Linear.Matrix:fromQuaternion from linear-1.19.1.3, A

Percentage Accurate: 95.7% → 100.0%

Time: 6.4s

Alternatives: 5

Speedup: 1.4×

• 28.1% 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: 95.7% 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, 1.4× 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 95.3%

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

   1. distribute-lft-out-- N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

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

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

   5. --lowering--.f64 N/A

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

   6. *-commutative N/A

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

   7. *-lowering-*.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 83.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot -2\right)\\ \mathbf{if}\;y \leq -5.5 \cdot 10^{+86}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x -2.0))))
   (if (<= y -5.5e+86) t_0 (if (<= y 7.4e-9) (* x (* x 2.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = y * (x * -2.0);
	double tmp;
	if (y <= -5.5e+86) {
		tmp = t_0;
	} else if (y <= 7.4e-9) {
		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 <= (-5.5d+86)) then
        tmp = t_0
    else if (y <= 7.4d-9) 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 <= -5.5e+86) {
		tmp = t_0;
	} else if (y <= 7.4e-9) {
		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 <= -5.5e+86:
		tmp = t_0
	elif y <= 7.4e-9:
		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 <= -5.5e+86)
		tmp = t_0;
	elseif (y <= 7.4e-9)
		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 <= -5.5e+86)
		tmp = t_0;
	elseif (y <= 7.4e-9)
		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, -5.5e+86], t$95$0, If[LessEqual[y, 7.4e-9], N[(x * N[(x * 2.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.5000000000000002e86 or 7.4e-9 < y

   1. Initial program 89.2%

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

   3. Taylor expanded in x around 0

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{-2 \cdot \left(x \cdot y\right) + 0} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot -2} + 0 \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot -2\right)} + 0 \]

      4. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-2 \cdot y\right)} + 0 \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, -2 \cdot y, 0\right)} \]

      6. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{-2 \cdot y + 0}, 0\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot -2} + 0, 0\right) \]

      8. accelerator-lowering-fma.f64 84.3

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(y, -2, 0\right)}, 0\right) \]

   5. Simplified 84.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(y, -2, 0\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot \left(y \cdot -2 + 0\right)} \]

      2. +-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 85.1

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

   7. Applied egg-rr 85.1%

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

   if -5.5000000000000002e86 < y < 7.4e-9

   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

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

      1. +-rgt-identity N/A

         \[\leadsto 2 \cdot \color{blue}{\left({x}^{2} + 0\right)} \]

      2. unpow2 N/A

         \[\leadsto 2 \cdot \left(\color{blue}{x \cdot x} + 0\right) \]

      3. accelerator-lowering-fma.f64 85.4

         \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]

   5. Simplified 85.4%

      \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

      5. *-lowering-*.f64 85.4

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

   7. Applied egg-rr 85.4%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+86}:\\ \;\;\;\;y \cdot \left(x \cdot -2\right)\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.3%

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

3. Taylor expanded in x around 0

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

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{-2 \cdot \left(x \cdot y\right) + 0} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot -2} + 0 \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{x \cdot \left(y \cdot -2\right)} + 0 \]

   4. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(-2 \cdot y\right)} + 0 \]

   5. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -2 \cdot y, 0\right)} \]

   6. +-rgt-identity N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{-2 \cdot y + 0}, 0\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot -2} + 0, 0\right) \]

   8. accelerator-lowering-fma.f64 52.0

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(y, -2, 0\right)}, 0\right) \]

5. Simplified 52.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(y, -2, 0\right), 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{x \cdot \left(y \cdot -2 + 0\right)} \]

   2. +-rgt-identity N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 52.4

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

7. Applied egg-rr 52.4%

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

8. Final simplification 52.4%

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

Alternative 4: 4.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ x \cdot 2 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.3%

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

3. Taylor expanded in x around inf

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

   1. +-rgt-identity N/A

      \[\leadsto 2 \cdot \color{blue}{\left({x}^{2} + 0\right)} \]

   2. unpow2 N/A

      \[\leadsto 2 \cdot \left(\color{blue}{x \cdot x} + 0\right) \]

   3. accelerator-lowering-fma.f64 60.8

      \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]

5. Simplified 60.8%

   \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

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

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

   5. *-lowering-*.f64 60.8

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

7. Applied egg-rr 60.8%

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

9. Applied egg-rr 4.5%

   \[\leadsto \color{blue}{2} \cdot x \]

10. Final simplification 4.5%

    \[\leadsto x \cdot 2 \]
11. Add Preprocessing

Alternative 5: 3.8% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 2.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 2.0

Julia

function code(x, y)
	return 2.0
end

MATLAB

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

Wolfram

code[x_, y_] := 2.0

Derivation

1. Initial program 95.3%

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

3. Taylor expanded in x around inf

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

   1. +-rgt-identity N/A

      \[\leadsto 2 \cdot \color{blue}{\left({x}^{2} + 0\right)} \]

   2. unpow2 N/A

      \[\leadsto 2 \cdot \left(\color{blue}{x \cdot x} + 0\right) \]

   3. accelerator-lowering-fma.f64 60.8

      \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]

5. Simplified 60.8%

   \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

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

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

   5. *-lowering-*.f64 60.8

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

7. Applied egg-rr 60.8%

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

9. Applied egg-rr 3.8%

   \[\leadsto \color{blue}{2} \]
10. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.4× 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 2024195 
(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))))