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

Percentage Accurate: 99.9% → 100.0%

Time: 5.8s

Alternatives: 5

Speedup: 1.7×

• 43.0% 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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[(x * x), $MachinePrecision] + N[(y * N[(y * 2.0), $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. +-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. Step-by-step derivation

   1. fma-udef 99.9%

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

   2. +-commutative 99.9%

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

   3. count-2 99.9%

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

   4. fma-def 99.9%

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

   5. associate-+l+ 99.9%

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

   6. *-un-lft-identity 99.9%

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

   7. *-un-lft-identity 99.9%

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

   8. associate-+l+ 99.9%

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

   9. associate-+l+ 99.9%

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

   10. count-2 99.9%

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

   11. associate-*r* 99.9%

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

   12. *-commutative 99.9%

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

   13. fma-def 100.0%

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

   14. *-commutative 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(y * y), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] + 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. Final simplification 99.9%

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

Alternative 4: 99.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * x) + (y * (y * 3.0d0))
end function

Java

public static double code(double x, double y) {
	return (x * x) + (y * (y * 3.0));
}

Python

def code(x, y):
	return (x * x) + (y * (y * 3.0))

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = (x * x) + (y * (y * 3.0));
end

Wolfram

code[x_, y_] := N[(N[(x * x), $MachinePrecision] + N[(y * N[(y * 3.0), $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. associate-+l+ 99.9%

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

   2. +-commutative 99.9%

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

   3. +-commutative 99.9%

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

   4. associate-+r+ 99.9%

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

   5. count-2 99.9%

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

   6. distribute-lft1-in 99.9%

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

   7. metadata-eval 99.9%

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

   8. *-commutative 99.9%

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

   9. fma-def 99.9%

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

   10. metadata-eval 99.9%

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

   11. metadata-eval 99.9%

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

3. Simplified 99.9%

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

   1. fma-udef 99.9%

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

   2. associate-*l* 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 5: 57.0% accurate, 3.0× speedup

Math

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

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(Float64(y * y) * 3.0)
end

MATLAB

function tmp = code(x, y)
	tmp = (y * y) * 3.0;
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around 0 57.9%

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

4. Simplified 57.9%

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

5. Final simplification 57.9%

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

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 2023279 
(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)))