Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, G

Percentage Accurate: 100.0% → 100.0%

Time: 4.6s

Alternatives: 3

Speedup: 2.2×

Specification

Math

\[\begin{array}{l} \\ \frac{1}{2} \cdot \left(x + y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (/ 1.0 2.0) (+ x y)))

C

double code(double x, double y) {
	return (1.0 / 2.0) * (x + y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 / 2.0d0) * (x + y)
end function

Java

public static double code(double x, double y) {
	return (1.0 / 2.0) * (x + y);
}

Python

def code(x, y):
	return (1.0 / 2.0) * (x + y)

Julia

function code(x, y)
	return Float64(Float64(1.0 / 2.0) * Float64(x + y))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 / 2.0) * (x + y);
end

Wolfram

code[x_, y_] := N[(N[(1.0 / 2.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2} \cdot \left(x + y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (/ 1.0 2.0) (+ x y)))

C

double code(double x, double y) {
	return (1.0 / 2.0) * (x + y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 / 2.0d0) * (x + y)
end function

Java

public static double code(double x, double y) {
	return (1.0 / 2.0) * (x + y);
}

Python

def code(x, y):
	return (1.0 / 2.0) * (x + y)

Julia

function code(x, y)
	return Float64(Float64(1.0 / 2.0) * Float64(x + y))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 / 2.0) * (x + y);
end

Wolfram

code[x_, y_] := N[(N[(1.0 / 2.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(y + x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* 0.5 (+ y x)))

C

double code(double x, double y) {
	return 0.5 * (y + x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.5d0 * (y + x)
end function

Java

public static double code(double x, double y) {
	return 0.5 * (y + x);
}

Python

def code(x, y):
	return 0.5 * (y + x)

Julia

function code(x, y)
	return Float64(0.5 * Float64(y + x))
end

MATLAB

function tmp = code(x, y)
	tmp = 0.5 * (y + x);
end

Wolfram

code[x_, y_] := N[(0.5 * N[(y + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{1}{2} \cdot \left(x + y\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot x + \frac{1}{2} \cdot y} \]
4. Step-by-step derivation

   1. *-lft-identity N/A

      \[\leadsto \frac{1}{2} \cdot x + \frac{1}{2} \cdot \color{blue}{\left(1 \cdot y\right)} \]

   2. *-inverses N/A

      \[\leadsto \frac{1}{2} \cdot x + \frac{1}{2} \cdot \left(\color{blue}{\frac{x}{x}} \cdot y\right) \]

   3. associate-*l/ N/A

      \[\leadsto \frac{1}{2} \cdot x + \frac{1}{2} \cdot \color{blue}{\frac{x \cdot y}{x}} \]

   4. associate-*r/ N/A

      \[\leadsto \frac{1}{2} \cdot x + \frac{1}{2} \cdot \color{blue}{\left(x \cdot \frac{y}{x}\right)} \]

   5. distribute-lft-out N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(x + x \cdot \frac{y}{x}\right)} \]

   6. *-rgt-identity N/A

      \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{x \cdot 1} + x \cdot \frac{y}{x}\right) \]

   7. distribute-lft-in N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot \left(1 + \frac{y}{x}\right)\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(x \cdot \left(1 + \frac{y}{x}\right)\right)} \]

   9. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot \color{blue}{\left(\frac{y}{x} + 1\right)}\right) \]

   10. distribute-lft-in N/A

       \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot \frac{y}{x} + x \cdot 1\right)} \]

   11. associate-*r/ N/A

       \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\frac{x \cdot y}{x}} + x \cdot 1\right) \]

   12. associate-*l/ N/A

       \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\frac{x}{x} \cdot y} + x \cdot 1\right) \]

   13. *-inverses N/A

       \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{1} \cdot y + x \cdot 1\right) \]

   14. *-lft-identity N/A

       \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{y} + x \cdot 1\right) \]

   15. *-rgt-identity N/A

       \[\leadsto \frac{1}{2} \cdot \left(y + \color{blue}{x}\right) \]

   16. lower-+.f64 100.0

       \[\leadsto 0.5 \cdot \color{blue}{\left(y + x\right)} \]

5. Simplified 100.0%

   \[\leadsto \color{blue}{0.5 \cdot \left(y + x\right)} \]
6. Add Preprocessing

Alternative 2: 51.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + x \leq -2 \cdot 10^{-272}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ y x) -2e-272) (* 0.5 x) (* 0.5 y)))

C

double code(double x, double y) {
	double tmp;
	if ((y + x) <= -2e-272) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y + x) <= (-2d-272)) then
        tmp = 0.5d0 * x
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y + x) <= -2e-272) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y + x) <= -2e-272:
		tmp = 0.5 * x
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y + x) <= -2e-272)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y + x) <= -2e-272)
		tmp = 0.5 * x;
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y + x), $MachinePrecision], -2e-272], N[(0.5 * x), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -1.99999999999999986e-272

   1. Initial program 100.0%

      \[\frac{1}{2} \cdot \left(x + y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x} \]
   4. Step-by-step derivation

      1. lower-*.f64 47.2

         \[\leadsto \color{blue}{0.5 \cdot x} \]

   5. Simplified 47.2%

      \[\leadsto \color{blue}{0.5 \cdot x} \]

   if -1.99999999999999986e-272 < (+.f64 x y)

   1. Initial program 100.0%

      \[\frac{1}{2} \cdot \left(x + y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 48.4

         \[\leadsto \color{blue}{0.5 \cdot y} \]

   5. Simplified 48.4%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + x \leq -2 \cdot 10^{-272}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* 0.5 x))

C

double code(double x, double y) {
	return 0.5 * x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.5d0 * x
end function

Java

public static double code(double x, double y) {
	return 0.5 * x;
}

Python

def code(x, y):
	return 0.5 * x

Julia

function code(x, y)
	return Float64(0.5 * x)
end

MATLAB

function tmp = code(x, y)
	tmp = 0.5 * x;
end

Wolfram

code[x_, y_] := N[(0.5 * x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{1}{2} \cdot \left(x + y\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot x} \]
4. Step-by-step derivation

   1. lower-*.f64 50.4

      \[\leadsto \color{blue}{0.5 \cdot x} \]

5. Simplified 50.4%

   \[\leadsto \color{blue}{0.5 \cdot x} \]
6. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{x + y}{2} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (+ x y) 2.0))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return (x + y) / 2.0

Julia

function code(x, y)
	return Float64(Float64(x + y) / 2.0)
end

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024215 
(FPCore (x y)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, G"
  :precision binary64

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

  (* (/ 1.0 2.0) (+ x y)))