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

Percentage Accurate: 99.9% → 99.9%

Time: 6.1s

Alternatives: 7

Speedup: 1.0×

• 43.5% 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. +-commutative 99.9%

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

   3. sqr-neg 99.9%

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

   4. +-commutative 99.9%

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

   5. sqr-neg 99.9%

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

   6. fma-def 100.0%

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

   7. sqr-neg 100.0%

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

   8. sqr-neg 100.0%

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

   9. associate-+l+ 100.0%

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

   10. fma-def 100.0%

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

   11. count-2 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 3: 57.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(y * y), $MachinePrecision] + N[(y * 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. add-cube-cbrt 99.3%

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

   2. pow3 99.2%

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

   3. add-sqr-sqrt 99.3%

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

   4. pow2 99.3%

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

   5. hypot-def 99.3%

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

3. Applied egg-rr 99.3%

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

4. Taylor expanded in x around 0 52.3%

   \[\leadsto \left({\color{blue}{\left({\left({y}^{2}\right)}^{0.3333333333333333}\right)}}^{3} + y \cdot y\right) + y \cdot y \]

6. Simplified 54.1%

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

   1. sqr-pow 54.1%

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

   2. rem-cube-cbrt 54.0%

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

   3. pow2 54.0%

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

   4. sqr-pow 54.0%

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

   5. distribute-lft-out 54.0%

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

   6. pow2 54.0%

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

   7. sqrt-pow1 54.0%

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

   8. rem-cube-cbrt 54.1%

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

   9. sqrt-prod 32.3%

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

   10. add-sqr-sqrt 35.8%

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

   11. pow2 35.8%

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

   12. sqrt-pow1 35.8%

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

   13. rem-cube-cbrt 35.9%

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

   14. sqrt-prod 32.4%

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

   15. add-sqr-sqrt 38.1%

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

   16. pow2 38.1%

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

   17. sqrt-pow1 38.1%

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

   18. rem-cube-cbrt 38.1%

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

8. Applied egg-rr 54.3%

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

9. Final simplification 54.3%

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

Alternative 4: 57.9% 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. Step-by-step derivation

   1. add-cube-cbrt 99.3%

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

   2. pow3 99.2%

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

   3. add-sqr-sqrt 99.3%

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

   4. pow2 99.3%

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

   5. hypot-def 99.3%

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

3. Applied egg-rr 99.3%

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

4. Taylor expanded in x around 0 52.3%

   \[\leadsto \left({\color{blue}{\left({\left({y}^{2}\right)}^{0.3333333333333333}\right)}}^{3} + y \cdot y\right) + y \cdot y \]

6. Simplified 54.1%

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

   1. sqr-pow 54.1%

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

   2. rem-cube-cbrt 54.0%

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

   3. pow2 54.0%

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

   4. sqr-pow 54.0%

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

   5. distribute-lft-out 54.0%

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

   6. fma-def 54.0%

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

8. Applied egg-rr 54.4%

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

   1. fma-udef 54.3%

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

   2. pow2 54.3%

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

   3. distribute-lft-out 54.3%

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

   4. *-commutative 54.3%

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

   5. count-2 54.3%

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

   6. *-un-lft-identity 54.3%

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

   7. distribute-rgt-out 54.3%

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

   8. metadata-eval 54.3%

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

10. Applied egg-rr 54.3%

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

11. Final simplification 54.3%

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

Alternative 5: 37.3% 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. add-cube-cbrt 99.3%

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

   2. pow3 99.2%

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

   3. add-sqr-sqrt 99.3%

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

   4. pow2 99.3%

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

   5. hypot-def 99.3%

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

3. Applied egg-rr 99.3%

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

4. Taylor expanded in x around 0 52.3%

   \[\leadsto \left({\color{blue}{\left({\left({y}^{2}\right)}^{0.3333333333333333}\right)}}^{3} + y \cdot y\right) + y \cdot y \]

6. Simplified 54.1%

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

   1. sqr-pow 54.1%

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

   2. rem-cube-cbrt 54.0%

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

   3. pow2 54.0%

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

   4. sqr-pow 54.0%

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

   5. distribute-lft-out 54.0%

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

   6. fma-def 54.0%

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

8. Applied egg-rr 54.4%

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

10. Applied egg-rr 33.1%

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

11. Final simplification 33.1%

    \[\leadsto y \cdot y \]

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 99.9%

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

   1. add-cube-cbrt 99.3%

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

   2. pow3 99.2%

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

   3. add-sqr-sqrt 99.3%

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

   4. pow2 99.3%

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

   5. hypot-def 99.3%

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

3. Applied egg-rr 99.3%

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

4. Taylor expanded in x around 0 52.3%

   \[\leadsto \left({\color{blue}{\left({\left({y}^{2}\right)}^{0.3333333333333333}\right)}}^{3} + y \cdot y\right) + y \cdot y \]

6. Simplified 54.1%

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

7. Taylor expanded in y around 0 33.7%

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

9. Simplified 1.2%

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

10. Final simplification 1.2%

    \[\leadsto -2 \]

Alternative 7: 1.2% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -1.0)

C

double code(double x, double y) {
	return -1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -1.0d0
end function

Java

public static double code(double x, double y) {
	return -1.0;
}

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

function tmp = code(x, y)
	tmp = -1.0;
end

Wolfram

code[x_, y_] := -1.0

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. add-cube-cbrt 99.3%

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

   2. pow3 99.2%

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

   3. add-sqr-sqrt 99.3%

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

   4. pow2 99.3%

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

   5. hypot-def 99.3%

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

3. Applied egg-rr 99.3%

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

4. Taylor expanded in x around 0 52.3%

   \[\leadsto \left({\color{blue}{\left({\left({y}^{2}\right)}^{0.3333333333333333}\right)}}^{3} + y \cdot y\right) + y \cdot y \]

6. Simplified 54.1%

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

   1. sqr-pow 54.1%

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

   2. rem-cube-cbrt 54.0%

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

   3. pow2 54.0%

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

   4. sqr-pow 54.0%

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

   5. distribute-lft-out 54.0%

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

   6. fma-def 54.0%

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

8. Applied egg-rr 54.4%

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

9. Taylor expanded in y around 0 54.3%

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

11. Simplified 1.2%

    \[\leadsto \color{blue}{-1} \]

12. Final simplification 1.2%

    \[\leadsto -1 \]

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 2023340 
(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)))