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

Percentage Accurate: 100.0% → 100.0%

Time: 6.0s

Alternatives: 6

Speedup: 1.0×

• 21.0% 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: 74.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + z \leq -5 \cdot 10^{+120}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;1 + z \leq 0.9998450520860446:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{elif}\;1 + z \leq 50000:\\ \;\;\;\;y + x\\ \mathbf{elif}\;1 + z \leq 10^{+142}:\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ 1.0 z) -5e+120)
   (* z y)
   (if (<= (+ 1.0 z) 0.9998450520860446)
     (fma z x x)
     (if (<= (+ 1.0 z) 50000.0)
       (+ y x)
       (if (<= (+ 1.0 z) 1e+142) (* z y) (* z x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((1.0 + z) <= -5e+120) {
		tmp = z * y;
	} else if ((1.0 + z) <= 0.9998450520860446) {
		tmp = fma(z, x, x);
	} else if ((1.0 + z) <= 50000.0) {
		tmp = y + x;
	} else if ((1.0 + z) <= 1e+142) {
		tmp = z * y;
	} else {
		tmp = z * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(1.0 + z) <= -5e+120)
		tmp = Float64(z * y);
	elseif (Float64(1.0 + z) <= 0.9998450520860446)
		tmp = fma(z, x, x);
	elseif (Float64(1.0 + z) <= 50000.0)
		tmp = Float64(y + x);
	elseif (Float64(1.0 + z) <= 1e+142)
		tmp = Float64(z * y);
	else
		tmp = Float64(z * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(1.0 + z), $MachinePrecision], -5e+120], N[(z * y), $MachinePrecision], If[LessEqual[N[(1.0 + z), $MachinePrecision], 0.9998450520860446], N[(z * x + x), $MachinePrecision], If[LessEqual[N[(1.0 + z), $MachinePrecision], 50000.0], N[(y + x), $MachinePrecision], If[LessEqual[N[(1.0 + z), $MachinePrecision], 1e+142], N[(z * y), $MachinePrecision], N[(z * x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 z #s(literal 1 binary64)) < -5.00000000000000019e120 or 5e4 < (+.f64 z #s(literal 1 binary64)) < 1.00000000000000005e142

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 62.8%

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

   if -5.00000000000000019e120 < (+.f64 z #s(literal 1 binary64)) < 0.999845052086044572

   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-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 73.5

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

   5. Applied rewrites 73.5%

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

   if 0.999845052086044572 < (+.f64 z #s(literal 1 binary64)) < 5e4

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 4.1

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

   5. Applied rewrites 4.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 3.7%

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

   9. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. lower-+.f64 98.3

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

   11. Applied rewrites 98.3%

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

   if 1.00000000000000005e142 < (+.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 z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 55.6%

      \[\leadsto x \cdot \color{blue}{z} \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + z \leq -5 \cdot 10^{+120}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;1 + z \leq 0.9998450520860446:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{elif}\;1 + z \leq 50000:\\ \;\;\;\;y + x\\ \mathbf{elif}\;1 + z \leq 10^{+142}:\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + z \leq -20000:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;1 + z \leq 50000:\\ \;\;\;\;y + x\\ \mathbf{elif}\;1 + z \leq 10^{+142}:\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ 1.0 z) -20000.0)
   (* z y)
   (if (<= (+ 1.0 z) 50000.0)
     (+ y x)
     (if (<= (+ 1.0 z) 1e+142) (* z y) (* z x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((1.0 + z) <= -20000.0) {
		tmp = z * y;
	} else if ((1.0 + z) <= 50000.0) {
		tmp = y + x;
	} else if ((1.0 + z) <= 1e+142) {
		tmp = z * y;
	} else {
		tmp = z * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((1.0d0 + z) <= (-20000.0d0)) then
        tmp = z * y
    else if ((1.0d0 + z) <= 50000.0d0) then
        tmp = y + x
    else if ((1.0d0 + z) <= 1d+142) then
        tmp = z * y
    else
        tmp = z * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((1.0 + z) <= -20000.0) {
		tmp = z * y;
	} else if ((1.0 + z) <= 50000.0) {
		tmp = y + x;
	} else if ((1.0 + z) <= 1e+142) {
		tmp = z * y;
	} else {
		tmp = z * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (1.0 + z) <= -20000.0:
		tmp = z * y
	elif (1.0 + z) <= 50000.0:
		tmp = y + x
	elif (1.0 + z) <= 1e+142:
		tmp = z * y
	else:
		tmp = z * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(1.0 + z) <= -20000.0)
		tmp = Float64(z * y);
	elseif (Float64(1.0 + z) <= 50000.0)
		tmp = Float64(y + x);
	elseif (Float64(1.0 + z) <= 1e+142)
		tmp = Float64(z * y);
	else
		tmp = Float64(z * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((1.0 + z) <= -20000.0)
		tmp = z * y;
	elseif ((1.0 + z) <= 50000.0)
		tmp = y + x;
	elseif ((1.0 + z) <= 1e+142)
		tmp = z * y;
	else
		tmp = z * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(1.0 + z), $MachinePrecision], -20000.0], N[(z * y), $MachinePrecision], If[LessEqual[N[(1.0 + z), $MachinePrecision], 50000.0], N[(y + x), $MachinePrecision], If[LessEqual[N[(1.0 + z), $MachinePrecision], 1e+142], N[(z * y), $MachinePrecision], N[(z * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 z #s(literal 1 binary64)) < -2e4 or 5e4 < (+.f64 z #s(literal 1 binary64)) < 1.00000000000000005e142

