Linear.Matrix:fromQuaternion from linear-1.19.1.3, A

Percentage Accurate: 95.6% → 100.0%

Time: 3.9s

Alternatives: 4

Speedup: 1.3×

• 27.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 2.0d0 * ((x * x) - (x * y))
end function

Java

public static double code(double x, double y) {
	return 2.0 * ((x * x) - (x * y));
}

Python

def code(x, y):
	return 2.0 * ((x * x) - (x * y))

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = 2.0 * ((x * x) - (x * y));
end

Wolfram

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

Initial Program: 95.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 2.0d0 * ((x * x) - (x * y))
end function

Java

public static double code(double x, double y) {
	return 2.0 * ((x * x) - (x * y));
}

Python

def code(x, y):
	return 2.0 * ((x * x) - (x * y))

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = 2.0 * ((x * x) - (x * y));
end

Wolfram

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

Alternative 1: 100.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x - y) * (x * 2.0d0)
end function

Java

public static double code(double x, double y) {
	return (x - y) * (x * 2.0);
}

Python

def code(x, y):
	return (x - y) * (x * 2.0)

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = (x - y) * (x * 2.0);
end

Wolfram

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

Derivation

1. Initial program 94.5%

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

   1. distribute-lft-out-- 100.0%

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

   2. associate-*r* 100.0%

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

   3. *-commutative 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \left(x - y\right) \cdot \left(x \cdot 2\right) \]

Alternative 2: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+48} \lor \neg \left(y \leq 1.9 \cdot 10^{+45}\right):\\ \;\;\;\;y \cdot \left(x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.4e+48) (not (<= y 1.9e+45))) (* y (* x -2.0)) (* x (+ x x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.4e+48) || !(y <= 1.9e+45)) {
		tmp = y * (x * -2.0);
	} else {
		tmp = x * (x + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.4d+48)) .or. (.not. (y <= 1.9d+45))) then
        tmp = y * (x * (-2.0d0))
    else
        tmp = x * (x + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.4e+48) || !(y <= 1.9e+45)) {
		tmp = y * (x * -2.0);
	} else {
		tmp = x * (x + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.4e+48) or not (y <= 1.9e+45):
		tmp = y * (x * -2.0)
	else:
		tmp = x * (x + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.4e+48) || !(y <= 1.9e+45))
		tmp = Float64(y * Float64(x * -2.0));
	else
		tmp = Float64(x * Float64(x + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.4e+48) || ~((y <= 1.9e+45)))
		tmp = y * (x * -2.0);
	else
		tmp = x * (x + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.4e+48], N[Not[LessEqual[y, 1.9e+45]], $MachinePrecision]], N[(y * N[(x * -2.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(x + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.40000000000000006e48 or 1.9000000000000001e45 < y

   1. Initial program 87.0%

      \[2 \cdot \left(x \cdot x - x \cdot y\right) \]

   2. Taylor expanded in x around 0 87.0%

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

      1. *-commutative 87.0%

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

      2. associate-*r* 87.0%

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

   4. Simplified 87.0%

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

   if -1.40000000000000006e48 < y < 1.9000000000000001e45

   1. Initial program 100.0%

      \[2 \cdot \left(x \cdot x - x \cdot y\right) \]

   2. Taylor expanded in x around inf 87.0%

      \[\leadsto \color{blue}{2 \cdot {x}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 87.0%

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

      2. count-2 87.0%

         \[\leadsto \color{blue}{x \cdot x + x \cdot x} \]

      3. distribute-lft-in 87.0%

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

   4. Simplified 87.0%

      \[\leadsto \color{blue}{x \cdot \left(x + x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+48} \lor \neg \left(y \leq 1.9 \cdot 10^{+45}\right):\\ \;\;\;\;y \cdot \left(x \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + x\right)\\ \end{array} \]

Alternative 3: 59.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

   \[2 \cdot \left(x \cdot x - x \cdot y\right) \]

2. Taylor expanded in x around inf 60.7%

   \[\leadsto \color{blue}{2 \cdot {x}^{2}} \]
3. Step-by-step derivation

   1. unpow2 60.7%

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

   2. count-2 60.7%

      \[\leadsto \color{blue}{x \cdot x + x \cdot x} \]

