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

Percentage Accurate: 100.0% → 100.0%

Time: 4.4s

Alternatives: 8

Speedup: 1.0×

• 20.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

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) * (1.0d0 - z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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) * (1.0d0 - z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (1.0 - 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 = (1.0d0 - z) * (x + y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 2: 75.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(-z\right)\\ \mathbf{if}\;1 - z \leq -1 \cdot 10^{+157}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;1 - z \leq -1 \cdot 10^{+94}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{elif}\;1 - z \leq -40:\\ \;\;\;\;t_0\\ \mathbf{elif}\;1 - z \leq 1:\\ \;\;\;\;x + y\\ \mathbf{elif}\;1 - z \leq 5 \cdot 10^{+59}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- z))))
   (if (<= (- 1.0 z) -1e+157)
     t_0
     (if (<= (- 1.0 z) -1e+94)
       (* z (- y))
       (if (<= (- 1.0 z) -40.0)
         t_0
         (if (<= (- 1.0 z) 1.0)
           (+ x y)
           (if (<= (- 1.0 z) 5e+59) (* y (- 1.0 z)) t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = x * -z;
	double tmp;
	if ((1.0 - z) <= -1e+157) {
		tmp = t_0;
	} else if ((1.0 - z) <= -1e+94) {
		tmp = z * -y;
	} else if ((1.0 - z) <= -40.0) {
		tmp = t_0;
	} else if ((1.0 - z) <= 1.0) {
		tmp = x + y;
	} else if ((1.0 - z) <= 5e+59) {
		tmp = y * (1.0 - z);
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = x * -z
    if ((1.0d0 - z) <= (-1d+157)) then
        tmp = t_0
    else if ((1.0d0 - z) <= (-1d+94)) then
        tmp = z * -y
    else if ((1.0d0 - z) <= (-40.0d0)) then
        tmp = t_0
    else if ((1.0d0 - z) <= 1.0d0) then
        tmp = x + y
    else if ((1.0d0 - z) <= 5d+59) then
        tmp = y * (1.0d0 - z)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * -z;
	double tmp;
	if ((1.0 - z) <= -1e+157) {
		tmp = t_0;
	} else if ((1.0 - z) <= -1e+94) {
		tmp = z * -y;
	} else if ((1.0 - z) <= -40.0) {
		tmp = t_0;
	} else if ((1.0 - z) <= 1.0) {
		tmp = x + y;
	} else if ((1.0 - z) <= 5e+59) {
		tmp = y * (1.0 - z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * -z
	tmp = 0
	if (1.0 - z) <= -1e+157:
		tmp = t_0
	elif (1.0 - z) <= -1e+94:
		tmp = z * -y
	elif (1.0 - z) <= -40.0:
		tmp = t_0
	elif (1.0 - z) <= 1.0:
		tmp = x + y
	elif (1.0 - z) <= 5e+59:
		tmp = y * (1.0 - z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(-z))
	tmp = 0.0
	if (Float64(1.0 - z) <= -1e+157)
		tmp = t_0;
	elseif (Float64(1.0 - z) <= -1e+94)
		tmp = Float64(z * Float64(-y));
	elseif (Float64(1.0 - z) <= -40.0)
		tmp = t_0;
	elseif (Float64(1.0 - z) <= 1.0)
		tmp = Float64(x + y);
	elseif (Float64(1.0 - z) <= 5e+59)
		tmp = Float64(y * Float64(1.0 - z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * -z;
	tmp = 0.0;
	if ((1.0 - z) <= -1e+157)
		tmp = t_0;
	elseif ((1.0 - z) <= -1e+94)
		tmp = z * -y;
	elseif ((1.0 - z) <= -40.0)
		tmp = t_0;
	elseif ((1.0 - z) <= 1.0)
		tmp = x + y;
	elseif ((1.0 - z) <= 5e+59)
		tmp = y * (1.0 - z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * (-z)), $MachinePrecision]}, If[LessEqual[N[(1.0 - z), $MachinePrecision], -1e+157], t$95$0, If[LessEqual[N[(1.0 - z), $MachinePrecision], -1e+94], N[(z * (-y)), $MachinePrecision], If[LessEqual[N[(1.0 - z), $MachinePrecision], -40.0], t$95$0, If[LessEqual[N[(1.0 - z), $MachinePrecision], 1.0], N[(x + y), $MachinePrecision], If[LessEqual[N[(1.0 - z), $MachinePrecision], 5e+59], N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 1 z) < -9.99999999999999983e156 or -1e94 < (-.f64 1 z) < -40 or 4.9999999999999997e59 < (-.f64 1 z)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 51.6%

