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

Percentage Accurate: 99.9% → 100.0%

Time: 6.8s

Alternatives: 6

Speedup: 1.0×

• 44.2% 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: 100.0% 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 99.8%

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

   1. +-commutative 99.8%

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

   2. sqr-neg 99.8%

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

   3. +-commutative 99.8%

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

   4. sqr-neg 99.8%

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

   5. +-commutative 99.8%

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

   6. fma-def 99.9%

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

   7. sqr-neg 99.9%

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

   8. sqr-neg 99.9%

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

   9. associate-+l+ 99.9%

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

   10. fma-def 99.9%

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

   11. 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. Add Preprocessing

5. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0 99.8%

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

5. Simplified 99.8%

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

   1. unpow2 99.8%

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

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

Alternative 3: 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 99.8%

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

3. Final simplification 99.8%

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

Alternative 4: 58.2% 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 99.8%

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

   1. associate-+l+ 99.8%

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

   2. count-2 99.8%

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

   3. fma-udef 99.8%

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

   4. +-commutative 99.8%

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

   5. expm1-log1p-u 97.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot y + \mathsf{fma}\left(x, x, 2 \cdot \left(y \cdot y\right)\right)\right)\right)} \]

   6. fma-udef 97.2%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(y, y, \mathsf{fma}\left(x, x, 2 \cdot \left(y \cdot y\right)\right)\right)}\right)\right) \]

   7. expm1-udef 78.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(y, y, \mathsf{fma}\left(x, x, 2 \cdot \left(y \cdot y\right)\right)\right)\right)} - 1} \]

4. Applied egg-rr 78.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(y, \mathsf{hypot}\left(y, \mathsf{hypot}\left(x, y\right)\right)\right)\right)}^{2}\right)} - 1} \]

6. Simplified 99.8%

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

7. Taylor expanded in y around inf 56.8%

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

   1. unpow2 56.8%

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

   2. hypot-udef 56.8%

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

   3. unpow2 56.8%

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

   4. fma-udef 56.8%

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

   5. hypot-udef 56.8%

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

   6. unpow2 56.8%

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

   7. fma-udef 56.8%

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

   8. add-sqr-sqrt 56.8%

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

   9. *-commutative 56.8%

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

   10. unpow-prod-down 56.7%

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

   11. sqrt-pow2 57.0%

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

   12. metadata-eval 57.0%

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

   13. metadata-eval 57.0%

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

   14. pow2 57.0%

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

   15. fma-def 56.9%

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

   16. distribute-rgt1-in 56.9%

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

   17. metadata-eval 56.9%

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

   18. rem-cube-cbrt 56.5%

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

9. Applied egg-rr 56.9%

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

10. Final simplification 56.9%

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

Alternative 5: 37.7% 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 99.8%

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

   1. +-commutative 99.8%

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

   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. add-sqr-sqrt 99.8%

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

   4. pow2 99.8%

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

   5. +-commutative 99.8%

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

   6. add-sqr-sqrt 99.8%

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

   7. hypot-def 99.8%

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

   8. hypot-def 99.8%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in y around inf 56.8%

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

   1. unpow2 56.8%

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

   2. swap-sqr 56.7%

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

   3. rem-square-sqrt 57.0%

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

   4. *-commutative 57.0%

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

   5. count-2 57.0%

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

   6. distribute-lft-out 57.0%

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

7. Applied egg-rr 57.0%

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

   1. distribute-lft-in 57.0%

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

   2. fma-def 56.9%

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

   3. flip-+ 0.0%

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

   4. distribute-lft-out-- 0.0%

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

   5. +-inverses 0.0%

      \[\leadsto y \cdot y + \frac{\left(y \cdot y\right) \cdot \color{blue}{0}}{y \cdot y - y \cdot y} \]

   6. metadata-eval 0.0%

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

   7. distribute-rgt-out-- 0.0%

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

   8. distribute-lft-out-- 0.0%

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

   9. +-inverses 0.0%

      \[\leadsto y \cdot y + \frac{1 \cdot \color{blue}{0}}{y \cdot y - y \cdot y} \]

   10. metadata-eval 0.0%

       \[\leadsto y \cdot y + \frac{\color{blue}{0}}{y \cdot y - y \cdot y} \]

   11. metadata-eval 0.0%

       \[\leadsto y \cdot y + \frac{\color{blue}{0 - 0}}{y \cdot y - y \cdot y} \]

   12. metadata-eval 0.0%

       \[\leadsto y \cdot y + \frac{\color{blue}{0 \cdot 0} - 0}{y \cdot y - y \cdot y} \]

   13. metadata-eval 0.0%

       \[\leadsto y \cdot y + \frac{0 \cdot 0 - \color{blue}{0 \cdot 0}}{y \cdot y - y \cdot y} \]

   14. +-inverses 0.0%

       \[\leadsto y \cdot y + \frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0}} \]

   15. metadata-eval 0.0%

       \[\leadsto y \cdot y + \frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0 + 0}} \]

   16. flip-- 36.6%

       \[\leadsto y \cdot y + \color{blue}{\left(0 - 0\right)} \]

   17. metadata-eval 36.6%

       \[\leadsto y \cdot y + \color{blue}{0} \]

   18. +-inverses 13.6%

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

   19. associate-+r- 13.9%

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

   20. count-2 13.9%

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

   21. *-un-lft-identity 13.9%

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

   22. distribute-rgt-out-- 36.6%

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

   23. metadata-eval 36.6%

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

   24. *-commutative 36.6%

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

9. Applied egg-rr 36.6%

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

10. Final simplification 36.6%

    \[\leadsto y \cdot y \]
11. Add Preprocessing

Alternative 6: 1.2% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -2.0)

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -2.0d0
end function

Java

public static double code(double x, double y) {
	return -2.0;
}

Python

def code(x, y):
	return -2.0

Julia

function code(x, y)
	return -2.0
end

MATLAB

function tmp = code(x, y)
	tmp = -2.0;
end

Wolfram

code[x_, y_] := -2.0

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0 99.8%

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

5. Simplified 99.8%

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

   1. unpow2 99.8%

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

7. Applied egg-rr 99.8%

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

8. Taylor expanded in y around inf 56.9%

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

10. Simplified 1.3%

    \[\leadsto \color{blue}{-2} \]

11. Final simplification 1.3%

    \[\leadsto -2 \]
12. 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 2024019 
(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)))