   3. distribute-lft-in 60.7%

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

4. Simplified 60.7%

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

5. Final simplification 60.7%

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

Alternative 4: 11.7% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.0)

C

double code(double x, double y) {
	return 0.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.0d0
end function

Java

public static double code(double x, double y) {
	return 0.0;
}

Python

def code(x, y):
	return 0.0

Julia

function code(x, y)
	return 0.0
end

MATLAB

function tmp = code(x, y)
	tmp = 0.0;
end

Wolfram

code[x_, y_] := 0.0

Derivation

1. Initial program 94.5%

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

   1. fma-neg 96.5%

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

   2. distribute-rgt-neg-in 96.5%

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

3. Applied egg-rr 96.5%

   \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(x, x, x \cdot \left(-y\right)\right)} \]
4. Step-by-step derivation

   1. fma-udef 94.5%

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

   2. distribute-lft-out 100.0%

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

   3. sub-neg 100.0%

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

   4. associate-*l* 100.0%

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

   5. count-2 100.0%

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

   6. /-rgt-identity 100.0%

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

   7. associate-/r/ 99.7%

      \[\leadsto \color{blue}{\frac{x + x}{\frac{1}{x - y}}} \]

   8. div-inv 99.8%

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

   9. flip-+ 0.0%

      \[\leadsto \color{blue}{\frac{x \cdot x - x \cdot x}{x - x}} \cdot \frac{1}{\frac{1}{x - y}} \]

   10. frac-times 0.0%

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

   11. +-inverses 0.0%

       \[\leadsto \frac{\color{blue}{0} \cdot 1}{\left(x - x\right) \cdot \frac{1}{x - y}} \]

   12. metadata-eval 0.0%

       \[\leadsto \frac{\color{blue}{0}}{\left(x - x\right) \cdot \frac{1}{x - y}} \]

   13. +-inverses 0.0%

       \[\leadsto \frac{0}{\color{blue}{0} \cdot \frac{1}{x - y}} \]

   14. sub-neg 0.0%

       \[\leadsto \frac{0}{0 \cdot \frac{1}{\color{blue}{x + \left(-y\right)}}} \]

   15. add-sqr-sqrt 0.0%

       \[\leadsto \frac{0}{0 \cdot \frac{1}{x + \color{blue}{\sqrt{-y} \cdot \sqrt{-y}}}} \]

   16. sqrt-unprod 0.0%

       \[\leadsto \frac{0}{0 \cdot \frac{1}{x + \color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}}} \]

   17. sqr-neg 0.0%

       \[\leadsto \frac{0}{0 \cdot \frac{1}{x + \sqrt{\color{blue}{y \cdot y}}}} \]

   18. sqrt-unprod 0.0%

       \[\leadsto \frac{0}{0 \cdot \frac{1}{x + \color{blue}{\sqrt{y} \cdot \sqrt{y}}}} \]

   19. add-sqr-sqrt 0.0%

       \[\leadsto \frac{0}{0 \cdot \frac{1}{x + \color{blue}{y}}} \]

5. Applied egg-rr 0.0%

   \[\leadsto \color{blue}{\frac{0}{0 \cdot \frac{1}{x + y}}} \]
6. Step-by-step derivation

   1. div0 14.9%

      \[\leadsto \color{blue}{0} \]

7. Simplified 14.9%

   \[\leadsto \color{blue}{0} \]

8. Final simplification 14.9%

   \[\leadsto 0 \]

Developer target: 100.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * 2.0d0) * (x - y)
end function

Java

public static double code(double x, double y) {
	return (x * 2.0) * (x - y);
}

Python

def code(x, y):
	return (x * 2.0) * (x - y)

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = (x * 2.0) * (x - y);
end

Wolfram

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

Reproduce

herbie shell --seed 2023221 
(FPCore (x y)
  :name "Linear.Matrix:fromQuaternion from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (* (* x 2.0) (- x y))

  (* 2.0 (- (* x x) (* x y))))