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

   3. Taylor expanded in z around inf 50.0%

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

      1. mul-1-neg 50.0%

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

      2. distribute-lft-neg-out 50.0%

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

      3. *-commutative 50.0%

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

   5. Simplified 50.0%

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

   if -9.99999999999999983e156 < (-.f64 1 z) < -1e94

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. distribute-lft-in 100.0%

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

      3. fma-def 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in z around inf 99.9%

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

      1. neg-mul-1 99.9%

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

      2. +-commutative 99.9%

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

      3. mul-1-neg 99.9%

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

      4. unsub-neg 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in y around inf 52.9%

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

      1. associate-*r* 52.9%

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

      2. mul-1-neg 52.9%

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

   9. Simplified 52.9%

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

   if -40 < (-.f64 1 z) < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 99.2%

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

   if 1 < (-.f64 1 z) < 4.9999999999999997e59

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 41.0%

      \[\leadsto \color{blue}{y \cdot \left(1 - z\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - z \leq -1 \cdot 10^{+157}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{elif}\;1 - z \leq -1 \cdot 10^{+94}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{elif}\;1 - z \leq -40:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{elif}\;1 - z \leq 1:\\ \;\;\;\;x + y\\ \mathbf{elif}\;1 - z \leq 5 \cdot 10^{+59}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \end{array} \]

Alternative 3: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - z \leq -40 \lor \neg \left(1 - z \leq 2\right):\\ \;\;\;\;z \cdot \left(\left(-x\right) - y\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (- 1.0 z) -40.0) (not (<= (- 1.0 z) 2.0)))
   (* z (- (- x) y))
   (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((1.0 - z) <= -40.0) || !((1.0 - z) <= 2.0)) {
		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 (((1.0d0 - z) <= (-40.0d0)) .or. (.not. ((1.0d0 - z) <= 2.0d0))) 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 (((1.0 - z) <= -40.0) || !((1.0 - z) <= 2.0)) {
		tmp = z * (-x - y);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((1.0 - z) <= -40.0) or not ((1.0 - z) <= 2.0):
		tmp = z * (-x - y)
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(1.0 - z) <= -40.0) || !(Float64(1.0 - z) <= 2.0))
		tmp = Float64(z * Float64(Float64(-x) - y));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((1.0 - z) <= -40.0) || ~(((1.0 - z) <= 2.0)))
		tmp = z * (-x - y);
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(1.0 - z), $MachinePrecision], -40.0], N[Not[LessEqual[N[(1.0 - z), $MachinePrecision], 2.0]], $MachinePrecision]], N[(z * N[((-x) - y), $MachinePrecision]), $MachinePrecision], N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 z) < -40 or 2 < (-.f64 1 z)

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 96.4%

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

      1. mul-1-neg 96.4%

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

      2. +-commutative 96.4%

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

      3. distribute-rgt-neg-out 96.4%

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

      4. +-commutative 96.4%

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

   4. Simplified 96.4%

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

   if -40 < (-.f64 1 z) < 2

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 98.8%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - z \leq -40 \lor \neg \left(1 - z \leq 2\right):\\ \;\;\;\;z \cdot \left(\left(-x\right) - y\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 4: 74.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(-z\right)\\ \mathbf{if}\;z \leq -260000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+92} \lor \neg \left(z \leq 1.55 \cdot 10^{+153}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- z))))
   (if (<= z -260000000.0)
     t_0
     (if (<= z 1.0)
       (+ x y)
       (if (or (<= z 4e+92) (not (<= z 1.55e+153))) t_0 (* z (- y)))))))

