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

Percentage Accurate: 100.0% → 100.0%

Time: 5.8s

Alternatives: 10

Speedup: 1.0×

• 20.5% 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, 0.8× speedup

Math

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

FPCore

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

C

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

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 * (x + y)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. distribute-rgt-in 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. +-commutative 100.0%

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

   4. associate-+r+ 100.0%

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

   5. *-commutative 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 51.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.6:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq -1.95 \cdot 10^{-118}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-137}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq -2.5 \cdot 10^{-275}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-285}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-223}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{-116}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-81}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 0.245:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.6)
   (* x z)
   (if (<= z -1.95e-118)
     x
     (if (<= z -1.2e-137)
       y
       (if (<= z -2.5e-275)
         x
         (if (<= z 2.9e-285)
           y
           (if (<= z 2.3e-223)
             x
             (if (<= z 7.4e-116)
               y
               (if (<= z 7e-81) x (if (<= z 0.245) y (* x z)))))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.6) {
		tmp = x * z;
	} else if (z <= -1.95e-118) {
		tmp = x;
	} else if (z <= -1.2e-137) {
		tmp = y;
	} else if (z <= -2.5e-275) {
		tmp = x;
	} else if (z <= 2.9e-285) {
		tmp = y;
	} else if (z <= 2.3e-223) {
		tmp = x;
	} else if (z <= 7.4e-116) {
		tmp = y;
	} else if (z <= 7e-81) {
		tmp = x;
	} else if (z <= 0.245) {
		tmp = y;
	} else {
		tmp = x * z;
	}
	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 (z <= (-3.6d0)) then
        tmp = x * z
    else if (z <= (-1.95d-118)) then
        tmp = x
    else if (z <= (-1.2d-137)) then
        tmp = y
    else if (z <= (-2.5d-275)) then
        tmp = x
    else if (z <= 2.9d-285) then
        tmp = y
    else if (z <= 2.3d-223) then
        tmp = x
    else if (z <= 7.4d-116) then
        tmp = y
    else if (z <= 7d-81) then
        tmp = x
    else if (z <= 0.245d0) then
        tmp = y
    else
        tmp = x * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.6) {
		tmp = x * z;
	} else if (z <= -1.95e-118) {
		tmp = x;
	} else if (z <= -1.2e-137) {
		tmp = y;
	} else if (z <= -2.5e-275) {
		tmp = x;
	} else if (z <= 2.9e-285) {
		tmp = y;
	} else if (z <= 2.3e-223) {
		tmp = x;
	} else if (z <= 7.4e-116) {
		tmp = y;
	} else if (z <= 7e-81) {
		tmp = x;
	} else if (z <= 0.245) {
		tmp = y;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.6:
		tmp = x * z
	elif z <= -1.95e-118:
		tmp = x
	elif z <= -1.2e-137:
		tmp = y
	elif z <= -2.5e-275:
		tmp = x
	elif z <= 2.9e-285:
		tmp = y
	elif z <= 2.3e-223:
		tmp = x
	elif z <= 7.4e-116:
		tmp = y
	elif z <= 7e-81:
		tmp = x
	elif z <= 0.245:
		tmp = y
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.6)
		tmp = Float64(x * z);
	elseif (z <= -1.95e-118)
		tmp = x;
	elseif (z <= -1.2e-137)
		tmp = y;
	elseif (z <= -2.5e-275)
		tmp = x;
	elseif (z <= 2.9e-285)
		tmp = y;
	elseif (z <= 2.3e-223)
		tmp = x;
	elseif (z <= 7.4e-116)
		tmp = y;
	elseif (z <= 7e-81)
		tmp = x;
	elseif (z <= 0.245)
		tmp = y;
	else
		tmp = Float64(x * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -3.6)
		tmp = x * z;
	elseif (z <= -1.95e-118)
		tmp = x;
	elseif (z <= -1.2e-137)
		tmp = y;
	elseif (z <= -2.5e-275)
		tmp = x;
	elseif (z <= 2.9e-285)
		tmp = y;
	elseif (z <= 2.3e-223)
		tmp = x;
	elseif (z <= 7.4e-116)
		tmp = y;
	elseif (z <= 7e-81)
		tmp = x;
	elseif (z <= 0.245)
		tmp = y;
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.6], N[(x * z), $MachinePrecision], If[LessEqual[z, -1.95e-118], x, If[LessEqual[z, -1.2e-137], y, If[LessEqual[z, -2.5e-275], x, If[LessEqual[z, 2.9e-285], y, If[LessEqual[z, 2.3e-223], x, If[LessEqual[z, 7.4e-116], y, If[LessEqual[z, 7e-81], x, If[LessEqual[z, 0.245], y, N[(x * z), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.60000000000000009 or 0.245 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 48.3%

