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

Percentage Accurate: 100.0% → 100.0%

Time: 4.2s

Alternatives: 9

Speedup: 1.0×

• Unsound rule application detected in e-graph. Results may not be sound.

• 20.4% 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

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

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. Final simplification 100.0%

   \[\leadsto \left(x + y\right) \cdot \left(z + 1\right) \]

Alternative 2: 50.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -1.16 \cdot 10^{-139}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-249}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 5.7 \cdot 10^{-214}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-184}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 10^{-128}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 510000:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+29}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0)
   (* y z)
   (if (<= z -1.16e-139)
     x
     (if (<= z 5.8e-249)
       y
       (if (<= z 5.7e-214)
         x
         (if (<= z 4.2e-184)
           y
           (if (<= z 1e-128)
             x
             (if (<= z 510000.0) y (if (<= z 6e+29) (* y z) (* x z))))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = y * z;
	} else if (z <= -1.16e-139) {
		tmp = x;
	} else if (z <= 5.8e-249) {
		tmp = y;
	} else if (z <= 5.7e-214) {
		tmp = x;
	} else if (z <= 4.2e-184) {
		tmp = y;
	} else if (z <= 1e-128) {
		tmp = x;
	} else if (z <= 510000.0) {
		tmp = y;
	} else if (z <= 6e+29) {
		tmp = y * z;
	} 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 <= (-1.0d0)) then
        tmp = y * z
    else if (z <= (-1.16d-139)) then
        tmp = x
    else if (z <= 5.8d-249) then
        tmp = y
    else if (z <= 5.7d-214) then
        tmp = x
    else if (z <= 4.2d-184) then
        tmp = y
    else if (z <= 1d-128) then
        tmp = x
    else if (z <= 510000.0d0) then
        tmp = y
    else if (z <= 6d+29) then
        tmp = y * z
    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 <= -1.0) {
		tmp = y * z;
	} else if (z <= -1.16e-139) {
		tmp = x;
	} else if (z <= 5.8e-249) {
		tmp = y;
	} else if (z <= 5.7e-214) {
		tmp = x;
	} else if (z <= 4.2e-184) {
		tmp = y;
	} else if (z <= 1e-128) {
		tmp = x;
	} else if (z <= 510000.0) {
		tmp = y;
	} else if (z <= 6e+29) {
		tmp = y * z;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = y * z
	elif z <= -1.16e-139:
		tmp = x
	elif z <= 5.8e-249:
		tmp = y
	elif z <= 5.7e-214:
		tmp = x
	elif z <= 4.2e-184:
		tmp = y
	elif z <= 1e-128:
		tmp = x
	elif z <= 510000.0:
		tmp = y
	elif z <= 6e+29:
		tmp = y * z
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(y * z);
	elseif (z <= -1.16e-139)
		tmp = x;
	elseif (z <= 5.8e-249)
		tmp = y;
	elseif (z <= 5.7e-214)
		tmp = x;
	elseif (z <= 4.2e-184)
		tmp = y;
	elseif (z <= 1e-128)
		tmp = x;
	elseif (z <= 510000.0)
		tmp = y;
	elseif (z <= 6e+29)
		tmp = Float64(y * z);
	else
		tmp = Float64(x * 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 <= -1.16e-139)
		tmp = x;
	elseif (z <= 5.8e-249)
		tmp = y;
	elseif (z <= 5.7e-214)
		tmp = x;
	elseif (z <= 4.2e-184)
		tmp = y;
	elseif (z <= 1e-128)
		tmp = x;
	elseif (z <= 510000.0)
		tmp = y;
	elseif (z <= 6e+29)
		tmp = y * z;
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.0], N[(y * z), $MachinePrecision], If[LessEqual[z, -1.16e-139], x, If[LessEqual[z, 5.8e-249], y, If[LessEqual[z, 5.7e-214], x, If[LessEqual[z, 4.2e-184], y, If[LessEqual[z, 1e-128], x, If[LessEqual[z, 510000.0], y, If[LessEqual[z, 6e+29], N[(y * z), $MachinePrecision], N[(x * z), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1 or 5.1e5 < z < 5.9999999999999998e29

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. distribute-lft-in 99.9%

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

      3. *-rgt-identity 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around 0 53.7%

