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

Percentage Accurate: 99.9% → 99.9%

Time: 7.6s

Alternatives: 6

Speedup: 1.8×

• 43.3% 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, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(y * N[(y + y), $MachinePrecision] + N[(y * 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. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

4. Applied rewrites 100.0%

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

Alternative 2: 69.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-57}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 4.5e-57) (* x x) (* 3.0 (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.5e-57) {
		tmp = x * x;
	} else {
		tmp = 3.0 * (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 <= 4.5d-57) then
        tmp = x * x
    else
        tmp = 3.0d0 * (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.5e-57) {
		tmp = x * x;
	} else {
		tmp = 3.0 * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.5e-57:
		tmp = x * x
	else:
		tmp = 3.0 * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.5e-57)
		tmp = Float64(x * x);
	else
		tmp = Float64(3.0 * Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.5e-57)
		tmp = x * x;
	else
		tmp = 3.0 * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.5e-57], N[(x * x), $MachinePrecision], N[(3.0 * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.49999999999999973e-57

   1. Initial program 99.9%

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

   4. Applied rewrites 59.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-frac N/A

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

      5. associate-*r/ N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

         \[\leadsto 1 \cdot {x}^{2} + \frac{{x}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{0} \cdot y\right)\right)}{x} \]

      8. mul0-lft N/A

         \[\leadsto 1 \cdot {x}^{2} + \frac{{x}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{0}\right)\right)}{x} \]

      9. metadata-eval N/A

         \[\leadsto 1 \cdot {x}^{2} + \frac{{x}^{2} \cdot \color{blue}{0}}{x} \]

      10. mul0-rgt N/A

          \[\leadsto 1 \cdot {x}^{2} + \frac{\color{blue}{0}}{x} \]

      11. mul0-lft N/A

          \[\leadsto 1 \cdot {x}^{2} + \frac{\color{blue}{0 \cdot y}}{x} \]

      12. metadata-eval N/A

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

      13. distribute-rgt1-in N/A

          \[\leadsto 1 \cdot {x}^{2} + \frac{\color{blue}{y + -1 \cdot y}}{x} \]

      14. *-lft-identity N/A

          \[\leadsto 1 \cdot {x}^{2} + \color{blue}{1 \cdot \frac{y + -1 \cdot y}{x}} \]

      15. *-lft-identity N/A

          \[\leadsto \color{blue}{{x}^{2}} + 1 \cdot \frac{y + -1 \cdot y}{x} \]

      16. metadata-eval N/A

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

      17. fp-cancel-sub-sign-inv N/A

          \[\leadsto \color{blue}{{x}^{2} - -1 \cdot \frac{y + -1 \cdot y}{x}} \]

      18. mul-1-neg N/A

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

   10. Applied rewrites 61.5%

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

   if 4.49999999999999973e-57 < y

   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 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 84.4

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

   5. Applied rewrites 84.4%

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

Alternative 3: 99.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(3.0 * N[(y * y), $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. Add Preprocessing

4. Applied rewrites 52.2%

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

5. Taylor expanded in y around 0

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

7. Applied rewrites 99.9%

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

Alternative 4: 99.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. distribute-lft1-in N/A

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

   4. metadata-eval N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 99.9

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

5. Applied rewrites 99.9%

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

Alternative 5: 64.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.4 \cdot 10^{+86}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 8.4e+86) (* x x) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8.4e+86) {
		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 <= 8.4d+86) 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 <= 8.4e+86) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8.4e+86:
		tmp = x * x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8.4e+86)
		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 <= 8.4e+86)
		tmp = x * x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8.4e+86], N[(x * x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.3999999999999996e86

   1. Initial program 99.9%

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

   4. Applied rewrites 62.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-frac N/A

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

      5. associate-*r/ N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

         \[\leadsto 1 \cdot {x}^{2} + \frac{{x}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{0} \cdot y\right)\right)}{x} \]

      8. mul0-lft N/A

         \[\leadsto 1 \cdot {x}^{2} + \frac{{x}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{0}\right)\right)}{x} \]

      9. metadata-eval N/A

         \[\leadsto 1 \cdot {x}^{2} + \frac{{x}^{2} \cdot \color{blue}{0}}{x} \]

      10. mul0-rgt N/A

          \[\leadsto 1 \cdot {x}^{2} + \frac{\color{blue}{0}}{x} \]

