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

Percentage Accurate: 99.9% → 99.9%

Time: 6.9s

Alternatives: 4

Speedup: 1.0×

• 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. +-commutative 99.9%

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

   2. fma-def 99.9%

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

   3. associate-+l+ 99.9%

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

   4. fma-def 99.9%

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

   5. count-2 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 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: 57.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return (y * 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 * y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(y * y), $MachinePrecision] + N[(N[(y * y), $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. 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 99.4%

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

   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 99.4%

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

3. Applied egg-rr 99.4%

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

4. Taylor expanded in x around 0 60.7%

   \[\leadsto \left({\color{blue}{\left({\left({y}^{2}\right)}^{0.3333333333333333}\right)}}^{3} + y \cdot y\right) + y \cdot y \]
5. Step-by-step derivation

   1. unpow1/3 62.4%

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

6. Simplified 62.4%

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

   1. rem-cube-cbrt 62.4%

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

   2. unpow2 62.4%

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

8. Applied egg-rr 62.4%

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

9. Final simplification 62.4%

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

Alternative 4: 36.7% 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 99.9%

   \[\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y \]
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 99.4%

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

   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 99.4%

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

3. Applied egg-rr 99.4%

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

4. Taylor expanded in x around 0 60.7%

   \[\leadsto \left({\color{blue}{\left({\left({y}^{2}\right)}^{0.3333333333333333}\right)}}^{3} + y \cdot y\right) + y \cdot y \]
5. Step-by-step derivation

   1. unpow1/3 62.4%

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

6. Simplified 62.4%

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

   1. rem-cube-cbrt 62.4%

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

   2. unpow2 62.4%

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

8. Applied egg-rr 62.4%

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

10. Applied egg-rr 41.5%

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

11. Final simplification 41.5%

    \[\leadsto y \cdot y \]

Developer target: 99.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023301 
(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)))