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

Percentage Accurate: 93.3% → 96.2%

Time: 7.1s

Alternatives: 8

Speedup: 1.0×

• 25.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{+285}:\\ \;\;\;\;t_1 + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)))
   (if (<= t_1 2e+285) (+ t_1 x) (+ x (/ y (/ a (- z t)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double tmp;
	if (t_1 <= 2e+285) {
		tmp = t_1 + x;
	} else {
		tmp = x + (y / (a / (z - t)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y * (z - t)) / a
    if (t_1 <= 2d+285) then
        tmp = t_1 + x
    else
        tmp = x + (y / (a / (z - t)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double tmp;
	if (t_1 <= 2e+285) {
		tmp = t_1 + x;
	} else {
		tmp = x + (y / (a / (z - t)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y * (z - t)) / a
	tmp = 0
	if t_1 <= 2e+285:
		tmp = t_1 + x
	else:
		tmp = x + (y / (a / (z - t)))
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(z - t)) / a)
	tmp = 0.0
	if (t_1 <= 2e+285)
		tmp = Float64(t_1 + x);
	else
		tmp = Float64(x + Float64(y / Float64(a / Float64(z - t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y * (z - t)) / a;
	tmp = 0.0;
	if (t_1 <= 2e+285)
		tmp = t_1 + x;
	else
		tmp = x + (y / (a / (z - t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+285], N[(t$95$1 + x), $MachinePrecision], N[(x + N[(y / N[(a / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < 2e285

   1. Initial program 98.2%

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

   if 2e285 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 72.7%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

         \[\leadsto x + \color{blue}{\frac{y}{\frac{a}{z - t}}} \]

   3. Simplified 100.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq 2 \cdot 10^{+285}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a} + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Alternative 2: 85.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{+40} \lor \neg \left(t \leq 2.5 \cdot 10^{+14}\right):\\ \;\;\;\;x - t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -5.6e+40) (not (<= t 2.5e+14)))
   (- x (* t (/ y a)))
   (+ x (/ y (/ a z)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -5.6e+40) || !(t <= 2.5e+14)) {
		tmp = x - (t * (y / a));
	} else {
		tmp = x + (y / (a / z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((t <= (-5.6d+40)) .or. (.not. (t <= 2.5d+14))) then
        tmp = x - (t * (y / a))
    else
        tmp = x + (y / (a / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -5.6e+40) || !(t <= 2.5e+14)) {
		tmp = x - (t * (y / a));
	} else {
		tmp = x + (y / (a / z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (t <= -5.6e+40) or not (t <= 2.5e+14):
		tmp = x - (t * (y / a))
	else:
		tmp = x + (y / (a / z))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -5.6e+40) || !(t <= 2.5e+14))
		tmp = Float64(x - Float64(t * Float64(y / a)));
	else
		tmp = Float64(x + Float64(y / Float64(a / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((t <= -5.6e+40) || ~((t <= 2.5e+14)))
		tmp = x - (t * (y / a));
	else
		tmp = x + (y / (a / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -5.6e+40], N[Not[LessEqual[t, 2.5e+14]], $MachinePrecision]], N[(x - N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.6000000000000003e40 or 2.5e14 < t

   1. Initial program 92.1%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   2. Step-by-step derivation

      1. associate-*l/ 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in z around 0 85.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot t}{a} + x} \]
   5. Step-by-step derivation

      1. mul-1-neg 85.4%

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

      2. associate-*l/ 90.2%

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

      3. distribute-rgt-neg-out 90.2%

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

      4. +-commutative 90.2%

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

      5. *-commutative 90.2%

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

      6. distribute-lft-neg-out 90.2%

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

      7. unsub-neg 90.2%

         \[\leadsto \color{blue}{x - t \cdot \frac{y}{a}} \]

