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

Percentage Accurate: 99.9% → 99.9%

Time: 4.2s

Alternatives: 6

Speedup: 1.0×

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

• 26.5% 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. Final simplification 100.0%

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

Alternative 2: 46.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.1 \cdot 10^{+102}:\\ \;\;\;\;\frac{x \cdot 0.5}{t}\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-262} \lor \neg \left(x \leq -2.5 \cdot 10^{-285}\right) \land x \leq 1.7 \cdot 10^{-260}:\\ \;\;\;\;\frac{z}{t} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -7.1e+102)
   (/ (* x 0.5) t)
   (if (or (<= x -2.3e-262) (and (not (<= x -2.5e-285)) (<= x 1.7e-260)))
     (* (/ z t) -0.5)
     (* 0.5 (/ y t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -7.1e+102) {
		tmp = (x * 0.5) / t;
	} else if ((x <= -2.3e-262) || (!(x <= -2.5e-285) && (x <= 1.7e-260))) {
		tmp = (z / t) * -0.5;
	} else {
		tmp = 0.5 * (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 (x <= (-7.1d+102)) then
        tmp = (x * 0.5d0) / t
    else if ((x <= (-2.3d-262)) .or. (.not. (x <= (-2.5d-285))) .and. (x <= 1.7d-260)) then
        tmp = (z / t) * (-0.5d0)
    else
        tmp = 0.5d0 * (y / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -7.1e+102) {
		tmp = (x * 0.5) / t;
	} else if ((x <= -2.3e-262) || (!(x <= -2.5e-285) && (x <= 1.7e-260))) {
		tmp = (z / t) * -0.5;
	} else {
		tmp = 0.5 * (y / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -7.1e+102:
		tmp = (x * 0.5) / t
	elif (x <= -2.3e-262) or (not (x <= -2.5e-285) and (x <= 1.7e-260)):
		tmp = (z / t) * -0.5
	else:
		tmp = 0.5 * (y / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -7.1e+102)
		tmp = Float64(Float64(x * 0.5) / t);
	elseif ((x <= -2.3e-262) || (!(x <= -2.5e-285) && (x <= 1.7e-260)))
		tmp = Float64(Float64(z / t) * -0.5);
	else
		tmp = Float64(0.5 * Float64(y / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -7.1e+102)
		tmp = (x * 0.5) / t;
	elseif ((x <= -2.3e-262) || (~((x <= -2.5e-285)) && (x <= 1.7e-260)))
		tmp = (z / t) * -0.5;
	else
		tmp = 0.5 * (y / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -7.1e+102], N[(N[(x * 0.5), $MachinePrecision] / t), $MachinePrecision], If[Or[LessEqual[x, -2.3e-262], And[N[Not[LessEqual[x, -2.5e-285]], $MachinePrecision], LessEqual[x, 1.7e-260]]], N[(N[(z / t), $MachinePrecision] * -0.5), $MachinePrecision], N[(0.5 * N[(y / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.0999999999999998e102

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 74.3%

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

      1. associate-*r/ 74.3%

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

   4. Simplified 74.3%

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

   if -7.0999999999999998e102 < x < -2.3000000000000001e-262 or -2.50000000000000009e-285 < x < 1.6999999999999999e-260

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 54.2%

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

      1. *-commutative 54.2%

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

   4. Simplified 54.2%

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

   if -2.3000000000000001e-262 < x < -2.50000000000000009e-285 or 1.6999999999999999e-260 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 38.9%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.1 \cdot 10^{+102}:\\ \;\;\;\;\frac{x \cdot 0.5}{t}\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-262} \lor \neg \left(x \leq -2.5 \cdot 10^{-285}\right) \land x \leq 1.7 \cdot 10^{-260}:\\ \;\;\;\;\frac{z}{t} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{t}\\ \end{array} \]

