Linear.Matrix:fromQuaternion from linear-1.19.1.3, B

Percentage Accurate: 95.5% → 100.0%

Time: 3.6s

Alternatives: 3

Speedup: 1.3×

• 27.3% 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

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

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 98.4%

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

   1. +-commutative 98.4%

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

   2. distribute-lft-out 100.0%

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

   3. associate-*r* 100.0%

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

   4. *-commutative 100.0%

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

   5. +-commutative 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 57.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

   1. +-commutative 98.4%

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

   2. distribute-lft-out 100.0%

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

   3. associate-*r* 100.0%

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

   4. *-commutative 100.0%

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

   5. +-commutative 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 59.5%

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

6. Final simplification 59.5%

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

Alternative 3: 57.0% accurate, 1.8× 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 98.4%

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

   1. +-commutative 98.4%

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

   2. distribute-lft-out 100.0%

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

   3. associate-*r* 100.0%

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

   4. *-commutative 100.0%

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

   5. +-commutative 100.0%

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

3. Simplified 100.0%

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

   1. *-commutative 100.0%

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

   2. +-commutative 100.0%

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

   3. distribute-lft-in 98.4%

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

   4. fma-define 98.8%

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

   5. *-commutative 98.8%

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

   6. *-commutative 98.8%

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

6. Applied egg-rr 98.8%

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

7. Taylor expanded in x around 0 59.5%

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

   1. associate-*r* 59.5%

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

   2. *-commutative 59.5%

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

9. Simplified 59.5%

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

10. Final simplification 59.5%

    \[\leadsto y \cdot \left(x \cdot 2\right) \]
11. 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 2024053 
(FPCore (x y)
  :name "Linear.Matrix:fromQuaternion from linear-1.19.1.3, B"
  :precision binary64

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

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