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

Average Error: 6.3 → 0.8

Time: 13.1s

Precision: binary64

Cost: 1608

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+274}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+288}:\\ \;\;\;\;t_1 + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

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)) a)))
   (if (<= t_1 -5e+274)
     (+ x (* (- z t) (/ y a)))
     (if (<= t_1 5e+288) (+ t_1 x) (+ x (/ y (/ a (- z t))))))))

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)) / a;
	double tmp;
	if (t_1 <= -5e+274) {
		tmp = x + ((z - t) * (y / a));
	} else if (t_1 <= 5e+288) {
		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
    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)) / a
    if (t_1 <= (-5d+274)) then
        tmp = x + ((z - t) * (y / a))
    else if (t_1 <= 5d+288) 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) {
	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)) / a;
	double tmp;
	if (t_1 <= -5e+274) {
		tmp = x + ((z - t) * (y / a));
	} else if (t_1 <= 5e+288) {
		tmp = t_1 + x;
	} else {
		tmp = x + (y / (a / (z - t)));
	}
	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)) / a
	tmp = 0
	if t_1 <= -5e+274:
		tmp = x + ((z - t) * (y / a))
	elif t_1 <= 5e+288:
		tmp = t_1 + x
	else:
		tmp = x + (y / (a / (z - t)))
	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(Float64(y * Float64(z - t)) / a)
	tmp = 0.0
	if (t_1 <= -5e+274)
		tmp = Float64(x + Float64(Float64(z - t) * Float64(y / a)));
	elseif (t_1 <= 5e+288)
		tmp = Float64(t_1 + x);
	else
		tmp = Float64(x + Float64(y / Float64(a / Float64(z - t))));
	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)) / a;
	tmp = 0.0;
	if (t_1 <= -5e+274)
		tmp = x + ((z - t) * (y / a));
	elseif (t_1 <= 5e+288)
		tmp = t_1 + x;
	else
		tmp = x + (y / (a / (z - t)));
	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[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+274], N[(x + N[(N[(z - t), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+288], N[(t$95$1 + x), $MachinePrecision], N[(x + N[(y / N[(a / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	6.3
Target	0.7
Herbie	0.8

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -4.9999999999999998e274

   1. Initial program 46.1

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

   2. Simplified 2.8

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

      [Start] 46.1	\[ x + \frac{y \cdot \left(z - t\right)}{a} \]
      associate-*l/ [<=] 2.8	\[ x + \color{blue}{\frac{y}{a} \cdot \left(z - t\right)} \]

   if -4.9999999999999998e274 < (/.f64 (*.f64 y (-.f64 z t)) a) < 5.0000000000000003e288

   1. Initial program 0.4

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

   if 5.0000000000000003e288 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 53.0

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

   2. Simplified 4.7

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

      [Start] 53.0	\[ x + \frac{y \cdot \left(z - t\right)}{a} \]
      associate-/l* [=>] 4.7	\[ x + \color{blue}{\frac{y}{\frac{a}{z - t}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq -5 \cdot 10^{+274}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 5 \cdot 10^{+288}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a} + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Alternatives

Alternative 1
Error	20.8
Cost	1505

\[\begin{array}{l} t_1 := y \cdot \frac{z - t}{a}\\ t_2 := x + y \cdot \frac{z}{a}\\ t_3 := -\frac{y \cdot t}{a}\\ \mathbf{if}\;a \leq -5.4 \cdot 10^{+63}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -5.5 \cdot 10^{+43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -3.2 \cdot 10^{-184}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{-286}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{-266}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{-195}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq 1.5 \cdot 10^{+27} \lor \neg \left(a \leq 2.7 \cdot 10^{+82}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	1.3
Cost	1097

\[\begin{array}{l} \mathbf{if}\;z - t \leq -1 \cdot 10^{+37} \lor \neg \left(z - t \leq 2 \cdot 10^{+47}\right):\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Alternative 3
Error	28.2
Cost	716

\[\begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-91}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-256}:\\ \;\;\;\;y \cdot \frac{-t}{a}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-132}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	28.1
Cost	716

\[\begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-87}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-260}:\\ \;\;\;\;t \cdot \frac{-y}{a}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-132}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	14.2
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-109} \lor \neg \left(x \leq 1.25 \cdot 10^{-135}\right):\\ \;\;\;\;x + \frac{z}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - t}{a}\\ \end{array} \]

Alternative 6
Error	14.3
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-109} \lor \neg \left(x \leq 8.5 \cdot 10^{-136}\right):\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - t}{a}\\ \end{array} \]

Alternative 7
Error	9.4
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-25} \lor \neg \left(z \leq 0.017\right):\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x - t \cdot \frac{y}{a}\\ \end{array} \]

Alternative 8
Error	20.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-79}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-66}:\\ \;\;\;\;y \cdot \frac{z - t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	28.1
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.52 \cdot 10^{-109}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-132}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	27.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-77}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-138}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Error	28.1
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-108}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{-133}:\\ \;\;\;\;\frac{y}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12
Error	2.5
Cost	576

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

Alternative 13
Error	31.1
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023187 
(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)))