C

double code(double x, double y, double z) {
	double t_0 = x * -z;
	double tmp;
	if (z <= -260000000.0) {
		tmp = t_0;
	} else if (z <= 1.0) {
		tmp = x + y;
	} else if ((z <= 4e+92) || !(z <= 1.55e+153)) {
		tmp = t_0;
	} else {
		tmp = z * -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) :: t_0
    real(8) :: tmp
    t_0 = x * -z
    if (z <= (-260000000.0d0)) then
        tmp = t_0
    else if (z <= 1.0d0) then
        tmp = x + y
    else if ((z <= 4d+92) .or. (.not. (z <= 1.55d+153))) then
        tmp = t_0
    else
        tmp = z * -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * -z;
	double tmp;
	if (z <= -260000000.0) {
		tmp = t_0;
	} else if (z <= 1.0) {
		tmp = x + y;
	} else if ((z <= 4e+92) || !(z <= 1.55e+153)) {
		tmp = t_0;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * -z
	tmp = 0
	if z <= -260000000.0:
		tmp = t_0
	elif z <= 1.0:
		tmp = x + y
	elif (z <= 4e+92) or not (z <= 1.55e+153):
		tmp = t_0
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(-z))
	tmp = 0.0
	if (z <= -260000000.0)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = Float64(x + y);
	elseif ((z <= 4e+92) || !(z <= 1.55e+153))
		tmp = t_0;
	else
		tmp = Float64(z * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * -z;
	tmp = 0.0;
	if (z <= -260000000.0)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = x + y;
	elseif ((z <= 4e+92) || ~((z <= 1.55e+153)))
		tmp = t_0;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * (-z)), $MachinePrecision]}, If[LessEqual[z, -260000000.0], t$95$0, If[LessEqual[z, 1.0], N[(x + y), $MachinePrecision], If[Or[LessEqual[z, 4e+92], N[Not[LessEqual[z, 1.55e+153]], $MachinePrecision]], t$95$0, N[(z * (-y)), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.6e8 or 1 < z < 4.0000000000000002e92 or 1.55e153 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 52.8%

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

   3. Taylor expanded in z around inf 51.1%

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

      1. mul-1-neg 51.1%

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

      2. distribute-lft-neg-out 51.1%

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

      3. *-commutative 51.1%

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

   5. Simplified 51.1%

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

   if -2.6e8 < z < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 96.0%

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

   if 4.0000000000000002e92 < z < 1.55e153

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. distribute-lft-in 100.0%

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

      3. fma-def 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in z around inf 99.9%

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

      1. neg-mul-1 99.9%

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

      2. +-commutative 99.9%

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

      3. mul-1-neg 99.9%

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

      4. unsub-neg 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in y around inf 52.9%

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

      1. associate-*r* 52.9%

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

      2. mul-1-neg 52.9%

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

   9. Simplified 52.9%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -260000000:\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+92} \lor \neg \left(z \leq 1.55 \cdot 10^{+153}\right):\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]

Alternative 5: 74.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -260000000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -260000000.0) (not (<= z 1.0))) (* x (- z)) (+ x y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -260000000.0) || !(z <= 1.0)) {
		tmp = x * -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 <= (-260000000.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = x * -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 <= -260000000.0) || !(z <= 1.0)) {
		tmp = x * -z;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -260000000.0) or not (z <= 1.0):
		tmp = x * -z
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -260000000.0) || !(z <= 1.0))
		tmp = Float64(x * Float64(-z));
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -260000000.0) || ~((z <= 1.0)))
		tmp = x * -z;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -260000000.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(x * (-z)), $MachinePrecision], N[(x + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.6e8 or 1 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 52.3%

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

   3. Taylor expanded in z around inf 50.8%

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

      1. mul-1-neg 50.8%

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

      2. distribute-lft-neg-out 50.8%

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

      3. *-commutative 50.8%

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

   5. Simplified 50.8%

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

   if -2.6e8 < z < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 96.0%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -260000000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 6: 63.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-54}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.6e-54) (* x (- 1.0 z)) (* y (- 1.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.6e-54) {
		tmp = x * (1.0 - z);
	} else {
		tmp = y * (1.0 - 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 (x <= (-2.6d-54)) then
        tmp = x * (1.0d0 - z)
    else
        tmp = y * (1.0d0 - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.6e-54) {
		tmp = x * (1.0 - z);
	} else {
		tmp = y * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.6e-54:
		tmp = x * (1.0 - z)
	else:
		tmp = y * (1.0 - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.6e-54)
		tmp = Float64(x * Float64(1.0 - z));
	else
		tmp = Float64(y * Float64(1.0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.6e-54)
		tmp = x * (1.0 - z);
	else
		tmp = y * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.6e-54], N[(x * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.60000000000000002e-54

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 78.2%

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

   if -2.60000000000000002e-54 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 62.2%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-54}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \end{array} \]

Alternative 7: 51.4% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in z around 0 48.8%

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

3. Final simplification 48.8%

   \[\leadsto x + y \]

Alternative 8: 26.2% 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%

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

2. Taylor expanded in x around inf 51.9%

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

3. Taylor expanded in z around 0 25.5%

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

4. Final simplification 25.5%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023200 
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, H"
  :precision binary64
  (* (+ x y) (- 1.0 z)))