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

Percentage Accurate: 99.9% → 99.9%

Time: 5.6s

Alternatives: 8

Speedup: 1.0×

• 26.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{\left(x + y\right) - z}{t \cdot 2} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ (- (+ x y) z) (* t 2.0)))

C

double code(double x, double y, double z, double t) {
	return ((x + y) - z) / (t * 2.0);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x + y) - z) / (t * 2.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x + y) - z) / (t * 2.0);
}

Python

def code(x, y, z, t):
	return ((x + y) - z) / (t * 2.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x + y) - z) / Float64(t * 2.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x + y) - z) / (t * 2.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(x + y\right) - z}{t \cdot 2} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ (- (+ x y) z) (* t 2.0)))

C

double code(double x, double y, double z, double t) {
	return ((x + y) - z) / (t * 2.0);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x + y) - z) / (t * 2.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x + y) - z) / (t * 2.0);
}

Python

def code(x, y, z, t):
	return ((x + y) - z) / (t * 2.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x + y) - z) / Float64(t * 2.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x + y) - z) / (t * 2.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(x + y\right) - z}{t \cdot 2} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ (- (+ x y) z) (* t 2.0)))

C

double code(double x, double y, double z, double t) {
	return ((x + y) - z) / (t * 2.0);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x + y) - z) / (t * 2.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x + y) - z) / (t * 2.0);
}

Python

def code(x, y, z, t):
	return ((x + y) - z) / (t * 2.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x + y) - z) / Float64(t * 2.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x + y) - z) / (t * 2.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto \frac{\left(x + y\right) - z}{t \cdot 2} \]
4. Add Preprocessing

Alternative 2: 54.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.65 \cdot 10^{+119} \lor \neg \left(z \leq -2.75 \cdot 10^{+45} \lor \neg \left(z \leq -3.15 \cdot 10^{-27}\right) \land z \leq 6 \cdot 10^{+130}\right):\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2.65e+119)
         (not
          (or (<= z -2.75e+45) (and (not (<= z -3.15e-27)) (<= z 6e+130)))))
   (* z (/ -0.5 t))
   (* 0.5 (/ x t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.65e+119) || !((z <= -2.75e+45) || (!(z <= -3.15e-27) && (z <= 6e+130)))) {
		tmp = z * (-0.5 / t);
	} else {
		tmp = 0.5 * (x / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-2.65d+119)) .or. (.not. (z <= (-2.75d+45)) .or. (.not. (z <= (-3.15d-27))) .and. (z <= 6d+130))) then
        tmp = z * ((-0.5d0) / t)
    else
        tmp = 0.5d0 * (x / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.65e+119) || !((z <= -2.75e+45) || (!(z <= -3.15e-27) && (z <= 6e+130)))) {
		tmp = z * (-0.5 / t);
	} else {
		tmp = 0.5 * (x / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -2.65e+119) or not ((z <= -2.75e+45) or (not (z <= -3.15e-27) and (z <= 6e+130))):
		tmp = z * (-0.5 / t)
	else:
		tmp = 0.5 * (x / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2.65e+119) || !((z <= -2.75e+45) || (!(z <= -3.15e-27) && (z <= 6e+130))))
		tmp = Float64(z * Float64(-0.5 / t));
	else
		tmp = Float64(0.5 * Float64(x / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -2.65e+119) || ~(((z <= -2.75e+45) || (~((z <= -3.15e-27)) && (z <= 6e+130)))))
		tmp = z * (-0.5 / t);
	else
		tmp = 0.5 * (x / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -2.65e+119], N[Not[Or[LessEqual[z, -2.75e+45], And[N[Not[LessEqual[z, -3.15e-27]], $MachinePrecision], LessEqual[z, 6e+130]]]], $MachinePrecision]], N[(z * N[(-0.5 / t), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.64999999999999986e119 or -2.75e45 < z < -3.15000000000000005e-27 or 5.9999999999999999e130 < z

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 83.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{z}{t}} \]
   4. Step-by-step derivation

      1. *-commutative 83.2%

         \[\leadsto \color{blue}{\frac{z}{t} \cdot -0.5} \]

      2. associate-*l/ 83.2%

         \[\leadsto \color{blue}{\frac{z \cdot -0.5}{t}} \]

      3. associate-*r/ 82.9%

         \[\leadsto \color{blue}{z \cdot \frac{-0.5}{t}} \]

   5. Simplified 82.9%

      \[\leadsto \color{blue}{z \cdot \frac{-0.5}{t}} \]

   if -2.64999999999999986e119 < z < -2.75e45 or -3.15000000000000005e-27 < z < 5.9999999999999999e130

