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

Average Error: 7.3 → 3.5

Time: 22.5s

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 r458718 = x;
        double r458719 = y;
        double r458720 = z;
        double r458721 = r458719 * r458720;
        double r458722 = r458721 - r458718;
        double r458723 = t;
        double r458724 = r458723 * r458720;
        double r458725 = r458724 - r458718;
        double r458726 = r458722 / r458725;
        double r458727 = r458718 + r458726;
        double r458728 = 1.0;
        double r458729 = r458718 + r458728;
        double r458730 = r458727 / r458729;
        return r458730;
}

↓

double f(double x, double y, double z, double t) {
        double r458731 = z;
        double r458732 = -3.2224235875553088e+72;
        bool r458733 = r458731 <= r458732;
        double r458734 = 3.0678918401680893e+102;
        bool r458735 = r458731 <= r458734;
        double r458736 = !r458735;
        bool r458737 = r458733 || r458736;
        double r458738 = x;
        double r458739 = y;
        double r458740 = t;
        double r458741 = r458739 / r458740;
        double r458742 = r458738 + r458741;
        double r458743 = 1.0;
        double r458744 = r458738 + r458743;
        double r458745 = r458742 / r458744;
        double r458746 = r458739 * r458731;
        double r458747 = r458746 - r458738;
        double r458748 = r458740 * r458731;
        double r458749 = r458748 - r458738;
        double r458750 = r458747 / r458749;
        double r458751 = r458738 + r458750;
        double r458752 = 1.0;
        double r458753 = r458752 / r458744;
        double r458754 = r458751 * r458753;
        double r458755 = r458737 ? r458745 : r458754;
        return r458755;
}

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)))