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

Average Error: 6.1 → 0.6

Time: 12.0s

Precision: binary64

Cost: 1352

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+162}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+165}:\\ \;\;\;\;x + \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (- z t))))
   (if (<= t_1 -5e+162)
     (+ x (/ y (/ a (- z t))))
     (if (<= t_1 2e+165) (+ x (/ t_1 a)) (+ x (* (- z t) (/ y a)))))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (z - t);
	double tmp;
	if (t_1 <= -5e+162) {
		tmp = x + (y / (a / (z - t)));
	} else if (t_1 <= 2e+165) {
		tmp = x + (t_1 / a);
	} 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
    code = x + ((y * (z - t)) / a)
end function

↓

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)
    if (t_1 <= (-5d+162)) then
        tmp = x + (y / (a / (z - t)))
    else if (t_1 <= 2d+165) then
        tmp = x + (t_1 / a)
    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) {
	return x + ((y * (z - t)) / a);
}

↓

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (z - t);
	double tmp;
	if (t_1 <= -5e+162) {
		tmp = x + (y / (a / (z - t)));
	} else if (t_1 <= 2e+165) {
		tmp = x + (t_1 / a);
	} else {
		tmp = x + ((z - t) * (y / a));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t, a):
	t_1 = y * (z - t)
	tmp = 0
	if t_1 <= -5e+162:
		tmp = x + (y / (a / (z - t)))
	elif t_1 <= 2e+165:
		tmp = x + (t_1 / a)
	else:
		tmp = x + ((z - t) * (y / a))
	return tmp

Julia

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

↓

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(z - t))
	tmp = 0.0
	if (t_1 <= -5e+162)
		tmp = Float64(x + Float64(y / Float64(a / Float64(z - t))));
	elseif (t_1 <= 2e+165)
		tmp = Float64(x + Float64(t_1 / a));
	else
		tmp = Float64(x + Float64(Float64(z - t) * Float64(y / a)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z, t, a)
	t_1 = y * (z - t);
	tmp = 0.0;
	if (t_1 <= -5e+162)
		tmp = x + (y / (a / (z - t)));
	elseif (t_1 <= 2e+165)
		tmp = x + (t_1 / a);
	else
		tmp = x + ((z - t) * (y / a));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+162], N[(x + N[(y / N[(a / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+165], N[(x + N[(t$95$1 / a), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(z - t), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	6.1
Target	0.7
Herbie	0.6

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (-.f64 z t)) < -4.9999999999999997e162

   1. Initial program 23.2

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

   2. Simplified 1.6

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

      [Start] 23.2	\[ x + \frac{y \cdot \left(z - t\right)}{a} \]
      associate-/l* [=>] 1.6	\[ x + \color{blue}{\frac{y}{\frac{a}{z - t}}} \]

   if -4.9999999999999997e162 < (*.f64 y (-.f64 z t)) < 1.9999999999999998e165

   1. Initial program 0.4

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

   if 1.9999999999999998e165 < (*.f64 y (-.f64 z t))

   1. Initial program 22.7

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

   2. Simplified 0.9

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

      [Start] 22.7	\[ x + \frac{y \cdot \left(z - t\right)}{a} \]
      associate-*l/ [<=] 0.9	\[ x + \color{blue}{\frac{y}{a} \cdot \left(z - t\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -5 \cdot 10^{+162}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq 2 \cdot 10^{+165}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	20.1
Cost	1240

\[\begin{array}{l} t_1 := x + y \cdot \frac{z}{a}\\ t_2 := -t \cdot \frac{y}{a}\\ \mathbf{if}\;x \leq -4 \cdot 10^{-121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-166}:\\ \;\;\;\;\frac{-y \cdot t}{a}\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-291}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-276}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-218}:\\ \;\;\;\;\frac{z}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-191}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	1.6
Cost	1097

\[\begin{array}{l} \mathbf{if}\;z - t \leq -2 \cdot 10^{-102} \lor \neg \left(z - t \leq 10^{+37}\right):\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Alternative 3
Error	27.9
Cost	980

\[\begin{array}{l} t_1 := \frac{z}{\frac{a}{y}}\\ t_2 := -t \cdot \frac{y}{a}\\ \mathbf{if}\;x \leq -8.8 \cdot 10^{-94}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-276}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-218}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-190}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-39}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	28.1
Cost	849

\[\begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-92}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-230} \lor \neg \left(x \leq -1.65 \cdot 10^{-279}\right) \land x \leq 6.8 \cdot 10^{-39}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	28.1
Cost	849

\[\begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{-90}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-230} \lor \neg \left(x \leq -1.65 \cdot 10^{-279}\right) \land x \leq 2.3 \cdot 10^{-38}:\\ \;\;\;\;\frac{z}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	2.7
Cost	841

\[\begin{array}{l} \mathbf{if}\;t \leq -3.1 \cdot 10^{-239} \lor \neg \left(t \leq 10^{-259}\right):\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{a}\\ \end{array} \]

Alternative 7
Error	17.4
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-290} \lor \neg \left(x \leq 9.2 \cdot 10^{-278}\right):\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;-t \cdot \frac{y}{a}\\ \end{array} \]

Alternative 8
Error	17.3
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-290} \lor \neg \left(x \leq 5.4 \cdot 10^{-281}\right):\\ \;\;\;\;x + \frac{z}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;-t \cdot \frac{y}{a}\\ \end{array} \]

Alternative 9
Error	10.0
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{-41} \lor \neg \left(t \leq 4.5 \cdot 10^{-85}\right):\\ \;\;\;\;x - \frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{\frac{a}{y}}\\ \end{array} \]

Alternative 10
Error	9.9
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{-20} \lor \neg \left(t \leq 4.5 \cdot 10^{-85}\right):\\ \;\;\;\;x - t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{\frac{a}{y}}\\ \end{array} \]

Alternative 11
Error	30.7
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023016 
(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)))