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

   5. Taylor expanded in z around inf 50.3%

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

   if -1 < z < -1.15999999999999999e-139 or 5.80000000000000044e-249 < z < 5.6999999999999996e-214 or 4.1999999999999998e-184 < z < 1.00000000000000005e-128

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]

   2. Taylor expanded in x around inf 59.7%

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

   3. Taylor expanded in z around 0 56.5%

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

   if -1.15999999999999999e-139 < z < 5.80000000000000044e-249 or 5.6999999999999996e-214 < z < 4.1999999999999998e-184 or 1.00000000000000005e-128 < z < 5.1e5

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]

   2. Taylor expanded in x around 0 40.5%

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

   3. Taylor expanded in z around 0 38.9%

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

   if 5.9999999999999998e29 < z

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. *-rgt-identity 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate-+l+ 100.0%

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

      2. flip-+ 29.3%

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

      3. +-commutative 29.3%

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

      4. fma-def 29.3%

         \[\leadsto \frac{x \cdot x - \color{blue}{\mathsf{fma}\left(x + y, z, y\right)} \cdot \left(y + \left(x + y\right) \cdot z\right)}{x - \left(y + \left(x + y\right) \cdot z\right)} \]

      5. +-commutative 29.3%

         \[\leadsto \frac{x \cdot x - \mathsf{fma}\left(x + y, z, y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot z + y\right)}}{x - \left(y + \left(x + y\right) \cdot z\right)} \]

      6. fma-def 29.3%

         \[\leadsto \frac{x \cdot x - \mathsf{fma}\left(x + y, z, y\right) \cdot \color{blue}{\mathsf{fma}\left(x + y, z, y\right)}}{x - \left(y + \left(x + y\right) \cdot z\right)} \]

      7. +-commutative 29.3%

         \[\leadsto \frac{x \cdot x - \mathsf{fma}\left(x + y, z, y\right) \cdot \mathsf{fma}\left(x + y, z, y\right)}{x - \color{blue}{\left(\left(x + y\right) \cdot z + y\right)}} \]

      8. fma-def 29.3%

         \[\leadsto \frac{x \cdot x - \mathsf{fma}\left(x + y, z, y\right) \cdot \mathsf{fma}\left(x + y, z, y\right)}{x - \color{blue}{\mathsf{fma}\left(x + y, z, y\right)}} \]

   5. Applied egg-rr 29.3%

      \[\leadsto \color{blue}{\frac{x \cdot x - \mathsf{fma}\left(x + y, z, y\right) \cdot \mathsf{fma}\left(x + y, z, y\right)}{x - \mathsf{fma}\left(x + y, z, y\right)}} \]
   6. Step-by-step derivation

      1. difference-of-squares 29.3%

         \[\leadsto \frac{\color{blue}{\left(x + \mathsf{fma}\left(x + y, z, y\right)\right) \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      2. +-commutative 29.3%

         \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(x + y, z, y\right) + x\right)} \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      3. fma-udef 29.3%

         \[\leadsto \frac{\left(\color{blue}{\left(\left(x + y\right) \cdot z + y\right)} + x\right) \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      4. associate-+r+ 29.3%

         \[\leadsto \frac{\color{blue}{\left(\left(x + y\right) \cdot z + \left(y + x\right)\right)} \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      5. +-commutative 29.3%

         \[\leadsto \frac{\left(\left(x + y\right) \cdot z + \color{blue}{\left(x + y\right)}\right) \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      6. *-rgt-identity 29.3%

         \[\leadsto \frac{\left(\left(x + y\right) \cdot z + \color{blue}{\left(x + y\right) \cdot 1}\right) \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      7. distribute-lft-in 29.3%

         \[\leadsto \frac{\color{blue}{\left(\left(x + y\right) \cdot \left(z + 1\right)\right)} \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      8. associate-/l* 61.5%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) \cdot \left(z + 1\right)}{\frac{x - \mathsf{fma}\left(x + y, z, y\right)}{x - \mathsf{fma}\left(x + y, z, y\right)}}} \]

      9. *-inverses 100.0%

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

      10. associate-/l* 99.6%

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

      11. +-commutative 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in z around inf 99.6%

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

   9. Taylor expanded in y around 0 39.5%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -1.16 \cdot 10^{-139}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-249}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 5.7 \cdot 10^{-214}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-184}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 10^{-128}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 510000:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+29}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]