Alternative 3: 56.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+88} \lor \neg \left(z \leq 3.1 \cdot 10^{+47}\right):\\ \;\;\;\;\frac{z}{t} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -6.2e+88) (not (<= z 3.1e+47)))
   (* (/ z t) -0.5)
   (* 0.5 (/ y t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -6.2e+88) || !(z <= 3.1e+47)) {
		tmp = (z / t) * -0.5;
	} else {
		tmp = 0.5 * (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 <= (-6.2d+88)) .or. (.not. (z <= 3.1d+47))) then
        tmp = (z / t) * (-0.5d0)
    else
        tmp = 0.5d0 * (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 <= -6.2e+88) || !(z <= 3.1e+47)) {
		tmp = (z / t) * -0.5;
	} else {
		tmp = 0.5 * (y / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -6.2e+88) or not (z <= 3.1e+47):
		tmp = (z / t) * -0.5
	else:
		tmp = 0.5 * (y / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -6.2e+88) || !(z <= 3.1e+47))
		tmp = Float64(Float64(z / t) * -0.5);
	else
		tmp = Float64(0.5 * Float64(y / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -6.2e+88) || ~((z <= 3.1e+47)))
		tmp = (z / t) * -0.5;
	else
		tmp = 0.5 * (y / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -6.2e+88], N[Not[LessEqual[z, 3.1e+47]], $MachinePrecision]], N[(N[(z / t), $MachinePrecision] * -0.5), $MachinePrecision], N[(0.5 * N[(y / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.2000000000000003e88 or 3.1000000000000001e47 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 74.2%

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

      1. *-commutative 74.2%

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

   4. Simplified 74.2%

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

   if -6.2000000000000003e88 < z < 3.1000000000000001e47

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 51.3%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+88} \lor \neg \left(z \leq 3.1 \cdot 10^{+47}\right):\\ \;\;\;\;\frac{z}{t} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{t}\\ \end{array} \]

Alternative 4: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.1 \cdot 10^{+157}:\\ \;\;\;\;0.5 \cdot \frac{x - z}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 7.1e+157) (* 0.5 (/ (- x z) t)) (* 0.5 (/ y t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 7.1e+157) {
		tmp = 0.5 * ((x - z) / t);
	} else {
		tmp = 0.5 * (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 <= 7.1d+157) then
        tmp = 0.5d0 * ((x - z) / t)
    else
        tmp = 0.5d0 * (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 <= 7.1e+157) {
		tmp = 0.5 * ((x - z) / t);
	} else {
		tmp = 0.5 * (y / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 7.1e+157:
		tmp = 0.5 * ((x - z) / t)
	else:
		tmp = 0.5 * (y / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 7.1e+157)
		tmp = Float64(0.5 * Float64(Float64(x - z) / t));
	else
		tmp = Float64(0.5 * Float64(y / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 7.1e+157)
		tmp = 0.5 * ((x - z) / t);
	else
		tmp = 0.5 * (y / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 7.1e+157], N[(0.5 * N[(N[(x - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.10000000000000026e157

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 74.4%

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

   if 7.10000000000000026e157 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 81.4%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.1 \cdot 10^{+157}:\\ \;\;\;\;0.5 \cdot \frac{x - z}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{t}\\ \end{array} \]

Alternative 5: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-59}:\\ \;\;\;\;0.5 \cdot \frac{x - z}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y - z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -3.4e-59) (* 0.5 (/ (- x z) t)) (* 0.5 (/ (- y z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -3.4e-59) {
		tmp = 0.5 * ((x - z) / t);
	} else {
		tmp = 0.5 * ((y - z) / 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 <= (-3.4d-59)) then
        tmp = 0.5d0 * ((x - z) / t)
    else
        tmp = 0.5d0 * ((y - z) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -3.4e-59) {
		tmp = 0.5 * ((x - z) / t);
	} else {
		tmp = 0.5 * ((y - z) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -3.4e-59:
		tmp = 0.5 * ((x - z) / t)
	else:
		tmp = 0.5 * ((y - z) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -3.4e-59)
		tmp = Float64(0.5 * Float64(Float64(x - z) / t));
	else
		tmp = Float64(0.5 * Float64(Float64(y - z) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -3.4e-59)
		tmp = 0.5 * ((x - z) / t);
	else
		tmp = 0.5 * ((y - z) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -3.4e-59], N[(0.5 * N[(N[(x - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(y - z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.40000000000000018e-59

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 81.3%

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

   if -3.40000000000000018e-59 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 79.7%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-59}:\\ \;\;\;\;0.5 \cdot \frac{x - z}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y - z}{t}\\ \end{array} \]

Alternative 6: 37.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in y around inf 36.0%

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

3. Final simplification 36.0%

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

Reproduce

herbie shell --seed 2023173 
(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)))