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

Average Error: 7.2 → 3.6

Time: 44.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -8.287165386060872715453133833603909602049 \cdot 10^{86}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{1 + x}\\ \mathbf{elif}\;z \le 3.348305293084260034623912739354303673528 \cdot 10^{56}:\\ \;\;\;\;\frac{x + \frac{1}{\frac{t \cdot z - x}{y \cdot z - x}}}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{1 + x}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r32019592 = x;
        double r32019593 = y;
        double r32019594 = z;
        double r32019595 = r32019593 * r32019594;
        double r32019596 = r32019595 - r32019592;
        double r32019597 = t;
        double r32019598 = r32019597 * r32019594;
        double r32019599 = r32019598 - r32019592;
        double r32019600 = r32019596 / r32019599;
        double r32019601 = r32019592 + r32019600;
        double r32019602 = 1.0;
        double r32019603 = r32019592 + r32019602;
        double r32019604 = r32019601 / r32019603;
        return r32019604;
}

↓

double f(double x, double y, double z, double t) {
        double r32019605 = z;
        double r32019606 = -8.287165386060873e+86;
        bool r32019607 = r32019605 <= r32019606;
        double r32019608 = x;
        double r32019609 = y;
        double r32019610 = t;
        double r32019611 = r32019609 / r32019610;
        double r32019612 = r32019608 + r32019611;
        double r32019613 = 1.0;
        double r32019614 = r32019613 + r32019608;
        double r32019615 = r32019612 / r32019614;
        double r32019616 = 3.34830529308426e+56;
        bool r32019617 = r32019605 <= r32019616;
        double r32019618 = 1.0;
        double r32019619 = r32019610 * r32019605;
        double r32019620 = r32019619 - r32019608;
        double r32019621 = r32019609 * r32019605;
        double r32019622 = r32019621 - r32019608;
        double r32019623 = r32019620 / r32019622;
        double r32019624 = r32019618 / r32019623;
        double r32019625 = r32019608 + r32019624;
        double r32019626 = r32019625 / r32019614;
        double r32019627 = r32019617 ? r32019626 : r32019615;
        double r32019628 = r32019607 ? r32019615 : r32019627;
        return r32019628;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.2
Target	0.4
Herbie	3.6

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

Derivation

1. Split input into 2 regimes

2. if z < -8.287165386060873e+86 or 3.34830529308426e+56 < z

   1. Initial program 18.0

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

   2. Taylor expanded around inf 8.3

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

   if -8.287165386060873e+86 < z < 3.34830529308426e+56

   1. Initial program 0.8

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

   3. Applied clear-num 0.8

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

4. Final simplification 3.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -8.287165386060872715453133833603909602049 \cdot 10^{86}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{1 + x}\\ \mathbf{elif}\;z \le 3.348305293084260034623912739354303673528 \cdot 10^{56}:\\ \;\;\;\;\frac{x + \frac{1}{\frac{t \cdot z - x}{y \cdot z - x}}}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{1 + x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +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)))