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

Percentage Accurate: 99.9% → 99.9%

Time: 6.3s

Alternatives: 7

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: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[(x * x), $MachinePrecision] + N[(y * N[(y * 2.0), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. fma-udef 99.9%

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

   2. +-commutative 99.9%

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

   3. count-2 99.9%

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

   4. fma-def 99.9%

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

   5. associate-+l+ 99.9%

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

   6. *-un-lft-identity 99.9%

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

   7. *-un-lft-identity 99.9%

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

   8. associate-+l+ 99.9%

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

   9. associate-+l+ 99.9%

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

   10. count-2 99.9%

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

   11. associate-*r* 99.9%

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

   12. *-commutative 99.9%

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

   13. fma-def 99.9%

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

   14. *-commutative 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(x * x + 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. 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. associate-+l+ 99.9%

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

   3. fma-def 99.9%

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

   4. cancel-sign-sub 99.9%

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

   5. neg-mul-1 99.9%

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

   6. associate-*l* 99.9%

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

   7. count-2 99.9%

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

   8. distribute-rgt-out-- 99.9%

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

   9. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 3: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[(y * y), $MachinePrecision] * 3.0 + 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. Final simplification 99.9%

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

Alternative 4: 79.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.08 \cdot 10^{-138} \lor \neg \left(x \cdot x \leq 90000000\right) \land x \cdot x \leq 1.7 \cdot 10^{+116}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* x x) 1.08e-138)
         (and (not (<= (* x x) 90000000.0)) (<= (* x x) 1.7e+116)))
   (* y (* y 3.0))
   (* x x)))

C

double code(double x, double y) {
	double tmp;
	if (((x * x) <= 1.08e-138) || (!((x * x) <= 90000000.0) && ((x * x) <= 1.7e+116))) {
		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) <= 1.08d-138) .or. (.not. ((x * x) <= 90000000.0d0)) .and. ((x * x) <= 1.7d+116)) 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) <= 1.08e-138) || (!((x * x) <= 90000000.0) && ((x * x) <= 1.7e+116))) {
		tmp = y * (y * 3.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * x) <= 1.08e-138) or (not ((x * x) <= 90000000.0) and ((x * x) <= 1.7e+116)):
		tmp = y * (y * 3.0)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(x * x) <= 1.08e-138) || (!(Float64(x * x) <= 90000000.0) && (Float64(x * x) <= 1.7e+116)))
		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) <= 1.08e-138) || (~(((x * x) <= 90000000.0)) && ((x * x) <= 1.7e+116)))
		tmp = y * (y * 3.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(x * x), $MachinePrecision], 1.08e-138], And[N[Not[LessEqual[N[(x * x), $MachinePrecision], 90000000.0]], $MachinePrecision], LessEqual[N[(x * x), $MachinePrecision], 1.7e+116]]], N[(y * N[(y * 3.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.0799999999999999e-138 or 9e7 < (*.f64 x x) < 1.70000000000000011e116

   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 87.7%

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

   4. Simplified 87.7%

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

   if 1.0799999999999999e-138 < (*.f64 x x) < 9e7 or 1.70000000000000011e116 < (*.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. +-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. Step-by-step derivation

      1. fma-udef 99.9%

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

      2. +-commutative 99.9%

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

      3. count-2 99.9%

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

      4. fma-def 99.9%

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

      5. associate-+l+ 99.9%

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

      6. *-un-lft-identity 99.9%

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

      7. *-un-lft-identity 99.9%

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

      8. associate-+l+ 99.9%

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

      9. associate-+l+ 99.9%

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

      10. count-2 99.9%

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

      11. associate-*r* 99.9%

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

      12. *-commutative 99.9%

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

      13. fma-def 99.9%

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

      14. *-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in y around 0 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around inf 84.7%