   6. Simplified 90.2%

      \[\leadsto \color{blue}{x - t \cdot \frac{y}{a}} \]

   if -5.6000000000000003e40 < t < 2.5e14

   1. Initial program 95.5%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   2. Step-by-step derivation

      1. associate-/l* 96.8%

         \[\leadsto x + \color{blue}{\frac{y}{\frac{a}{z - t}}} \]

   3. Simplified 96.8%

      \[\leadsto \color{blue}{x + \frac{y}{\frac{a}{z - t}}} \]

   4. Taylor expanded in z around inf 92.2%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{+40} \lor \neg \left(t \leq 2.5 \cdot 10^{+14}\right):\\ \;\;\;\;x - t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z}}\\ \end{array} \]

Alternative 3: 83.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{+99}:\\ \;\;\;\;x - y \cdot \frac{t}{a}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+24}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x - t \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -2.5e+99)
   (- x (* y (/ t a)))
   (if (<= t 1.25e+24) (+ x (/ y (/ a z))) (- x (* t (/ y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -2.5e+99) {
		tmp = x - (y * (t / a));
	} else if (t <= 1.25e+24) {
		tmp = x + (y / (a / z));
	} else {
		tmp = x - (t * (y / a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= (-2.5d+99)) then
        tmp = x - (y * (t / a))
    else if (t <= 1.25d+24) then
        tmp = x + (y / (a / z))
    else
        tmp = x - (t * (y / a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -2.5e+99) {
		tmp = x - (y * (t / a));
	} else if (t <= 1.25e+24) {
		tmp = x + (y / (a / z));
	} else {
		tmp = x - (t * (y / a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -2.5e+99:
		tmp = x - (y * (t / a))
	elif t <= 1.25e+24:
		tmp = x + (y / (a / z))
	else:
		tmp = x - (t * (y / a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -2.5e+99)
		tmp = Float64(x - Float64(y * Float64(t / a)));
	elseif (t <= 1.25e+24)
		tmp = Float64(x + Float64(y / Float64(a / z)));
	else
		tmp = Float64(x - Float64(t * Float64(y / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -2.5e+99)
		tmp = x - (y * (t / a));
	elseif (t <= 1.25e+24)
		tmp = x + (y / (a / z));
	else
		tmp = x - (t * (y / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -2.5e+99], N[(x - N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.25e+24], N[(x + N[(y / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.50000000000000004e99

   1. Initial program 88.3%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   2. Step-by-step derivation

      1. associate-*l/ 96.8%

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

   3. Simplified 96.8%

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

      1. *-commutative 96.8%

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

      2. clear-num 96.8%

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

      3. un-div-inv 96.8%

         \[\leadsto x + \color{blue}{\frac{z - t}{\frac{a}{y}}} \]

   5. Applied egg-rr 96.8%

      \[\leadsto x + \color{blue}{\frac{z - t}{\frac{a}{y}}} \]

   6. Taylor expanded in z around 0 85.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot t}{a} + x} \]
   7. Step-by-step derivation

      1. +-commutative 85.4%

         \[\leadsto \color{blue}{x + -1 \cdot \frac{y \cdot t}{a}} \]

      2. mul-1-neg 85.4%

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

      3. associate-*r/ 91.3%

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

      4. distribute-lft-neg-out 91.3%

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

      5. cancel-sign-sub-inv 91.3%

         \[\leadsto \color{blue}{x - y \cdot \frac{t}{a}} \]

   8. Simplified 91.3%

      \[\leadsto \color{blue}{x - y \cdot \frac{t}{a}} \]

   if -2.50000000000000004e99 < t < 1.25000000000000011e24

   1. Initial program 95.8%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   2. Step-by-step derivation

      1. associate-/l* 96.4%

         \[\leadsto x + \color{blue}{\frac{y}{\frac{a}{z - t}}} \]

   3. Simplified 96.4%

      \[\leadsto \color{blue}{x + \frac{y}{\frac{a}{z - t}}} \]

   4. Taylor expanded in z around inf 90.8%

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

   if 1.25000000000000011e24 < t

   1. Initial program 92.9%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 87.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot t}{a} + x} \]
   5. Step-by-step derivation