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

      1. +-commutative 48.3%

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

      2. distribute-lft-in 48.3%

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

      3. *-rgt-identity 48.3%

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

   5. Applied egg-rr 48.3%

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

   6. Taylor expanded in z around inf 48.2%

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

   if -3.60000000000000009 < z < -1.95e-118 or -1.2e-137 < z < -2.49999999999999992e-275 or 2.9e-285 < z < 2.3e-223 or 7.4000000000000005e-116 < z < 6.99999999999999973e-81

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 53.8%

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

   4. Taylor expanded in z around 0 52.1%

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

   if -1.95e-118 < z < -1.2e-137 or -2.49999999999999992e-275 < z < 2.9e-285 or 2.3e-223 < z < 7.4000000000000005e-116 or 6.99999999999999973e-81 < z < 0.245

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 46.8%

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

      1. +-commutative 46.8%

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

      2. distribute-lft-in 46.8%

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

      3. *-rgt-identity 46.8%

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

   5. Applied egg-rr 46.8%

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

   6. Taylor expanded in z around 0 46.8%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq -1.95 \cdot 10^{-118}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-137}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq -2.5 \cdot 10^{-275}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-285}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-223}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{-116}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-81}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 0.245:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -1.85 \cdot 10^{-118}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{-138}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq -1.35 \cdot 10^{-275}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-287}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-223}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4.65 \cdot 10^{-116}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-78}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0)
   (* y z)
   (if (<= z -1.85e-118)
     x
     (if (<= z -1.65e-138)
       y
       (if (<= z -1.35e-275)
         x
         (if (<= z 6e-287)
           y
           (if (<= z 1.65e-223)
             x
             (if (<= z 4.65e-116)
               y
               (if (<= z 1.15e-78) x (if (<= z 1.0) y (* y z)))))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = y * z;
	} else if (z <= -1.85e-118) {
		tmp = x;
	} else if (z <= -1.65e-138) {
		tmp = y;
	} else if (z <= -1.35e-275) {
		tmp = x;
	} else if (z <= 6e-287) {
		tmp = y;
	} else if (z <= 1.65e-223) {
		tmp = x;
	} else if (z <= 4.65e-116) {
		tmp = y;
	} else if (z <= 1.15e-78) {
		tmp = x;
	} else if (z <= 1.0) {
		tmp = y;
	} else {
		tmp = y * z;
	}
	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 (z <= (-1.0d0)) then
        tmp = y * z
    else if (z <= (-1.85d-118)) then
        tmp = x
    else if (z <= (-1.65d-138)) then
        tmp = y
    else if (z <= (-1.35d-275)) then
        tmp = x
    else if (z <= 6d-287) then
        tmp = y
    else if (z <= 1.65d-223) then
        tmp = x
    else if (z <= 4.65d-116) then
        tmp = y
    else if (z <= 1.15d-78) then
        tmp = x
    else if (z <= 1.0d0) then
        tmp = y
    else
        tmp = y * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = y * z;
	} else if (z <= -1.85e-118) {
		tmp = x;
	} else if (z <= -1.65e-138) {
		tmp = y;
	} else if (z <= -1.35e-275) {
		tmp = x;
	} else if (z <= 6e-287) {
		tmp = y;
	} else if (z <= 1.65e-223) {
		tmp = x;
	} else if (z <= 4.65e-116) {
		tmp = y;
	} else if (z <= 1.15e-78) {
		tmp = x;
	} else if (z <= 1.0) {
		tmp = y;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = y * z
	elif z <= -1.85e-118:
		tmp = x
	elif z <= -1.65e-138:
		tmp = y
	elif z <= -1.35e-275:
		tmp = x
	elif z <= 6e-287:
		tmp = y
	elif z <= 1.65e-223:
		tmp = x
	elif z <= 4.65e-116:
		tmp = y
	elif z <= 1.15e-78:
		tmp = x
	elif z <= 1.0:
		tmp = y
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(y * z);
	elseif (z <= -1.85e-118)
		tmp = x;
	elseif (z <= -1.65e-138)
		tmp = y;
	elseif (z <= -1.35e-275)
		tmp = x;
	elseif (z <= 6e-287)
		tmp = y;
	elseif (z <= 1.65e-223)
		tmp = x;
	elseif (z <= 4.65e-116)
		tmp = y;
	elseif (z <= 1.15e-78)
		tmp = x;
	elseif (z <= 1.0)
		tmp = y;
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = y * z;
	elseif (z <= -1.85e-118)
		tmp = x;
	elseif (z <= -1.65e-138)
		tmp = y;
	elseif (z <= -1.35e-275)
		tmp = x;
	elseif (z <= 6e-287)
		tmp = y;
	elseif (z <= 1.65e-223)
		tmp = x;
	elseif (z <= 4.65e-116)
		tmp = y;
	elseif (z <= 1.15e-78)
		tmp = x;
	elseif (z <= 1.0)
		tmp = y;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.0], N[(y * z), $MachinePrecision], If[LessEqual[z, -1.85e-118], x, If[LessEqual[z, -1.65e-138], y, If[LessEqual[z, -1.35e-275], x, If[LessEqual[z, 6e-287], y, If[LessEqual[z, 1.65e-223], x, If[LessEqual[z, 4.65e-116], y, If[LessEqual[z, 1.15e-78], x, If[LessEqual[z, 1.0], y, N[(y * z), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1 or 1 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 58.5%