Alternative 3: 50.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -2.3 \cdot 10^{-137}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.85 \cdot 10^{-248}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-214}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-178}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{-128}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 510000:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0)
   (* y z)
   (if (<= z -2.3e-137)
     x
     (if (<= z 2.85e-248)
       y
       (if (<= z 1.2e-214)
         x
         (if (<= z 3.2e-178)
           y
           (if (<= z 8.6e-128) x (if (<= z 510000.0) y (* y z)))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = y * z;
	} else if (z <= -2.3e-137) {
		tmp = x;
	} else if (z <= 2.85e-248) {
		tmp = y;
	} else if (z <= 1.2e-214) {
		tmp = x;
	} else if (z <= 3.2e-178) {
		tmp = y;
	} else if (z <= 8.6e-128) {
		tmp = x;
	} else if (z <= 510000.0) {
		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 <= (-2.3d-137)) then
        tmp = x
    else if (z <= 2.85d-248) then
        tmp = y
    else if (z <= 1.2d-214) then
        tmp = x
    else if (z <= 3.2d-178) then
        tmp = y
    else if (z <= 8.6d-128) then
        tmp = x
    else if (z <= 510000.0d0) 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 <= -2.3e-137) {
		tmp = x;
	} else if (z <= 2.85e-248) {
		tmp = y;
	} else if (z <= 1.2e-214) {
		tmp = x;
	} else if (z <= 3.2e-178) {
		tmp = y;
	} else if (z <= 8.6e-128) {
		tmp = x;
	} else if (z <= 510000.0) {
		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 <= -2.3e-137:
		tmp = x
	elif z <= 2.85e-248:
		tmp = y
	elif z <= 1.2e-214:
		tmp = x
	elif z <= 3.2e-178:
		tmp = y
	elif z <= 8.6e-128:
		tmp = x
	elif z <= 510000.0:
		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 <= -2.3e-137)
		tmp = x;
	elseif (z <= 2.85e-248)
		tmp = y;
	elseif (z <= 1.2e-214)
		tmp = x;
	elseif (z <= 3.2e-178)
		tmp = y;
	elseif (z <= 8.6e-128)
		tmp = x;
	elseif (z <= 510000.0)
		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 <= -2.3e-137)
		tmp = x;
	elseif (z <= 2.85e-248)
		tmp = y;
	elseif (z <= 1.2e-214)
		tmp = x;
	elseif (z <= 3.2e-178)
		tmp = y;
	elseif (z <= 8.6e-128)
		tmp = x;
	elseif (z <= 510000.0)
		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, -2.3e-137], x, If[LessEqual[z, 2.85e-248], y, If[LessEqual[z, 1.2e-214], x, If[LessEqual[z, 3.2e-178], y, If[LessEqual[z, 8.6e-128], x, If[LessEqual[z, 510000.0], y, N[(y * z), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1 or 5.1e5 < z

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. distribute-lft-in 99.9%

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

      3. *-rgt-identity 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around 0 57.4%

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

   5. Taylor expanded in z around inf 55.5%

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

   if -1 < z < -2.30000000000000008e-137 or 2.8499999999999999e-248 < z < 1.2000000000000001e-214 or 3.2000000000000001e-178 < z < 8.59999999999999988e-128

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]

   2. Taylor expanded in x around inf 59.7%

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

   3. Taylor expanded in z around 0 56.5%

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

   if -2.30000000000000008e-137 < z < 2.8499999999999999e-248 or 1.2000000000000001e-214 < z < 3.2000000000000001e-178 or 8.59999999999999988e-128 < z < 5.1e5

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]

