Linear.Matrix:fromQuaternion from linear-1.19.1.3, B

Percentage Accurate: 95.4% → 100.0%

Time: 7.4s

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: 95.4% 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

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

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 95.7%

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(x \cdot x + x \cdot y\right)}\right) \]

   2. distribute-lft-out N/A

      \[\leadsto \mathsf{*.f64}\left(2, \left(x \cdot \color{blue}{\left(x + y\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{\left(x + y\right)}\right)\right) \]

   4. +-lowering-+.f64 100.0%

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

3. Simplified 100.0%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(x + y\right), \color{blue}{\left(2 \cdot x\right)}\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \left(\color{blue}{2} \cdot x\right)\right) \]

   5. *-lowering-*.f64 100.0%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, y\right), \mathsf{*.f64}\left(2, \color{blue}{x}\right)\right) \]

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

   \[\leadsto \left(x + y\right) \cdot \left(x \cdot 2\right) \]
8. Add Preprocessing

Alternative 2: 83.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(x \cdot y\right)\\ \mathbf{if}\;y \leq -4.5 \cdot 10^{+77}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1150000000000:\\ \;\;\;\;2 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 2.0 (* x y))))
   (if (<= y -4.5e+77) t_0 (if (<= y 1150000000000.0) (* 2.0 (* x x)) t_0))))

C

double code(double x, double y) {
	double t_0 = 2.0 * (x * y);
	double tmp;
	if (y <= -4.5e+77) {
		tmp = t_0;
	} else if (y <= 1150000000000.0) {
		tmp = 2.0 * (x * x);
	} 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 = 2.0d0 * (x * y)
    if (y <= (-4.5d+77)) then
        tmp = t_0
    else if (y <= 1150000000000.0d0) then
        tmp = 2.0d0 * (x * x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 2.0 * (x * y);
	double tmp;
	if (y <= -4.5e+77) {
		tmp = t_0;
	} else if (y <= 1150000000000.0) {
		tmp = 2.0 * (x * x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 2.0 * (x * y)
	tmp = 0
	if y <= -4.5e+77:
		tmp = t_0
	elif y <= 1150000000000.0:
		tmp = 2.0 * (x * x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(2.0 * Float64(x * y))
	tmp = 0.0
	if (y <= -4.5e+77)
		tmp = t_0;
	elseif (y <= 1150000000000.0)
		tmp = Float64(2.0 * Float64(x * x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 2.0 * (x * y);
	tmp = 0.0;
	if (y <= -4.5e+77)
		tmp = t_0;
	elseif (y <= 1150000000000.0)
		tmp = 2.0 * (x * x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.5e+77], t$95$0, If[LessEqual[y, 1150000000000.0], N[(2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.50000000000000024e77 or 1.15e12 < y

   1. Initial program 89.0%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(x \cdot x + x \cdot y\right)}\right) \]

      2. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(x \cdot \color{blue}{\left(x + y\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{\left(x + y\right)}\right)\right) \]

      4. +-lowering-+.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   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

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(x \cdot y\right)}\right) \]

      2. *-lowering-*.f64 88.2%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right) \]

   7. Simplified 88.2%

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

   if -4.50000000000000024e77 < y < 1.15e12

   1. Initial program 99.9%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(x \cdot x + x \cdot y\right)}\right) \]

      2. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(x \cdot \color{blue}{\left(x + y\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{\left(x + y\right)}\right)\right) \]

      4. +-lowering-+.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(2, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 85.9%

         \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   7. Simplified 85.9%

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

Alternative 3: 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 95.7%

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(x \cdot x + x \cdot y\right)}\right) \]

   2. distribute-lft-out N/A

      \[\leadsto \mathsf{*.f64}\left(2, \left(x \cdot \color{blue}{\left(x + y\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{\left(x + y\right)}\right)\right) \]

   4. +-lowering-+.f64 100.0%

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

3. Simplified 100.0%

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

Alternative 4: 58.5% 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 95.7%

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left(x \cdot x + x \cdot y\right)}\right) \]

   2. distribute-lft-out N/A

      \[\leadsto \mathsf{*.f64}\left(2, \left(x \cdot \color{blue}{\left(x + y\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{\left(x + y\right)}\right)\right) \]

   4. +-lowering-+.f64 100.0%

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(x, \color{blue}{y}\right)\right)\right) \]

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

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \color{blue}{\left({x}^{2}\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(2, \left(x \cdot \color{blue}{x}\right)\right) \]

   3. *-lowering-*.f64 64.0%

      \[\leadsto \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

7. Simplified 64.0%

   \[\leadsto \color{blue}{2 \cdot \left(x \cdot x\right)} \]
8. 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 2024161 
(FPCore (x y)
  :name "Linear.Matrix:fromQuaternion from linear-1.19.1.3, B"
  :precision binary64

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

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