Optimisation.CirclePacking:place from circle-packing-0.1.0.4, G

Percentage Accurate: 100.0% → 100.0%

Time: 4.8s

Alternatives: 6

Speedup: 1.0×

• 21.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (x y z) :precision binary64 (* (+ x y) (+ z 1.0)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (* (+ x y) (+ z 1.0)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (* (+ 1.0 z) (+ y x)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

   \[\leadsto \left(1 + z\right) \cdot \left(y + x\right) \]
4. Add Preprocessing

Alternative 2: 75.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + z \leq -1 \cdot 10^{+28}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;1 + z \leq 0.999999:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{elif}\;1 + z \leq 1.0002:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ 1.0 z) -1e+28)
   (* z y)
   (if (<= (+ 1.0 z) 0.999999)
     (fma x z x)
     (if (<= (+ 1.0 z) 1.0002) (+ y x) (fma x z x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((1.0 + z) <= -1e+28) {
		tmp = z * y;
	} else if ((1.0 + z) <= 0.999999) {
		tmp = fma(x, z, x);
	} else if ((1.0 + z) <= 1.0002) {
		tmp = y + x;
	} else {
		tmp = fma(x, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(1.0 + z) <= -1e+28)
		tmp = Float64(z * y);
	elseif (Float64(1.0 + z) <= 0.999999)
		tmp = fma(x, z, x);
	elseif (Float64(1.0 + z) <= 1.0002)
		tmp = Float64(y + x);
	else
		tmp = fma(x, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(1.0 + z), $MachinePrecision], -1e+28], N[(z * y), $MachinePrecision], If[LessEqual[N[(1.0 + z), $MachinePrecision], 0.999999], N[(x * z + x), $MachinePrecision], If[LessEqual[N[(1.0 + z), $MachinePrecision], 1.0002], N[(y + x), $MachinePrecision], N[(x * z + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 z #s(literal 1 binary64)) < -9.99999999999999958e27

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y \cdot \left(1 + z\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{z \cdot y + 1 \cdot y} \]

      3. *-lft-identity N/A

         \[\leadsto z \cdot y + \color{blue}{y} \]

      4. lower-fma.f64 53.3

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

   5. Applied rewrites 53.3%

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

   6. Taylor expanded in z around inf

      \[\leadsto y \cdot \color{blue}{z} \]
   7. Step-by-step derivation

   8. Applied rewrites 53.3%

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

   if -9.99999999999999958e27 < (+.f64 z #s(literal 1 binary64)) < 0.999998999999999971 or 1.0002 < (+.f64 z #s(literal 1 binary64))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + z\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot z + x \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto x \cdot z + \color{blue}{x} \]

      4. lower-fma.f64 53.5

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

   5. Applied rewrites 53.5%

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

   if 0.999998999999999971 < (+.f64 z #s(literal 1 binary64)) < 1.0002

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. lower-+.f64 99.0

         \[\leadsto \color{blue}{x + y} \]

   5. Applied rewrites 99.0%

      \[\leadsto \color{blue}{x + y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + z \leq -1 \cdot 10^{+28}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;1 + z \leq 0.999999:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{elif}\;1 + z \leq 1.0002:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, G"
  :precision binary64
  (* (+ x y) (+ z 1.0)))