Linear.Matrix:fromQuaternion from linear-1.19.1.3, A

Percentage Accurate: 95.4% → 100.0%

Time: 4.5s

Alternatives: 3

Speedup: 1.3×

• 27.9% 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: 95.4% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{2 \cdot \left(x \cdot \left(x - y\right)\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 78.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-50} \lor \neg \left(x \leq 1.35 \cdot 10^{-88}\right):\\ \;\;\;\;2 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.7e-50) (not (<= x 1.35e-88)))
   (* 2.0 (* x x))
   (* y (* x -2.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.7e-50) || !(x <= 1.35e-88)) {
		tmp = 2.0 * (x * x);
	} else {
		tmp = y * (x * -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.7d-50)) .or. (.not. (x <= 1.35d-88))) then
        tmp = 2.0d0 * (x * x)
    else
        tmp = y * (x * (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.7e-50) || !(x <= 1.35e-88)) {
		tmp = 2.0 * (x * x);
	} else {
		tmp = y * (x * -2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.7e-50) or not (x <= 1.35e-88):
		tmp = 2.0 * (x * x)
	else:
		tmp = y * (x * -2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.7e-50) || !(x <= 1.35e-88))
		tmp = Float64(2.0 * Float64(x * x));
	else
		tmp = Float64(y * Float64(x * -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.7e-50) || ~((x <= 1.35e-88)))
		tmp = 2.0 * (x * x);
	else
		tmp = y * (x * -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.7e-50], N[Not[LessEqual[x, 1.35e-88]], $MachinePrecision]], N[(2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(y * N[(x * -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.70000000000000007e-50 or 1.34999999999999997e-88 < x

   1. Initial program 91.0%

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

      1. distribute-lft-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 81.8%

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

   if -1.70000000000000007e-50 < x < 1.34999999999999997e-88

   1. Initial program 100.0%

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

      1. distribute-lft-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 90.3%

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

      1. associate-*r* 90.3%

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

      2. *-commutative 90.3%

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

   7. Simplified 90.3%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-50} \lor \neg \left(x \leq 1.35 \cdot 10^{-88}\right):\\ \;\;\;\;2 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(2.0 * N[(x * x), $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%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around inf 63.3%

   \[\leadsto 2 \cdot \left(x \cdot \color{blue}{x}\right) \]
6. Add Preprocessing

Developer Target 1: 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 2024185 
(FPCore (x y)
  :name "Linear.Matrix:fromQuaternion from linear-1.19.1.3, A"
  :precision binary64

  :alt
  (! :herbie-platform default (* (* x 2) (- x y)))

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