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

Average Error: 6.8 → 3.4

Time: 21.2s

Precision: 64

Math

\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1.0}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1.705982485713477 \cdot 10^{+79}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1.0}\\ \mathbf{elif}\;z \le 2.2939420975234537 \cdot 10^{+108}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1.0}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r28466783 = x;
        double r28466784 = y;
        double r28466785 = z;
        double r28466786 = r28466784 * r28466785;
        double r28466787 = r28466786 - r28466783;
        double r28466788 = t;
        double r28466789 = r28466788 * r28466785;
        double r28466790 = r28466789 - r28466783;
        double r28466791 = r28466787 / r28466790;
        double r28466792 = r28466783 + r28466791;
        double r28466793 = 1.0;
        double r28466794 = r28466783 + r28466793;
        double r28466795 = r28466792 / r28466794;
        return r28466795;
}

↓

double f(double x, double y, double z, double t) {
        double r28466796 = z;
        double r28466797 = -1.705982485713477e+79;
        bool r28466798 = r28466796 <= r28466797;
        double r28466799 = x;
        double r28466800 = y;
        double r28466801 = t;
        double r28466802 = r28466800 / r28466801;
        double r28466803 = r28466799 + r28466802;
        double r28466804 = 1.0;
        double r28466805 = r28466799 + r28466804;
        double r28466806 = r28466803 / r28466805;
        double r28466807 = 2.2939420975234537e+108;
        bool r28466808 = r28466796 <= r28466807;
        double r28466809 = r28466800 * r28466796;
        double r28466810 = r28466809 - r28466799;
        double r28466811 = r28466801 * r28466796;
        double r28466812 = r28466811 - r28466799;
        double r28466813 = r28466810 / r28466812;
        double r28466814 = r28466799 + r28466813;
        double r28466815 = r28466814 / r28466805;
        double r28466816 = r28466808 ? r28466815 : r28466806;
        double r28466817 = r28466798 ? r28466806 : r28466816;
        return r28466817;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.8
Target	0.4
Herbie	3.4

\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1.0}\]

Derivation

1. Split input into 2 regimes

2. if z < -1.705982485713477e+79 or 2.2939420975234537e+108 < z

   1. Initial program 18.5

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1.0}\]

   2. Taylor expanded around inf 8.1

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1.0}\]

   if -1.705982485713477e+79 < z < 2.2939420975234537e+108

   1. Initial program 1.0

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1.0}\]

   2. Taylor expanded around 0 1.0

      \[\leadsto \frac{x + \frac{\color{blue}{z \cdot y - x}}{t \cdot z - x}}{x + 1.0}\]
3. Recombined 2 regimes into one program.

4. Final simplification 3.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.705982485713477 \cdot 10^{+79}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1.0}\\ \mathbf{elif}\;z \le 2.2939420975234537 \cdot 10^{+108}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"

  :herbie-target
  (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1.0))

  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))