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

Percentage Accurate: 99.9% → 100.0%

Time: 3.9s

Alternatives: 4

Speedup: 1.0×

• 44.6% 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 99.9%

   \[\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y \]
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 100.0%

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

   3. associate-+l+ 100.0%

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

   4. fma-def 100.0%

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

   5. 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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate-+l+ 99.9%

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

   2. associate-+l+ 99.9%

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

   3. fma-def 99.9%

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

   4. cancel-sign-sub 99.9%

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

   5. neg-mul-1 99.9%

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

   6. associate-*l* 99.9%

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

   7. count-2 99.9%

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

   8. distribute-rgt-out-- 99.9%

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

   9. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 3: 99.9% accurate, 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 4: 57.2% 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 99.9%

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

   1. associate-+l+ 99.9%

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

   2. associate-+l+ 99.9%

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

   3. fma-def 99.9%

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

   4. distribute-lft-out 99.9%

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

   5. distribute-lft-out 99.9%

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

   6. count-2 99.9%

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

   7. distribute-rgt1-in 99.9%

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

   8. metadata-eval 99.9%

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

   9. *-commutative 99.9%

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

   10. metadata-eval 99.9%

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

   11. metadata-eval 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 3\right)\right)} \]
4. Step-by-step derivation

   1. expm1-log1p-u 97.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 3\right)\right)\right)\right)} \]

   2. expm1-udef 82.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 3\right)\right)\right)} - 1} \]

   3. add-sqr-sqrt 82.0%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 3\right)\right)} \cdot \sqrt{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 3\right)\right)}}\right)} - 1 \]

   4. pow2 82.0%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\sqrt{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 3\right)\right)}\right)}^{2}}\right)} - 1 \]

   5. fma-udef 82.0%

      \[\leadsto e^{\mathsf{log1p}\left({\left(\sqrt{\color{blue}{x \cdot x + y \cdot \left(y \cdot 3\right)}}\right)}^{2}\right)} - 1 \]

   6. add-sqr-sqrt 82.0%

      \[\leadsto e^{\mathsf{log1p}\left({\left(\sqrt{x \cdot x + \color{blue}{\sqrt{y \cdot \left(y \cdot 3\right)} \cdot \sqrt{y \cdot \left(y \cdot 3\right)}}}\right)}^{2}\right)} - 1 \]

   7. hypot-def 82.0%

      \[\leadsto e^{\mathsf{log1p}\left({\color{blue}{\left(\mathsf{hypot}\left(x, \sqrt{y \cdot \left(y \cdot 3\right)}\right)\right)}}^{2}\right)} - 1 \]

   8. associate-*r* 82.0%

      \[\leadsto e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(x, \sqrt{\color{blue}{\left(y \cdot y\right) \cdot 3}}\right)\right)}^{2}\right)} - 1 \]

   9. sqrt-prod 82.0%

      \[\leadsto e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(x, \color{blue}{\sqrt{y \cdot y} \cdot \sqrt{3}}\right)\right)}^{2}\right)} - 1 \]

   10. sqrt-prod 38.2%

       \[\leadsto e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(x, \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \sqrt{3}\right)\right)}^{2}\right)} - 1 \]

   11. add-sqr-sqrt 82.0%

       \[\leadsto e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(x, \color{blue}{y} \cdot \sqrt{3}\right)\right)}^{2}\right)} - 1 \]

5. Applied egg-rr 82.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(x, y \cdot \sqrt{3}\right)\right)}^{2}\right)} - 1} \]

7. Simplified 99.7%

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

8. Taylor expanded in x around 0 61.6%

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

   1. unpow2 61.6%

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

   2. unpow2 61.6%

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

   3. swap-sqr 61.7%

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

   4. unpow2 61.7%

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

10. Simplified 61.7%

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

    1. pow-prod-down 61.6%

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

    2. *-commutative 61.6%

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

    3. unpow2 61.6%

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

    4. associate-*r* 61.6%

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

    5. sqrt-pow2 61.9%

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

    6. metadata-eval 61.9%

       \[\leadsto \left({3}^{\color{blue}{1}} \cdot y\right) \cdot y \]

    7. metadata-eval 61.9%

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

12. Applied egg-rr 61.9%

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

13. Final simplification 61.9%

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

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 2023309 
(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)))