(x * x) <= 1.42e-149) || !(Float64(x * x) <= 1.26e-83))
		tmp = Float64(Float64(y * y) + 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 ((x * x) <= 7e-187)
		tmp = y * (y * 3.0);
	elseif (((x * x) <= 1.42e-149) || ~(((x * x) <= 1.26e-83)))
		tmp = (y * y) + (x * x);
	else
		tmp = (y * y) * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 7e-187], N[(y * N[(y * 3.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(x * x), $MachinePrecision], 1.42e-149], N[Not[LessEqual[N[(x * x), $MachinePrecision], 1.26e-83]], $MachinePrecision]], N[(N[(y * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * 3.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 6.99999999999999958e-187

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 95.9%

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

   4. Simplified 95.9%

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

   if 6.99999999999999958e-187 < (*.f64 x x) < 1.42e-149 or 1.2600000000000001e-83 < (*.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. 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. count-2 99.9%

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

      3. flip-+ 20.3%

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

      4. pow2 20.3%

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

      5. pow2 20.3%

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

      6. pow-prod-up 20.2%

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

      7. metadata-eval 20.2%

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

      8. *-commutative 20.2%

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

      9. *-commutative 20.2%

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

      10. swap-sqr 20.2%

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

      11. pow2 20.2%

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

      12. pow2 20.2%

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

      13. pow-prod-up 20.2%

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

      14. metadata-eval 20.2%

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

      15. metadata-eval 20.2%

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

      16. *-commutative 20.2%

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

      17. associate-*l* 20.2%

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

   3. Applied egg-rr 20.2%

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

   4. Taylor expanded in x around inf 92.4%

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

      1. unpow2 92.4%

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

   6. Simplified 92.4%

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

   if 1.42e-149 < (*.f64 x x) < 1.2600000000000001e-83

   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 85.3%

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

   4. Simplified 85.3%

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

   5. Taylor expanded in y around 0 85.3%

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

      1. unpow2 85.3%

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

   7. Simplified 85.3%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 7 \cdot 10^{-187}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \mathbf{elif}\;x \cdot x \leq 1.42 \cdot 10^{-149} \lor \neg \left(x \cdot x \leq 1.26 \cdot 10^{-83}\right):\\ \;\;\;\;y \cdot y + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot 3\\ \end{array} \]

Alternative 3: 99.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(y * N[(y * 3.0), $MachinePrecision]), $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. Step-by-step derivation

   1. associate-+l+ 99.9%

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

   2. +-commutative 99.9%

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

   3. +-commutative 99.9%

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

   4. associate-+r+ 99.9%

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

   5. count-2 99.9%

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

   6. distribute-lft1-in 99.9%

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

   7. metadata-eval 99.9%

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

   8. *-commutative 99.9%

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

   9. fma-def 99.9%

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

   10. metadata-eval 99.9%

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

   11. metadata-eval 99.9%

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

3. Simplified 99.9%

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

   1. fma-udef 99.9%

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

   2. associate-*l* 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 4: 57.2% accurate, 3.0× speedup

Math

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

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(Float64(y * y) * 3.0)
end

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(y * y), $MachinePrecision] * 3.0), $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 61.8%

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

4. Simplified 61.8%

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

5. Taylor expanded in y around 0 61.8%

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

   1. unpow2 61.8%

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

7. Simplified 61.8%

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

8. Final simplification 61.8%

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

Alternative 5: 57.2% 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. Taylor expanded in x around 0 61.8%

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

4. Simplified 61.8%

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

5. Final simplification 61.8%

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

Alternative 6: 37.5% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y \]
2. 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. count-2 99.9%

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

   3. flip-+ 26.2%

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

   4. pow2 26.2%

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

   5. pow2 26.2%

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

   6. pow-prod-up 26.1%

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

   7. metadata-eval 26.1%

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

   8. *-commutative 26.1%

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

   9. *-commutative 26.1%

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

   10. swap-sqr 26.1%

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

   11. pow2 26.1%

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

   12. pow2 26.1%

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

   13. pow-prod-up 26.1%

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

   14. metadata-eval 26.1%

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

   15. metadata-eval 26.1%

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

   16. *-commutative 26.1%

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

   17. associate-*l* 26.1%

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

3. Applied egg-rr 26.1%

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

4. Taylor expanded in x around 0 16.3%

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

5. Taylor expanded in y around 0 39.8%

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

   1. unpow2 39.8%

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

7. Simplified 39.8%

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

8. Final simplification 39.8%

   \[\leadsto y \cdot y \]

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 2023287 
(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)))