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

      1. +-commutative 58.5%

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

      2. distribute-lft-in 58.4%

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

      3. *-rgt-identity 58.4%

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

   5. Applied egg-rr 58.4%

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

   6. Taylor expanded in z around inf 57.3%

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

      1. *-commutative 57.3%

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

   8. Simplified 57.3%

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

   if -1 < z < -1.85000000000000007e-118 or -1.64999999999999991e-138 < z < -1.34999999999999997e-275 or 5.99999999999999984e-287 < z < 1.64999999999999997e-223 or 4.64999999999999999e-116 < z < 1.1500000000000001e-78

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 53.2%

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

   4. Taylor expanded in z around 0 52.7%

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

   if -1.85000000000000007e-118 < z < -1.64999999999999991e-138 or -1.34999999999999997e-275 < z < 5.99999999999999984e-287 or 1.64999999999999997e-223 < z < 4.64999999999999999e-116 or 1.1500000000000001e-78 < z < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 46.8%

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

      1. +-commutative 46.8%

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

      2. distribute-lft-in 46.8%

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

      3. *-rgt-identity 46.8%

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

   5. Applied egg-rr 46.8%

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

   6. Taylor expanded in z around 0 46.8%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -1.85 \cdot 10^{-118}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{-138}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq -1.35 \cdot 10^{-275}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-287}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-223}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4.65 \cdot 10^{-116}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-78}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-6}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+24}:\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0)
   (* y z)
   (if (<= z 4.8e-6) (+ x y) (if (<= z 1.1e+24) (* x (+ z 1.0)) (* y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = y * z;
	} else if (z <= 4.8e-6) {
		tmp = x + y;
	} else if (z <= 1.1e+24) {
		tmp = x * (z + 1.0);
	} else {
		tmp = y * z;
	}
	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 (z <= (-1.0d0)) then
        tmp = y * z
    else if (z <= 4.8d-6) then
        tmp = x + y
    else if (z <= 1.1d+24) then
        tmp = x * (z + 1.0d0)
    else
        tmp = y * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = y * z;
	} else if (z <= 4.8e-6) {
		tmp = x + y;
	} else if (z <= 1.1e+24) {
		tmp = x * (z + 1.0);
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = y * z
	elif z <= 4.8e-6:
		tmp = x + y
	elif z <= 1.1e+24:
		tmp = x * (z + 1.0)
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(y * z);
	elseif (z <= 4.8e-6)
		tmp = Float64(x + y);
	elseif (z <= 1.1e+24)
		tmp = Float64(x * Float64(z + 1.0));
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = y * z;
	elseif (z <= 4.8e-6)
		tmp = x + y;
	elseif (z <= 1.1e+24)
		tmp = x * (z + 1.0);
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.0], N[(y * z), $MachinePrecision], If[LessEqual[z, 4.8e-6], N[(x + y), $MachinePrecision], If[LessEqual[z, 1.1e+24], N[(x * N[(z + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1 or 1.10000000000000001e24 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 58.4%