      1. mul-1-neg 87.8%

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

      2. associate-*l/ 92.8%

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

      3. distribute-rgt-neg-out 92.8%

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

      4. +-commutative 92.8%

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

      5. *-commutative 92.8%

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

      6. distribute-lft-neg-out 92.8%

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

      7. unsub-neg 92.8%

         \[\leadsto \color{blue}{x - t \cdot \frac{y}{a}} \]

   6. Simplified 92.8%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{+99}:\\ \;\;\;\;x - y \cdot \frac{t}{a}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+24}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x - t \cdot \frac{y}{a}\\ \end{array} \]

Alternative 4: 95.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{-112}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 2.4e-112) (+ x (/ y (/ a (- z t)))) (+ x (* (- z t) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2.4e-112) {
		tmp = x + (y / (a / (z - t)));
	} else {
		tmp = x + ((z - t) * (y / a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 2.4d-112) then
        tmp = x + (y / (a / (z - t)))
    else
        tmp = x + ((z - t) * (y / a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2.4e-112) {
		tmp = x + (y / (a / (z - t)));
	} else {
		tmp = x + ((z - t) * (y / a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 2.4e-112:
		tmp = x + (y / (a / (z - t)))
	else:
		tmp = x + ((z - t) * (y / a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 2.4e-112)
		tmp = Float64(x + Float64(y / Float64(a / Float64(z - t))));
	else
		tmp = Float64(x + Float64(Float64(z - t) * Float64(y / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 2.4e-112)
		tmp = x + (y / (a / (z - t)));
	else
		tmp = x + ((z - t) * (y / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 2.4e-112], N[(x + N[(y / N[(a / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(z - t), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.4000000000000001e-112

   1. Initial program 93.9%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   2. Step-by-step derivation

      1. associate-/l* 97.2%

         \[\leadsto x + \color{blue}{\frac{y}{\frac{a}{z - t}}} \]

   3. Simplified 97.2%

      \[\leadsto \color{blue}{x + \frac{y}{\frac{a}{z - t}}} \]

   if 2.4000000000000001e-112 < t

   1. Initial program 94.5%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{-112}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \end{array} \]

Alternative 5: 69.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{-47}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z 2e-47) (+ x (/ y (/ a z))) (+ x (* z (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 2e-47) {
		tmp = x + (y / (a / z));
	} else {
		tmp = x + (z * (y / a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= 2d-47) then
        tmp = x + (y / (a / z))
    else
        tmp = x + (z * (y / a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 2e-47) {
		tmp = x + (y / (a / z));
	} else {
		tmp = x + (z * (y / a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= 2e-47:
		tmp = x + (y / (a / z))
	else:
		tmp = x + (z * (y / a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= 2e-47)
		tmp = Float64(x + Float64(y / Float64(a / z)));
	else
		tmp = Float64(x + Float64(z * Float64(y / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= 2e-47)
		tmp = x + (y / (a / z));
	else
		tmp = x + (z * (y / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, 2e-47], N[(x + N[(y / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.9999999999999999e-47

   1. Initial program 95.7%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   2. Step-by-step derivation

      1. associate-/l* 97.3%

         \[\leadsto x + \color{blue}{\frac{y}{\frac{a}{z - t}}} \]

   3. Simplified 97.3%

      \[\leadsto \color{blue}{x + \frac{y}{\frac{a}{z - t}}} \]

   4. Taylor expanded in z around inf 70.4%

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

   if 1.9999999999999999e-47 < z

   1. Initial program 90.3%

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in t around 0 78.8%

      \[\leadsto \color{blue}{\frac{y \cdot z}{a} + x} \]
   5. Step-by-step derivation

      1. associate-*l/ 84.0%

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

      2. *-commutative 84.0%

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

   6. Simplified 84.0%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{-47}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \end{array} \]

