Linear.Matrix:fromQuaternion from linear-1.19.1.3, A

Percentage Accurate: 95.1% → 100.0%

Time: 7.6s

Alternatives: 3

Speedup: 1.3×

• 28.9% 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: 95.1% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.4%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 84.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot -2\right)\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{-30}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.165:\\ \;\;\;\;x \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x -2.0))))
   (if (<= y -2.3e-30) t_0 (if (<= y 0.165) (* x (* x 2.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = y * (x * -2.0);
	double tmp;
	if (y <= -2.3e-30) {
		tmp = t_0;
	} else if (y <= 0.165) {
		tmp = x * (x * 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (x * (-2.0d0))
    if (y <= (-2.3d-30)) then
        tmp = t_0
    else if (y <= 0.165d0) then
        tmp = x * (x * 2.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (x * -2.0);
	double tmp;
	if (y <= -2.3e-30) {
		tmp = t_0;
	} else if (y <= 0.165) {
		tmp = x * (x * 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (x * -2.0)
	tmp = 0
	if y <= -2.3e-30:
		tmp = t_0
	elif y <= 0.165:
		tmp = x * (x * 2.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * -2.0))
	tmp = 0.0
	if (y <= -2.3e-30)
		tmp = t_0;
	elseif (y <= 0.165)
		tmp = Float64(x * Float64(x * 2.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (x * -2.0);
	tmp = 0.0;
	if (y <= -2.3e-30)
		tmp = t_0;
	elseif (y <= 0.165)
		tmp = x * (x * 2.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x * -2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.3e-30], t$95$0, If[LessEqual[y, 0.165], N[(x * N[(x * 2.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.29999999999999984e-30 or 0.165000000000000008 < y

   1. Initial program 92.7%

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

   3. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   if -2.29999999999999984e-30 < y < 0.165000000000000008

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 59.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.4%

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

3. Taylor expanded in x around inf 0

   \[\leadsto expr\]

5. Simplified 0

   \[\leadsto expr\]
6. Add Preprocessing

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 2024110 
(FPCore (x y)
  :name "Linear.Matrix:fromQuaternion from linear-1.19.1.3, A"
  :precision binary64

  :alt
  (* (* x 2.0) (- x y))

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