   2. Taylor expanded in x around 0 40.5%

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

   3. Taylor expanded in z around 0 38.9%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq -2.3 \cdot 10^{-137}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 2.85 \cdot 10^{-248}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-214}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-178}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{-128}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 510000:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 4: 75.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(z + 1\right)\\ \mathbf{if}\;z \leq -0.0023:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-5}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+26}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ z 1.0))))
   (if (<= z -0.0023)
     t_0
     (if (<= z 1.3e-5) (+ x y) (if (<= z 1.4e+26) t_0 (* x z))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (z + 1.0);
	double tmp;
	if (z <= -0.0023) {
		tmp = t_0;
	} else if (z <= 1.3e-5) {
		tmp = x + y;
	} else if (z <= 1.4e+26) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = y * (z + 1.0d0)
    if (z <= (-0.0023d0)) then
        tmp = t_0
    else if (z <= 1.3d-5) then
        tmp = x + y
    else if (z <= 1.4d+26) then
        tmp = t_0
    else
        tmp = x * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (z + 1.0);
	double tmp;
	if (z <= -0.0023) {
		tmp = t_0;
	} else if (z <= 1.3e-5) {
		tmp = x + y;
	} else if (z <= 1.4e+26) {
		tmp = t_0;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (z + 1.0)
	tmp = 0
	if z <= -0.0023:
		tmp = t_0
	elif z <= 1.3e-5:
		tmp = x + y
	elif z <= 1.4e+26:
		tmp = t_0
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(z + 1.0))
	tmp = 0.0
	if (z <= -0.0023)
		tmp = t_0;
	elseif (z <= 1.3e-5)
		tmp = Float64(x + y);
	elseif (z <= 1.4e+26)
		tmp = t_0;
	else
		tmp = Float64(x * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (z + 1.0);
	tmp = 0.0;
	if (z <= -0.0023)
		tmp = t_0;
	elseif (z <= 1.3e-5)
		tmp = x + y;
	elseif (z <= 1.4e+26)
		tmp = t_0;
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -0.0023], t$95$0, If[LessEqual[z, 1.3e-5], N[(x + y), $MachinePrecision], If[LessEqual[z, 1.4e+26], t$95$0, N[(x * z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.0023 or 1.29999999999999992e-5 < z < 1.4e26

   1. Initial program 99.9%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]

   2. Taylor expanded in x around 0 52.7%

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

   if -0.0023 < z < 1.29999999999999992e-5

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]

   2. Taylor expanded in z around 0 97.7%

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

   if 1.4e26 < z

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. *-rgt-identity 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate-+l+ 100.0%

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

      2. flip-+ 29.3%

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

      3. +-commutative 29.3%

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

      4. fma-def 29.3%

         \[\leadsto \frac{x \cdot x - \color{blue}{\mathsf{fma}\left(x + y, z, y\right)} \cdot \left(y + \left(x + y\right) \cdot z\right)}{x - \left(y + \left(x + y\right) \cdot z\right)} \]

      5. +-commutative 29.3%

         \[\leadsto \frac{x \cdot x - \mathsf{fma}\left(x + y, z, y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot z + y\right)}}{x - \left(y + \left(x + y\right) \cdot z\right)} \]

      6. fma-def 29.3%

         \[\leadsto \frac{x \cdot x - \mathsf{fma}\left(x + y, z, y\right) \cdot \color{blue}{\mathsf{fma}\left(x + y, z, y\right)}}{x - \left(y + \left(x + y\right) \cdot z\right)} \]

      7. +-commutative 29.3%

         \[\leadsto \frac{x \cdot x - \mathsf{fma}\left(x + y, z, y\right) \cdot \mathsf{fma}\left(x + y, z, y\right)}{x - \color{blue}{\left(\left(x + y\right) \cdot z + y\right)}} \]

      8. fma-def 29.3%

         \[\leadsto \frac{x \cdot x - \mathsf{fma}\left(x + y, z, y\right) \cdot \mathsf{fma}\left(x + y, z, y\right)}{x - \color{blue}{\mathsf{fma}\left(x + y, z, y\right)}} \]

   5. Applied egg-rr 29.3%

      \[\leadsto \color{blue}{\frac{x \cdot x - \mathsf{fma}\left(x + y, z, y\right) \cdot \mathsf{fma}\left(x + y, z, y\right)}{x - \mathsf{fma}\left(x + y, z, y\right)}} \]
   6. Step-by-step derivation

      1. difference-of-squares 29.3%

         \[\leadsto \frac{\color{blue}{\left(x + \mathsf{fma}\left(x + y, z, y\right)\right) \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      2. +-commutative 29.3%

