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

Percentage Accurate: 100.0% → 100.0%

Time: 4.1s

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\frac{1}{2} \cdot \left(x + y\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 100.0

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

   5. lift-/.f64 N/A

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

   6. metadata-eval 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 50.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-249}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ x y) -1e-249) (* x 0.5) (* y 0.5)))

C

double code(double x, double y) {
	double tmp;
	if ((x + y) <= -1e-249) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x + y) <= (-1d-249)) then
        tmp = x * 0.5d0
    else
        tmp = y * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x + y) <= -1e-249) {
		tmp = x * 0.5;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x + y) <= -1e-249:
		tmp = x * 0.5
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x + y) <= -1e-249)
		tmp = Float64(x * 0.5);
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x + y) <= -1e-249)
		tmp = x * 0.5;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x + y), $MachinePrecision], -1e-249], N[(x * 0.5), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -1.00000000000000005e-249

   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 57.5

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

   5. Applied rewrites 57.5%

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

   if -1.00000000000000005e-249 < (+.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 54.5

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

   5. Applied rewrites 54.5%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{-249}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(x * 0.5), $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 52.0

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

5. Applied rewrites 52.0%

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

6. Final simplification 52.0%

   \[\leadsto x \cdot 0.5 \]
7. 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 2024219 
(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)))