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

Percentage Accurate: 99.9% → 99.9%

Time: 4.4s

Alternatives: 4

Speedup: 1.0×

• 43.8% 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 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%

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

   8. sqr-neg 99.9%

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

   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. 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, 0.1× speedup

Math

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

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(y * Float64(y * 3.0)))
end

Wolfram

code[x_, y_] := N[(x * x + 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. 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. *-lft-identity 99.9%

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

   5. metadata-eval 99.9%

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

   6. count-2 99.9%

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

   7. distribute-rgt-out 99.9%

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

   8. metadata-eval 99.9%

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

   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. Step-by-step derivation

   1. add-sqr-sqrt 99.8%

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

   2. sqrt-unprod 88.7%

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

   3. swap-sqr 88.7%

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

   4. pow2 88.7%

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

   5. pow2 88.7%

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

   6. pow-prod-up 88.7%

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

   7. metadata-eval 88.7%

      \[\leadsto \mathsf{fma}\left(x, x, \sqrt{{y}^{\color{blue}{4}} \cdot \left(3 \cdot 3\right)}\right) \]

   8. metadata-eval 88.7%

      \[\leadsto \mathsf{fma}\left(x, x, \sqrt{{y}^{4} \cdot \color{blue}{9}}\right) \]

5. Applied egg-rr 88.7%

   \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\sqrt{{y}^{4} \cdot 9}}\right) \]
6. Step-by-step derivation

   1. sqrt-prod 88.6%

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\sqrt{{y}^{4}} \cdot \sqrt{9}}\right) \]

   2. metadata-eval 88.6%

      \[\leadsto \mathsf{fma}\left(x, x, \sqrt{{y}^{4}} \cdot \color{blue}{3}\right) \]

   3. sqrt-pow1 99.9%

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{{y}^{\left(\frac{4}{2}\right)}} \cdot 3\right) \]

   4. metadata-eval 99.9%

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

   5. pow2 99.9%

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

   6. associate-*l* 99.9%

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

7. Applied egg-rr 99.9%

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

8. Final simplification 99.9%

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

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: 4.3% 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. associate-+r+ 99.9%

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

   2. flip3-+ 20.1%

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

   3. flip-+ 0.0%

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

   4. frac-add 0.0%

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

3. Applied egg-rr 0.0%

   \[\leadsto \color{blue}{\frac{\left({x}^{6} + {y}^{6}\right) \cdot 0 + \left({x}^{4} + \left({y}^{4} - {\left(x \cdot y\right)}^{2}\right)\right) \cdot 0}{\left({x}^{4} + \left({y}^{4} - {\left(x \cdot y\right)}^{2}\right)\right) \cdot 0}} \]

5. Simplified 4.3%

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

6. Final simplification 4.3%

   \[\leadsto 2 \]

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 2023331 
(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)))