         \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(x + y, z, y\right) + x\right)} \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      3. fma-udef 29.3%

         \[\leadsto \frac{\left(\color{blue}{\left(\left(x + y\right) \cdot z + y\right)} + x\right) \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      4. associate-+r+ 29.3%

         \[\leadsto \frac{\color{blue}{\left(\left(x + y\right) \cdot z + \left(y + x\right)\right)} \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      5. +-commutative 29.3%

         \[\leadsto \frac{\left(\left(x + y\right) \cdot z + \color{blue}{\left(x + y\right)}\right) \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      6. *-rgt-identity 29.3%

         \[\leadsto \frac{\left(\left(x + y\right) \cdot z + \color{blue}{\left(x + y\right) \cdot 1}\right) \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      7. distribute-lft-in 29.3%

         \[\leadsto \frac{\color{blue}{\left(\left(x + y\right) \cdot \left(z + 1\right)\right)} \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      8. associate-/l* 61.5%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) \cdot \left(z + 1\right)}{\frac{x - \mathsf{fma}\left(x + y, z, y\right)}{x - \mathsf{fma}\left(x + y, z, y\right)}}} \]

      9. *-inverses 100.0%

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

      10. associate-/l* 99.6%

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

      11. +-commutative 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in z around inf 99.6%

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

   9. Taylor expanded in y around 0 39.5%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.0023:\\ \;\;\;\;y \cdot \left(z + 1\right)\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-5}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+26}:\\ \;\;\;\;y \cdot \left(z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]

Alternative 5: 74.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 510000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+33}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.0)
   (* y z)
   (if (<= z 510000.0) (+ x y) (if (<= z 6e+33) (* y z) (* x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.0) {
		tmp = y * z;
	} else if (z <= 510000.0) {
		tmp = x + y;
	} else if (z <= 6e+33) {
		tmp = y * z;
	} 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 <= (-1.0d0)) then
        tmp = y * z
    else if (z <= 510000.0d0) then
        tmp = x + y
    else if (z <= 6d+33) then
        tmp = y * z
    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 <= -1.0) {
		tmp = y * z;
	} else if (z <= 510000.0) {
		tmp = x + y;
	} else if (z <= 6e+33) {
		tmp = y * z;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.0:
		tmp = y * z
	elif z <= 510000.0:
		tmp = x + y
	elif z <= 6e+33:
		tmp = y * z
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(y * z);
	elseif (z <= 510000.0)
		tmp = Float64(x + y);
	elseif (z <= 6e+33)
		tmp = Float64(y * z);
	else
		tmp = Float64(x * 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 <= 510000.0)
		tmp = x + y;
	elseif (z <= 6e+33)
		tmp = y * z;
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.0], N[(y * z), $MachinePrecision], If[LessEqual[z, 510000.0], N[(x + y), $MachinePrecision], If[LessEqual[z, 6e+33], N[(y * z), $MachinePrecision], N[(x * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1 or 5.1e5 < z < 5.99999999999999967e33

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. distribute-lft-in 99.9%

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

      3. *-rgt-identity 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around 0 53.7%

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

   5. Taylor expanded in z around inf 50.3%

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

   if -1 < z < 5.1e5

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]

   2. Taylor expanded in z around 0 95.1%

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

   if 5.99999999999999967e33 < z

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. *-rgt-identity 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate-+l+ 100.0%

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

      2. flip-+ 29.3%

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

      3. +-commutative 29.3%

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

      4. fma-def 29.3%

         \[\leadsto \frac{x \cdot x - \color{blue}{\mathsf{fma}\left(x + y, z, y\right)} \cdot \left(y + \left(x + y\right) \cdot z\right)}{x - \left(y + \left(x + y\right) \cdot z\right)} \]

      5. +-commutative 29.3%

         \[\leadsto \frac{x \cdot x - \mathsf{fma}\left(x + y, z, y\right) \cdot \color{blue}{\left(\left(x + y\right) \cdot z + y\right)}}{x - \left(y + \left(x + y\right) \cdot z\right)} \]

      6. fma-def 29.3%

         \[\leadsto \frac{x \cdot x - \mathsf{fma}\left(x + y, z, y\right) \cdot \color{blue}{\mathsf{fma}\left(x + y, z, y\right)}}{x - \left(y + \left(x + y\right) \cdot z\right)} \]

      7. +-commutative 29.3%

         \[\leadsto \frac{x \cdot x - \mathsf{fma}\left(x + y, z, y\right) \cdot \mathsf{fma}\left(x + y, z, y\right)}{x - \color{blue}{\left(\left(x + y\right) \cdot z + y\right)}} \]

      8. fma-def 29.3%

         \[\leadsto \frac{x \cdot x - \mathsf{fma}\left(x + y, z, y\right) \cdot \mathsf{fma}\left(x + y, z, y\right)}{x - \color{blue}{\mathsf{fma}\left(x + y, z, y\right)}} \]