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 49.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.65 \cdot 10^{+119} \lor \neg \left(z \leq -2.75 \cdot 10^{+45} \lor \neg \left(z \leq -3.15 \cdot 10^{-27}\right) \land z \leq 6 \cdot 10^{+130}\right):\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 47.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.4 \cdot 10^{+53}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-279} \lor \neg \left(x \leq 8.5 \cdot 10^{-214}\right) \land x \leq 10^{-148}:\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot 0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -6.4e+53)
   (* 0.5 (/ x t))
   (if (or (<= x -1.7e-279) (and (not (<= x 8.5e-214)) (<= x 1e-148)))
     (/ (* z -0.5) t)
     (/ (* y 0.5) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -6.4e+53) {
		tmp = 0.5 * (x / t);
	} else if ((x <= -1.7e-279) || (!(x <= 8.5e-214) && (x <= 1e-148))) {
		tmp = (z * -0.5) / t;
	} else {
		tmp = (y * 0.5) / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-6.4d+53)) then
        tmp = 0.5d0 * (x / t)
    else if ((x <= (-1.7d-279)) .or. (.not. (x <= 8.5d-214)) .and. (x <= 1d-148)) then
        tmp = (z * (-0.5d0)) / t
    else
        tmp = (y * 0.5d0) / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -6.4e+53) {
		tmp = 0.5 * (x / t);
	} else if ((x <= -1.7e-279) || (!(x <= 8.5e-214) && (x <= 1e-148))) {
		tmp = (z * -0.5) / t;
	} else {
		tmp = (y * 0.5) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -6.4e+53:
		tmp = 0.5 * (x / t)
	elif (x <= -1.7e-279) or (not (x <= 8.5e-214) and (x <= 1e-148)):
		tmp = (z * -0.5) / t
	else:
		tmp = (y * 0.5) / t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -6.4e+53)
		tmp = Float64(0.5 * Float64(x / t));
	elseif ((x <= -1.7e-279) || (!(x <= 8.5e-214) && (x <= 1e-148)))
		tmp = Float64(Float64(z * -0.5) / t);
	else
		tmp = Float64(Float64(y * 0.5) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -6.4e+53)
		tmp = 0.5 * (x / t);
	elseif ((x <= -1.7e-279) || (~((x <= 8.5e-214)) && (x <= 1e-148)))
		tmp = (z * -0.5) / t;
	else
		tmp = (y * 0.5) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -6.4e+53], N[(0.5 * N[(x / t), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, -1.7e-279], And[N[Not[LessEqual[x, 8.5e-214]], $MachinePrecision], LessEqual[x, 1e-148]]], N[(N[(z * -0.5), $MachinePrecision] / t), $MachinePrecision], N[(N[(y * 0.5), $MachinePrecision] / t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.4e53

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 56.8%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x}{t}} \]

   if -6.4e53 < x < -1.70000000000000007e-279 or 8.5000000000000006e-214 < x < 9.99999999999999936e-149

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 53.4%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{z}{t}} \]
   4. Step-by-step derivation

      1. *-commutative 53.4%

         \[\leadsto \color{blue}{\frac{z}{t} \cdot -0.5} \]

      2. associate-*l/ 53.4%

         \[\leadsto \color{blue}{\frac{z \cdot -0.5}{t}} \]

   5. Simplified 53.4%

      \[\leadsto \color{blue}{\frac{z \cdot -0.5}{t}} \]

   if -1.70000000000000007e-279 < x < 8.5000000000000006e-214 or 9.99999999999999936e-149 < x

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 35.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{y}{t}} \]
   4. Step-by-step derivation

      1. associate-*r/ 35.5%

         \[\leadsto \color{blue}{\frac{0.5 \cdot y}{t}} \]

   5. Simplified 35.5%

      \[\leadsto \color{blue}{\frac{0.5 \cdot y}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.4 \cdot 10^{+53}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-279} \lor \neg \left(x \leq 8.5 \cdot 10^{-214}\right) \land x \leq 10^{-148}:\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot 0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{-11} \lor \neg \left(y \leq 14500000\right) \land y \leq 1.1 \cdot 10^{+64}:\\ \;\;\;\;0.5 \cdot \frac{x - z}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x + y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y 4.4e-11) (and (not (<= y 14500000.0)) (<= y 1.1e+64)))
   (* 0.5 (/ (- x z) t))
   (* 0.5 (/ (+ x y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= 4.4e-11) || (!(y <= 14500000.0) && (y <= 1.1e+64))) {
		tmp = 0.5 * ((x - z) / t);
	} else {
		tmp = 0.5 * ((x + y) / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y <= 4.4d-11) .or. (.not. (y <= 14500000.0d0)) .and. (y <= 1.1d+64)) then
        tmp = 0.5d0 * ((x - z) / t)
    else
        tmp = 0.5d0 * ((x + y) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= 4.4e-11) || (!(y <= 14500000.0) && (y <= 1.1e+64))) {
		tmp = 0.5 * ((x - z) / t);
	} else {
		tmp = 0.5 * ((x + y) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= 4.4e-11) or (not (y <= 14500000.0) and (y <= 1.1e+64)):
		tmp = 0.5 * ((x - z) / t)
	else:
		tmp = 0.5 * ((x + y) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= 4.4e-11) || (!(y <= 14500000.0) && (y <= 1.1e+64)))
		tmp = Float64(0.5 * Float64(Float64(x - z) / t));
	else
		tmp = Float64(0.5 * Float64(Float64(x + y) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= 4.4e-11) || (~((y <= 14500000.0)) && (y <= 1.1e+64)))
		tmp = 0.5 * ((x - z) / t);
	else
		tmp = 0.5 * ((x + y) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, 4.4e-11], And[N[Not[LessEqual[y, 14500000.0]], $MachinePrecision], LessEqual[y, 1.1e+64]]], N[(0.5 * N[(N[(x - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(x + y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.4000000000000003e-11 or 1.45e7 < y < 1.10000000000000001e64