Alternative 6: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

   \[x + \frac{y \cdot \left(z - t\right)}{a} \]
2. Step-by-step derivation

   1. associate-*l/ 95.9%

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

3. Simplified 95.9%

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

4. Final simplification 95.9%

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

Alternative 7: 68.3% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (/ y (/ a z))))

C

double code(double x, double y, double z, double t, double a) {
	return x + (y / (a / z));
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + (y / (a / z))
end function

Java

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

Python

def code(x, y, z, t, a):
	return x + (y / (a / z))

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + N[(y / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.1%

   \[x + \frac{y \cdot \left(z - t\right)}{a} \]
2. Step-by-step derivation

   1. associate-/l* 96.0%

      \[\leadsto x + \color{blue}{\frac{y}{\frac{a}{z - t}}} \]

3. Simplified 96.0%

   \[\leadsto \color{blue}{x + \frac{y}{\frac{a}{z - t}}} \]

4. Taylor expanded in z around inf 72.6%

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

5. Final simplification 72.6%

   \[\leadsto x + \frac{y}{\frac{a}{z}} \]

Alternative 8: 39.5% accurate, 9.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return x;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := x

Derivation

1. Initial program 94.1%

   \[x + \frac{y \cdot \left(z - t\right)}{a} \]
2. Step-by-step derivation

   1. associate-*l/ 95.9%

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

3. Simplified 95.9%

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

4. Taylor expanded in x around inf 41.4%

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

5. Final simplification 41.4%

   \[\leadsto x \]

Developer target: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{z - t}\\ \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{t_1}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ a (- z t))))
   (if (< y -1.0761266216389975e-10)
     (+ x (/ 1.0 (/ t_1 y)))
     (if (< y 2.894426862792089e-49)
       (+ x (/ (* y (- z t)) a))
       (+ x (/ y t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a / (z - t);
	double tmp;
	if (y < -1.0761266216389975e-10) {
		tmp = x + (1.0 / (t_1 / y));
	} else if (y < 2.894426862792089e-49) {
		tmp = x + ((y * (z - t)) / a);
	} else {
		tmp = x + (y / t_1);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a / (z - t)
    if (y < (-1.0761266216389975d-10)) then
        tmp = x + (1.0d0 / (t_1 / y))
    else if (y < 2.894426862792089d-49) then
        tmp = x + ((y * (z - t)) / a)
    else
        tmp = x + (y / t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a / (z - t);
	double tmp;
	if (y < -1.0761266216389975e-10) {
		tmp = x + (1.0 / (t_1 / y));
	} else if (y < 2.894426862792089e-49) {
		tmp = x + ((y * (z - t)) / a);
	} else {
		tmp = x + (y / t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a / (z - t)
	tmp = 0
	if y < -1.0761266216389975e-10:
		tmp = x + (1.0 / (t_1 / y))
	elif y < 2.894426862792089e-49:
		tmp = x + ((y * (z - t)) / a)
	else:
		tmp = x + (y / t_1)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a / Float64(z - t))
	tmp = 0.0
	if (y < -1.0761266216389975e-10)
		tmp = Float64(x + Float64(1.0 / Float64(t_1 / y)));
	elseif (y < 2.894426862792089e-49)
		tmp = Float64(x + Float64(Float64(y * Float64(z - t)) / a));
	else
		tmp = Float64(x + Float64(y / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a / (z - t);
	tmp = 0.0;
	if (y < -1.0761266216389975e-10)
		tmp = x + (1.0 / (t_1 / y));
	elseif (y < 2.894426862792089e-49)
		tmp = x + ((y * (z - t)) / a);
	else
		tmp = x + (y / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a / N[(z - t), $MachinePrecision]), $MachinePrecision]}, If[Less[y, -1.0761266216389975e-10], N[(x + N[(1.0 / N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Less[y, 2.894426862792089e-49], N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2023240 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :precision binary64

  :herbie-target
  (if (< y -1.0761266216389975e-10) (+ x (/ 1.0 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

  (+ x (/ (* y (- z t)) a)))