   5. Applied egg-rr 29.3%

      \[\leadsto \color{blue}{\frac{x \cdot x - \mathsf{fma}\left(x + y, z, y\right) \cdot \mathsf{fma}\left(x + y, z, y\right)}{x - \mathsf{fma}\left(x + y, z, y\right)}} \]
   6. Step-by-step derivation

      1. difference-of-squares 29.3%

         \[\leadsto \frac{\color{blue}{\left(x + \mathsf{fma}\left(x + y, z, y\right)\right) \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      2. +-commutative 29.3%

         \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(x + y, z, y\right) + x\right)} \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      3. fma-udef 29.3%

         \[\leadsto \frac{\left(\color{blue}{\left(\left(x + y\right) \cdot z + y\right)} + x\right) \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      4. associate-+r+ 29.3%

         \[\leadsto \frac{\color{blue}{\left(\left(x + y\right) \cdot z + \left(y + x\right)\right)} \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      5. +-commutative 29.3%

         \[\leadsto \frac{\left(\left(x + y\right) \cdot z + \color{blue}{\left(x + y\right)}\right) \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      6. *-rgt-identity 29.3%

         \[\leadsto \frac{\left(\left(x + y\right) \cdot z + \color{blue}{\left(x + y\right) \cdot 1}\right) \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      7. distribute-lft-in 29.3%

         \[\leadsto \frac{\color{blue}{\left(\left(x + y\right) \cdot \left(z + 1\right)\right)} \cdot \left(x - \mathsf{fma}\left(x + y, z, y\right)\right)}{x - \mathsf{fma}\left(x + y, z, y\right)} \]

      8. associate-/l* 61.5%

         \[\leadsto \color{blue}{\frac{\left(x + y\right) \cdot \left(z + 1\right)}{\frac{x - \mathsf{fma}\left(x + y, z, y\right)}{x - \mathsf{fma}\left(x + y, z, y\right)}}} \]

      9. *-inverses 100.0%

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

      10. associate-/l* 99.6%

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

      11. +-commutative 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in z around inf 99.6%

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

   9. Taylor expanded in y around 0 39.5%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 510000:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+33}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]

Alternative 6: 97.9% accurate, 0.8× 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. Taylor expanded in z around inf 96.5%

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

   if -1 < z < 1

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]

   2. Taylor expanded in z around 0 96.3%

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

4. Final simplification 96.4%

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

Alternative 7: 58.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-187}:\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.2e-187) (* x (+ z 1.0)) (* y (+ z 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.2e-187) {
		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 (x <= (-5.2d-187)) 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 (x <= -5.2e-187) {
		tmp = x * (z + 1.0);
	} else {
		tmp = y * (z + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5.2e-187:
		tmp = x * (z + 1.0)
	else:
		tmp = y * (z + 1.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.2e-187)
		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 (x <= -5.2e-187)
		tmp = x * (z + 1.0);
	else
		tmp = y * (z + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.2e-187], 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 x < -5.1999999999999999e-187

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]

   2. Taylor expanded in x around inf 68.3%

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

   if -5.1999999999999999e-187 < x

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]

   2. Taylor expanded in x around 0 57.7%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-187}:\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z + 1\right)\\ \end{array} \]

Alternative 8: 28.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-208}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= x -2.1e-208) x y))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.1e-208) {
		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 (x <= (-2.1d-208)) 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 (x <= -2.1e-208) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.1e-208:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.1e-208)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.1e-208)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.1e-208], x, y]

Derivation

1. Split input into 2 regimes

2. if x < -2.10000000000000012e-208

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]

   2. Taylor expanded in x around inf 67.4%

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

   3. Taylor expanded in z around 0 38.6%

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

   if -2.10000000000000012e-208 < x

   1. Initial program 100.0%

      \[\left(x + y\right) \cdot \left(z + 1\right) \]

   2. Taylor expanded in x around 0 57.5%

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

   3. Taylor expanded in z around 0 25.8%

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

4. Final simplification 30.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-208}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 9: 25.6% 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. Taylor expanded in x around inf 54.0%

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

3. Taylor expanded in z around 0 32.5%

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

4. Final simplification 32.5%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023174 
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, G"
  :precision binary64
  (* (+ x y) (+ z 1.0)))