Linear.Matrix:fromQuaternion from linear-1.19.1.3, B

Percentage Accurate: 95.5% → 100.0%

Time: 3.2s

Alternatives: 4

Speedup: 1.3×

• 27.6% 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: 95.5% 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 93.3%

   \[2 \cdot \left(x \cdot x + x \cdot y\right) \]
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. Final simplification 100.0%

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

Alternative 2: 77.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{-108} \lor \neg \left(x \leq 7.2 \cdot 10^{+58}\right):\\ \;\;\;\;2 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.05e-108) (not (<= x 7.2e+58)))
   (* 2.0 (* x x))
   (* 2.0 (* x y))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.05e-108) || !(x <= 7.2e+58)) {
		tmp = 2.0 * (x * x);
	} else {
		tmp = 2.0 * (x * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-2.05d-108)) .or. (.not. (x <= 7.2d+58))) then
        tmp = 2.0d0 * (x * x)
    else
        tmp = 2.0d0 * (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -2.05e-108) || !(x <= 7.2e+58)) {
		tmp = 2.0 * (x * x);
	} else {
		tmp = 2.0 * (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.05e-108) or not (x <= 7.2e+58):
		tmp = 2.0 * (x * x)
	else:
		tmp = 2.0 * (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.05e-108) || !(x <= 7.2e+58))
		tmp = Float64(2.0 * Float64(x * x));
	else
		tmp = Float64(2.0 * Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2.05e-108) || ~((x <= 7.2e+58)))
		tmp = 2.0 * (x * x);
	else
		tmp = 2.0 * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.05e-108], N[Not[LessEqual[x, 7.2e+58]], $MachinePrecision]], N[(2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.05000000000000018e-108 or 7.19999999999999993e58 < x

   1. Initial program 87.6%

      \[2 \cdot \left(x \cdot x + x \cdot y\right) \]
   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. Taylor expanded in x around inf 79.0%

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

      1. unpow2 79.0%

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

   6. Simplified 79.0%

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

   if -2.05000000000000018e-108 < x < 7.19999999999999993e58

   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. Taylor expanded in x around 0 87.0%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{-108} \lor \neg \left(x \leq 7.2 \cdot 10^{+58}\right):\\ \;\;\;\;2 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 3: 77.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-112} \lor \neg \left(x \leq 5.4 \cdot 10^{+61}\right):\\ \;\;\;\;2 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(2 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -2.15e-112) (not (<= x 5.4e+61)))
   (* 2.0 (* x x))
   (* y (* 2.0 x))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -2.15e-112) || !(x <= 5.4e+61)) {
		tmp = 2.0 * (x * x);
	} else {
		tmp = y * (2.0 * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-2.15d-112)) .or. (.not. (x <= 5.4d+61))) then
        tmp = 2.0d0 * (x * x)
    else
        tmp = y * (2.0d0 * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -2.15e-112) || !(x <= 5.4e+61)) {
		tmp = 2.0 * (x * x);
	} else {
		tmp = y * (2.0 * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -2.15e-112) or not (x <= 5.4e+61):
		tmp = 2.0 * (x * x)
	else:
		tmp = y * (2.0 * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -2.15e-112) || !(x <= 5.4e+61))
		tmp = Float64(2.0 * Float64(x * x));
	else
		tmp = Float64(y * Float64(2.0 * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -2.15e-112) || ~((x <= 5.4e+61)))
		tmp = 2.0 * (x * x);
	else
		tmp = y * (2.0 * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -2.15e-112], N[Not[LessEqual[x, 5.4e+61]], $MachinePrecision]], N[(2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(y * N[(2.0 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1499999999999999e-112 or 5.4000000000000004e61 < x

   1. Initial program 87.6%

      \[2 \cdot \left(x \cdot x + x \cdot y\right) \]
   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. Taylor expanded in x around inf 79.0%

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

      1. unpow2 79.0%

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

   6. Simplified 79.0%

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

   if -2.1499999999999999e-112 < x < 5.4000000000000004e61

   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. Step-by-step derivation

      1. associate-*r* 100.0%

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

      2. flip-+ 72.5%

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

      3. associate-*r/ 64.3%

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

   5. Applied egg-rr 64.3%

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

   6. Taylor expanded in x around 0 87.0%

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

      1. associate-*r* 87.0%

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

   8. Simplified 87.0%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-112} \lor \neg \left(x \leq 5.4 \cdot 10^{+61}\right):\\ \;\;\;\;2 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(2 \cdot x\right)\\ \end{array} \]

Alternative 4: 58.7% 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 93.3%

   \[2 \cdot \left(x \cdot x + x \cdot y\right) \]
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. Taylor expanded in x around inf 58.0%

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

   1. unpow2 58.0%

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

6. Simplified 58.0%

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

7. Final simplification 58.0%

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

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 2023271 
(FPCore (x y)
  :name "Linear.Matrix:fromQuaternion from linear-1.19.1.3, B"
  :precision binary64

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

  (* 2.0 (+ (* x x) (* x y))))