Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, A

Average Error: 7.3 → 3.5

Time: 21.9s

Precision: 64

Math

\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -3.222423587555308764778936609441059573037 \cdot 10^{72} \lor \neg \left(z \le 3.067891840168089344532597897815119238015 \cdot 10^{102}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{y \cdot z - x}{t \cdot z - x}\right) \cdot \frac{1}{x + 1}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r519685 = x;
        double r519686 = y;
        double r519687 = z;
        double r519688 = r519686 * r519687;
        double r519689 = r519688 - r519685;
        double r519690 = t;
        double r519691 = r519690 * r519687;
        double r519692 = r519691 - r519685;
        double r519693 = r519689 / r519692;
        double r519694 = r519685 + r519693;
        double r519695 = 1.0;
        double r519696 = r519685 + r519695;
        double r519697 = r519694 / r519696;
        return r519697;
}

↓

double f(double x, double y, double z, double t) {
        double r519698 = z;
        double r519699 = -3.2224235875553088e+72;
        bool r519700 = r519698 <= r519699;
        double r519701 = 3.0678918401680893e+102;
        bool r519702 = r519698 <= r519701;
        double r519703 = !r519702;
        bool r519704 = r519700 || r519703;
        double r519705 = x;
        double r519706 = y;
        double r519707 = t;
        double r519708 = r519706 / r519707;
        double r519709 = r519705 + r519708;
        double r519710 = 1.0;
        double r519711 = r519705 + r519710;
        double r519712 = r519709 / r519711;
        double r519713 = r519706 * r519698;
        double r519714 = r519713 - r519705;
        double r519715 = r519707 * r519698;
        double r519716 = r519715 - r519705;
        double r519717 = r519714 / r519716;
        double r519718 = r519705 + r519717;
        double r519719 = 1.0;
        double r519720 = r519719 / r519711;
        double r519721 = r519718 * r519720;
        double r519722 = r519704 ? r519712 : r519721;
        return r519722;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.3
Target	0.4
Herbie	3.5

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

Derivation

1. Split input into 2 regimes

2. if z < -3.2224235875553088e+72 or 3.0678918401680893e+102 < z

   1. Initial program 19.3

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]

   2. Taylor expanded around inf 8.2

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

   if -3.2224235875553088e+72 < z < 3.0678918401680893e+102

   1. Initial program 0.9

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
   2. Using strategy rm

   3. Applied div-inv 1.0

      \[\leadsto \color{blue}{\left(x + \frac{y \cdot z - x}{t \cdot z - x}\right) \cdot \frac{1}{x + 1}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 3.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.222423587555308764778936609441059573037 \cdot 10^{72} \lor \neg \left(z \le 3.067891840168089344532597897815119238015 \cdot 10^{102}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{y \cdot z - x}{t \cdot z - x}\right) \cdot \frac{1}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :herbie-target
  (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1))

  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1)))