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

      1. +-commutative 58.4%

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

      2. distribute-lft-in 58.4%

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

      3. *-rgt-identity 58.4%

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

   5. Applied egg-rr 58.4%

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

   6. Taylor expanded in z around inf 58.4%

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

      1. *-commutative 58.4%

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

   8. Simplified 58.4%

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

   if -1 < z < 4.7999999999999998e-6

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0 98.4%

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

      1. +-commutative 98.4%

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

   5. Simplified 98.4%

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

   if 4.7999999999999998e-6 < z < 1.10000000000000001e24

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 45.4%

      \[\leadsto \color{blue}{x \cdot \left(1 + z\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-6}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+24}:\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;z \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 1.0))) (* z (+ x y)) (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		tmp = z * (x + y);
	} else {
		tmp = x + y;
	}
	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 ((z <= (-1.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = z * (x + y)
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

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

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 1.0):
		tmp = z * (x + y)
	else:
		tmp = x + y
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 98.0%

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

      1. +-commutative 98.0%

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

   5. Simplified 98.0%

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

   if -1 < z < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0 98.4%

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

      1. +-commutative 98.4%

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

   5. Simplified 98.4%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;z \cdot \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 395000000\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 395000000.0))) (* y z) (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 395000000.0)) {
		tmp = y * z;
	} else {
		tmp = x + y;
	}
	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 ((z <= (-1.0d0)) .or. (.not. (z <= 395000000.0d0))) then
        tmp = y * z
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 395000000.0)) {
		tmp = y * z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 395000000.0):
		tmp = y * z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 395000000.0))
		tmp = Float64(y * z);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 395000000.0)))
		tmp = y * z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 395000000.0]], $MachinePrecision]], N[(y * z), $MachinePrecision], N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 3.95e8 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 58.2%

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

      1. +-commutative 58.2%

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

      2. distribute-lft-in 58.2%

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

      3. *-rgt-identity 58.2%

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

   5. Applied egg-rr 58.2%

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

   6. Taylor expanded in z around inf 58.2%

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

      1. *-commutative 58.2%

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

   8. Simplified 58.2%

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

   if -1 < z < 3.95e8

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0 96.7%

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

      1. +-commutative 96.7%

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

   5. Simplified 96.7%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 395000000\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-63}:\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2e-63) (* x (+ z 1.0)) (* y (+ z 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e-63) {
		tmp = x * (z + 1.0);
	} else {
		tmp = y * (z + 1.0);
	}
	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 (y <= 2d-63) then
        tmp = x * (z + 1.0d0)
    else
        tmp = y * (z + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e-63) {
		tmp = x * (z + 1.0);
	} else {
		tmp = y * (z + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2e-63:
		tmp = x * (z + 1.0)
	else:
		tmp = y * (z + 1.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2e-63)
		tmp = Float64(x * Float64(z + 1.0));
	else
		tmp = Float64(y * Float64(z + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2e-63)
		tmp = x * (z + 1.0);
	else
		tmp = y * (z + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2e-63], N[(x * N[(z + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.00000000000000013e-63

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 61.5%

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

   if 2.00000000000000013e-63 < 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 73.6%

      \[\leadsto \color{blue}{y \cdot \left(1 + z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-63}:\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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]

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 9: 32.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-50}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= y 2e-50) x y))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e-50) {
		tmp = x;
	} else {
		tmp = y;
	}
	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 (y <= 2d-50) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e-50) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2e-50:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2e-50)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2e-50)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2e-50], x, y]

Derivation

1. Split input into 2 regimes

2. if y < 2.00000000000000002e-50

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 61.5%

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

   4. Taylor expanded in z around 0 36.6%

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

   if 2.00000000000000002e-50 < 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 76.2%

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

      1. +-commutative 76.2%

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

      2. distribute-lft-in 76.2%

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

      3. *-rgt-identity 76.2%

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

   5. Applied egg-rr 76.2%

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

   6. Taylor expanded in z around 0 33.3%

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

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-50}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 26.5% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around inf 52.0%

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

4. Taylor expanded in z around 0 31.3%

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

5. Final simplification 31.3%

   \[\leadsto x \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024034 
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, G"
  :precision binary64
  (* (+ x y) (+ z 1.0)))