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 77.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x - z}{t}} \]

   if 4.4000000000000003e-11 < y < 1.45e7 or 1.10000000000000001e64 < y

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 84.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x + y}{t}} \]
   4. Step-by-step derivation

      1. +-commutative 84.6%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{y + x}}{t} \]

   5. Simplified 84.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{y + x}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{-11} \lor \neg \left(y \leq 14500000\right) \land y \leq 1.1 \cdot 10^{+64}:\\ \;\;\;\;0.5 \cdot \frac{x - z}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x + y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+122} \lor \neg \left(z \leq 1.1 \cdot 10^{+130}\right):\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x + y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -3.2e+122) (not (<= z 1.1e+130)))
   (/ (* z -0.5) t)
   (* 0.5 (/ (+ x y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -3.2e+122) || !(z <= 1.1e+130)) {
		tmp = (z * -0.5) / t;
	} else {
		tmp = 0.5 * ((x + y) / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-3.2d+122)) .or. (.not. (z <= 1.1d+130))) then
        tmp = (z * (-0.5d0)) / t
    else
        tmp = 0.5d0 * ((x + y) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -3.2e+122) || !(z <= 1.1e+130)) {
		tmp = (z * -0.5) / t;
	} else {
		tmp = 0.5 * ((x + y) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -3.2e+122) or not (z <= 1.1e+130):
		tmp = (z * -0.5) / t
	else:
		tmp = 0.5 * ((x + y) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -3.2e+122) || !(z <= 1.1e+130))
		tmp = Float64(Float64(z * -0.5) / t);
	else
		tmp = Float64(0.5 * Float64(Float64(x + y) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -3.2e+122) || ~((z <= 1.1e+130)))
		tmp = (z * -0.5) / t;
	else
		tmp = 0.5 * ((x + y) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -3.2e+122], N[Not[LessEqual[z, 1.1e+130]], $MachinePrecision]], N[(N[(z * -0.5), $MachinePrecision] / t), $MachinePrecision], N[(0.5 * N[(N[(x + y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.20000000000000012e122 or 1.09999999999999997e130 < z

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 90.4%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{z}{t}} \]
   4. Step-by-step derivation

      1. *-commutative 90.4%

         \[\leadsto \color{blue}{\frac{z}{t} \cdot -0.5} \]

      2. associate-*l/ 90.4%

         \[\leadsto \color{blue}{\frac{z \cdot -0.5}{t}} \]

   5. Simplified 90.4%

      \[\leadsto \color{blue}{\frac{z \cdot -0.5}{t}} \]

   if -3.20000000000000012e122 < z < 1.09999999999999997e130

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 86.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x + y}{t}} \]
   4. Step-by-step derivation

      1. +-commutative 86.9%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{y + x}}{t} \]

   5. Simplified 86.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{y + x}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+122} \lor \neg \left(z \leq 1.1 \cdot 10^{+130}\right):\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x + y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 47.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.9 \cdot 10^{+53}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-280}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot 0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -5.9e+53)
   (* 0.5 (/ x t))
   (if (<= x -2.1e-280) (* z (/ -0.5 t)) (/ (* y 0.5) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -5.9e+53) {
		tmp = 0.5 * (x / t);
	} else if (x <= -2.1e-280) {
		tmp = z * (-0.5 / t);
	} else {
		tmp = (y * 0.5) / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-5.9d+53)) then
        tmp = 0.5d0 * (x / t)
    else if (x <= (-2.1d-280)) then
        tmp = z * ((-0.5d0) / t)
    else
        tmp = (y * 0.5d0) / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -5.9e+53) {
		tmp = 0.5 * (x / t);
	} else if (x <= -2.1e-280) {
		tmp = z * (-0.5 / t);
	} else {
		tmp = (y * 0.5) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -5.9e+53:
		tmp = 0.5 * (x / t)
	elif x <= -2.1e-280:
		tmp = z * (-0.5 / t)
	else:
		tmp = (y * 0.5) / t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -5.9e+53)
		tmp = Float64(0.5 * Float64(x / t));
	elseif (x <= -2.1e-280)
		tmp = Float64(z * Float64(-0.5 / t));
	else
		tmp = Float64(Float64(y * 0.5) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -5.9e+53)
		tmp = 0.5 * (x / t);
	elseif (x <= -2.1e-280)
		tmp = z * (-0.5 / t);
	else
		tmp = (y * 0.5) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -5.9e+53], N[(0.5 * N[(x / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.1e-280], N[(z * N[(-0.5 / t), $MachinePrecision]), $MachinePrecision], N[(N[(y * 0.5), $MachinePrecision] / t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.8999999999999997e53