      11. mul0-lft N/A

          \[\leadsto 1 \cdot {x}^{2} + \frac{\color{blue}{0 \cdot y}}{x} \]

      12. metadata-eval N/A

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

      13. distribute-rgt1-in N/A

          \[\leadsto 1 \cdot {x}^{2} + \frac{\color{blue}{y + -1 \cdot y}}{x} \]

      14. *-lft-identity N/A

          \[\leadsto 1 \cdot {x}^{2} + \color{blue}{1 \cdot \frac{y + -1 \cdot y}{x}} \]

      15. *-lft-identity N/A

          \[\leadsto \color{blue}{{x}^{2}} + 1 \cdot \frac{y + -1 \cdot y}{x} \]

      16. metadata-eval N/A

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

      17. fp-cancel-sub-sign-inv N/A

          \[\leadsto \color{blue}{{x}^{2} - -1 \cdot \frac{y + -1 \cdot y}{x}} \]

      18. mul-1-neg N/A

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

   10. Applied rewrites 59.8%

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

   if 8.3999999999999996e86 < y

   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. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

   4. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

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

      5. pow2 N/A

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

      6. sqrt-prod N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-sqrt.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 97.9

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

   9. Applied rewrites 97.9%

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

       1. lift-fma.f64 N/A

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

       2. +-commutative N/A

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

       3. fp-cancel-sign-sub-inv N/A

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

       4. rem-square-sqrt N/A

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

       5. sqrt-prod N/A

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

       6. sqr-neg-rev N/A

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

       7. sqrt-unprod N/A

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

       8. rem-square-sqrt N/A

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

       9. lower--.f64 N/A

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

       10. lift-fma.f64 N/A

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

       11. lift-sqrt.f64 N/A

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

       12. lift-sqrt.f64 N/A

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

       13. rem-square-sqrt N/A

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

       14. distribute-lft-in N/A

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

       15. fp-cancel-sign-sub-inv N/A

           \[\leadsto y \cdot y - \color{blue}{\left(y \cdot y - \left(\mathsf{neg}\left(y\right)\right) \cdot y\right)} \]

   11. Applied rewrites 74.2%

       \[\leadsto \color{blue}{y \cdot y - 0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.4 \cdot 10^{+86}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.3% 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. Add Preprocessing

4. Applied rewrites 52.2%

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

5. Taylor expanded in y around 0

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in x around inf

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

   1. distribute-rgt-in N/A

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

   2. *-commutative N/A

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

   3. mul-1-neg N/A

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

   4. distribute-neg-frac N/A

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

   5. associate-*r/ N/A

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

   6. distribute-rgt1-in N/A

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

   7. metadata-eval N/A

      \[\leadsto 1 \cdot {x}^{2} + \frac{{x}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{0} \cdot y\right)\right)}{x} \]

   8. mul0-lft N/A

      \[\leadsto 1 \cdot {x}^{2} + \frac{{x}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{0}\right)\right)}{x} \]

   9. metadata-eval N/A

      \[\leadsto 1 \cdot {x}^{2} + \frac{{x}^{2} \cdot \color{blue}{0}}{x} \]

   10. mul0-rgt N/A

       \[\leadsto 1 \cdot {x}^{2} + \frac{\color{blue}{0}}{x} \]

   11. mul0-lft N/A

       \[\leadsto 1 \cdot {x}^{2} + \frac{\color{blue}{0 \cdot y}}{x} \]

   12. metadata-eval N/A

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

   13. distribute-rgt1-in N/A

       \[\leadsto 1 \cdot {x}^{2} + \frac{\color{blue}{y + -1 \cdot y}}{x} \]

   14. *-lft-identity N/A

       \[\leadsto 1 \cdot {x}^{2} + \color{blue}{1 \cdot \frac{y + -1 \cdot y}{x}} \]

   15. *-lft-identity N/A

       \[\leadsto \color{blue}{{x}^{2}} + 1 \cdot \frac{y + -1 \cdot y}{x} \]

   16. metadata-eval N/A

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

   17. fp-cancel-sub-sign-inv N/A

       \[\leadsto \color{blue}{{x}^{2} - -1 \cdot \frac{y + -1 \cdot y}{x}} \]

   18. mul-1-neg N/A

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

10. Applied rewrites 52.8%

    \[\leadsto \color{blue}{x \cdot x} \]
11. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.5× 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 2024339 
(FPCore (x y)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, E"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* x x) (* y (+ y (+ y y)))))

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