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

       1. unpow2 84.7%

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

   11. Simplified 84.7%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.08 \cdot 10^{-138} \lor \neg \left(x \cdot x \leq 90000000\right) \land x \cdot x \leq 1.7 \cdot 10^{+116}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 5: 79.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2.65 \cdot 10^{-135}:\\ \;\;\;\;\left(y \cdot y\right) \cdot 3\\ \mathbf{elif}\;x \cdot x \leq 31000000:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \cdot x \leq 3.2 \cdot 10^{+116}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 2.65e-135)
   (* (* y y) 3.0)
   (if (<= (* x x) 31000000.0)
     (* x x)
     (if (<= (* x x) 3.2e+116) (* y (* y 3.0)) (* x x)))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 2.65e-135) {
		tmp = (y * y) * 3.0;
	} else if ((x * x) <= 31000000.0) {
		tmp = x * x;
	} else if ((x * x) <= 3.2e+116) {
		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) <= 2.65d-135) then
        tmp = (y * y) * 3.0d0
    else if ((x * x) <= 31000000.0d0) then
        tmp = x * x
    else if ((x * x) <= 3.2d+116) 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) <= 2.65e-135) {
		tmp = (y * y) * 3.0;
	} else if ((x * x) <= 31000000.0) {
		tmp = x * x;
	} else if ((x * x) <= 3.2e+116) {
		tmp = y * (y * 3.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 2.65e-135:
		tmp = (y * y) * 3.0
	elif (x * x) <= 31000000.0:
		tmp = x * x
	elif (x * x) <= 3.2e+116:
		tmp = y * (y * 3.0)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 2.65e-135)
		tmp = Float64(Float64(y * y) * 3.0);
	elseif (Float64(x * x) <= 31000000.0)
		tmp = Float64(x * x);
	elseif (Float64(x * x) <= 3.2e+116)
		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) <= 2.65e-135)
		tmp = (y * y) * 3.0;
	elseif ((x * x) <= 31000000.0)
		tmp = x * x;
	elseif ((x * x) <= 3.2e+116)
		tmp = y * (y * 3.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 2.65e-135], N[(N[(y * y), $MachinePrecision] * 3.0), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 31000000.0], N[(x * x), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 3.2e+116], N[(y * N[(y * 3.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 2.65e-135

   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 90.4%

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

   4. Simplified 90.4%

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

      1. add-sqr-sqrt 90.2%

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

      2. pow2 90.2%

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

      3. associate-*r* 90.2%

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

      4. sqrt-prod 90.1%

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

      5. sqrt-prod 43.5%

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

      6. add-sqr-sqrt 90.1%

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

   6. Applied egg-rr 90.1%

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

      1. unpow2 90.1%

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

      2. swap-sqr 89.9%

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

      3. add-sqr-sqrt 90.4%

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

   8. Applied egg-rr 90.4%

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

   if 2.65e-135 < (*.f64 x x) < 3.1e7 or 3.2e116 < (*.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. +-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. Step-by-step derivation

      1. fma-udef 99.9%

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

      2. +-commutative 99.9%

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

      3. count-2 99.9%

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

      4. fma-def 99.9%

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

      5. associate-+l+ 99.9%

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

      6. *-un-lft-identity 99.9%

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

      7. *-un-lft-identity 99.9%

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

      8. associate-+l+ 99.9%

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

      9. associate-+l+ 99.9%

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

      10. count-2 99.9%

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

      11. associate-*r* 99.9%

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

      12. *-commutative 99.9%

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

      13. fma-def 99.9%

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

      14. *-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in y around 0 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around inf 84.7%

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

       1. unpow2 84.7%

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

   11. Simplified 84.7%

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

   if 3.1e7 < (*.f64 x x) < 3.2e116

   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 71.7%

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

   4. Simplified 71.7%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2.65 \cdot 10^{-135}:\\ \;\;\;\;\left(y \cdot y\right) \cdot 3\\ \mathbf{elif}\;x \cdot x \leq 31000000:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \cdot x \leq 3.2 \cdot 10^{+116}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 6: 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. 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. Step-by-step derivation

   1. fma-udef 99.9%

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

   2. +-commutative 99.9%

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

   3. count-2 99.9%

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

   4. fma-def 99.9%

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

   5. associate-+l+ 99.9%

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

   6. *-un-lft-identity 99.9%

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

   7. *-un-lft-identity 99.9%

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

   8. associate-+l+ 99.9%

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

   9. associate-+l+ 99.9%

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

   10. count-2 99.9%

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

   11. associate-*r* 99.9%

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

   12. *-commutative 99.9%

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

   13. fma-def 99.9%

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

   14. *-commutative 99.9%

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

5. Applied egg-rr 99.9%

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

6. Taylor expanded in y around 0 99.9%

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

8. Simplified 99.9%

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

9. Final simplification 99.9%

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

Alternative 7: 57.2% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. fma-udef 99.9%

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

   2. +-commutative 99.9%

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

   3. count-2 99.9%

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

   4. fma-def 99.9%

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

   5. associate-+l+ 99.9%

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

   6. *-un-lft-identity 99.9%

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

   7. *-un-lft-identity 99.9%

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

   8. associate-+l+ 99.9%

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

   9. associate-+l+ 99.9%

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

   10. count-2 99.9%

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

   11. associate-*r* 99.9%

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

   12. *-commutative 99.9%

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

   13. fma-def 99.9%

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

   14. *-commutative 99.9%

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

5. Applied egg-rr 99.9%

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

6. Taylor expanded in y around 0 99.9%

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

8. Simplified 99.9%

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

9. Taylor expanded in x around inf 58.5%

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

    1. unpow2 58.5%

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

11. Simplified 58.5%

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

12. Final simplification 58.5%

    \[\leadsto x \cdot x \]

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 2023271 
(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)))