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 56.8%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x}{t}} \]

   if -5.8999999999999997e53 < x < -2.10000000000000001e-280

   1. Initial program 99.9%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 48.4%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{z}{t}} \]
   4. Step-by-step derivation

      1. *-commutative 48.4%

         \[\leadsto \color{blue}{\frac{z}{t} \cdot -0.5} \]

      2. associate-*l/ 48.4%

         \[\leadsto \color{blue}{\frac{z \cdot -0.5}{t}} \]

      3. associate-*r/ 48.4%

         \[\leadsto \color{blue}{z \cdot \frac{-0.5}{t}} \]

   5. Simplified 48.4%

      \[\leadsto \color{blue}{z \cdot \frac{-0.5}{t}} \]

   if -2.10000000000000001e-280 < x

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 36.3%

      \[\leadsto \color{blue}{0.5 \cdot \frac{y}{t}} \]
   4. Step-by-step derivation

      1. associate-*r/ 36.3%

         \[\leadsto \color{blue}{\frac{0.5 \cdot y}{t}} \]

   5. Simplified 36.3%

      \[\leadsto \color{blue}{\frac{0.5 \cdot y}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.9 \cdot 10^{+53}:\\ \;\;\;\;0.5 \cdot \frac{x}{t}\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-280}:\\ \;\;\;\;z \cdot \frac{-0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot 0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.6 \cdot 10^{-18}:\\ \;\;\;\;0.5 \cdot \frac{x - z}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 9.6e-18) (* 0.5 (/ (- x z) t)) (* (- y z) (/ 0.5 t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 9.6e-18) {
		tmp = 0.5 * ((x - z) / t);
	} else {
		tmp = (y - z) * (0.5 / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 9.6d-18) then
        tmp = 0.5d0 * ((x - z) / t)
    else
        tmp = (y - z) * (0.5d0 / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 9.6e-18) {
		tmp = 0.5 * ((x - z) / t);
	} else {
		tmp = (y - z) * (0.5 / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 9.6e-18:
		tmp = 0.5 * ((x - z) / t)
	else:
		tmp = (y - z) * (0.5 / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 9.6e-18)
		tmp = Float64(0.5 * Float64(Float64(x - z) / t));
	else
		tmp = Float64(Float64(y - z) * Float64(0.5 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 9.6e-18)
		tmp = 0.5 * ((x - z) / t);
	else
		tmp = (y - z) * (0.5 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 9.6e-18], N[(0.5 * N[(N[(x - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(y - z), $MachinePrecision] * N[(0.5 / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.59999999999999976e-18

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 77.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x - z}{t}} \]

   if 9.59999999999999976e-18 < y

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 84.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{y - z}{t}} \]
   4. Step-by-step derivation

      1. *-commutative 84.0%

         \[\leadsto \color{blue}{\frac{y - z}{t} \cdot 0.5} \]

      2. associate-*l/ 84.0%

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 0.5}{t}} \]

      3. associate-*r/ 83.7%

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{0.5}{t}} \]

   5. Simplified 83.7%

      \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{0.5}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.6 \cdot 10^{-18}:\\ \;\;\;\;0.5 \cdot \frac{x - z}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 37.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{x}{t} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* 0.5 (/ x t)))

C

double code(double x, double y, double z, double t) {
	return 0.5 * (x / t);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 0.5d0 * (x / t)
end function

Java

public static double code(double x, double y, double z, double t) {
	return 0.5 * (x / t);
}

Python

def code(x, y, z, t):
	return 0.5 * (x / t)

Julia

function code(x, y, z, t)
	return Float64(0.5 * Float64(x / t))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 0.5 * (x / t);
end

Wolfram

code[x_, y_, z_, t_] := N[(0.5 * N[(x / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 34.0%

   \[\leadsto \color{blue}{0.5 \cdot \frac{x}{t}} \]

4. Final simplification 34.0%

   \[\leadsto 0.5 \cdot \frac{x}{t} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024066 
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, B"
  :precision binary64
  (/ (- (+ x y) z) (* t 2.0)))