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.8%

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

   if -2e4 < (+.f64 z #s(literal 1 binary64)) < 5e4

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 4.1

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

   5. Applied rewrites 4.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 3.6%

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

   9. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. lower-+.f64 97.8

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

   11. Applied rewrites 97.8%

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

   if 1.00000000000000005e142 < (+.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 z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 55.6%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + z \leq -20000:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;1 + z \leq 50000:\\ \;\;\;\;y + x\\ \mathbf{elif}\;1 + z \leq 10^{+142}:\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + z \leq -20000:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;1 + z \leq 2:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ 1.0 z) -20000.0) (* z x) (if (<= (+ 1.0 z) 2.0) (+ y x) (* z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((1.0 + z) <= -20000.0) {
		tmp = z * x;
	} else if ((1.0 + z) <= 2.0) {
		tmp = y + x;
	} else {
		tmp = z * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((1.0d0 + z) <= (-20000.0d0)) then
        tmp = z * x
    else if ((1.0d0 + z) <= 2.0d0) then
        tmp = y + x
    else
        tmp = z * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((1.0 + z) <= -20000.0) {
		tmp = z * x;
	} else if ((1.0 + z) <= 2.0) {
		tmp = y + x;
	} else {
		tmp = z * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (1.0 + z) <= -20000.0:
		tmp = z * x
	elif (1.0 + z) <= 2.0:
		tmp = y + x
	else:
		tmp = z * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(1.0 + z) <= -20000.0)
		tmp = Float64(z * x);
	elseif (Float64(1.0 + z) <= 2.0)
		tmp = Float64(y + x);
	else
		tmp = Float64(z * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((1.0 + z) <= -20000.0)
		tmp = z * x;
	elseif ((1.0 + z) <= 2.0)
		tmp = y + x;
	else
		tmp = z * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(1.0 + z), $MachinePrecision], -20000.0], N[(z * x), $MachinePrecision], If[LessEqual[N[(1.0 + z), $MachinePrecision], 2.0], N[(y + x), $MachinePrecision], N[(z * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 z #s(literal 1 binary64)) < -2e4 or 2 < (+.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 z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 98.2

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 51.8%

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

   if -2e4 < (+.f64 z #s(literal 1 binary64)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 3.8

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

   5. Applied rewrites 3.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 3.4%

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

   9. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. lower-+.f64 98.5

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

   11. Applied rewrites 98.5%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + z \leq -20000:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;1 + z \leq 2:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + x \leq -1 \cdot 10^{-247}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ y x) -1e-247) (fma z x x) (fma z y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y + x) <= -1e-247) {
		tmp = fma(z, x, x);
	} else {
		tmp = fma(z, y, y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(y + x) <= -1e-247)
		tmp = fma(z, x, x);
	else
		tmp = fma(z, y, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(y + x), $MachinePrecision], -1e-247], N[(z * x + x), $MachinePrecision], N[(z * y + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -1e-247

   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-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 50.4

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

   5. Applied rewrites 50.4%

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

   if -1e-247 < (+.f64 x y)

   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 51.3

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

   5. Applied rewrites 51.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, y, y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + x \leq -1 \cdot 10^{-247}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return 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 = y + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in z around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 47.0

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

5. Applied rewrites 47.0%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 25.5%

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

9. Taylor expanded in z around 0

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

    1. +-commutative N/A

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

    2. lower-+.f64 55.1

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

11. Applied rewrites 55.1%

    \[\leadsto \color{blue}{y + x} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024298 
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, G"
  :precision binary64
  (* (+ x y) (+ z 1.0)))