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

Average Error: 7.5 → 3.6

Time: 5.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -5.48299927335239672 \cdot 10^{83} \lor \neg \left(z \le 1.42013895838381203 \cdot 10^{77}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x + 1}{x + \frac{1}{\frac{t \cdot z - x}{y \cdot z - x}}}}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r698435 = x;
        double r698436 = y;
        double r698437 = z;
        double r698438 = r698436 * r698437;
        double r698439 = r698438 - r698435;
        double r698440 = t;
        double r698441 = r698440 * r698437;
        double r698442 = r698441 - r698435;
        double r698443 = r698439 / r698442;
        double r698444 = r698435 + r698443;
        double r698445 = 1.0;
        double r698446 = r698435 + r698445;
        double r698447 = r698444 / r698446;
        return r698447;
}

↓

double f(double x, double y, double z, double t) {
        double r698448 = z;
        double r698449 = -5.482999273352397e+83;
        bool r698450 = r698448 <= r698449;
        double r698451 = 1.420138958383812e+77;
        bool r698452 = r698448 <= r698451;
        double r698453 = !r698452;
        bool r698454 = r698450 || r698453;
        double r698455 = x;
        double r698456 = y;
        double r698457 = t;
        double r698458 = r698456 / r698457;
        double r698459 = r698455 + r698458;
        double r698460 = 1.0;
        double r698461 = r698455 + r698460;
        double r698462 = r698459 / r698461;
        double r698463 = 1.0;
        double r698464 = r698457 * r698448;
        double r698465 = r698464 - r698455;
        double r698466 = r698456 * r698448;
        double r698467 = r698466 - r698455;
        double r698468 = r698465 / r698467;
        double r698469 = r698463 / r698468;
        double r698470 = r698455 + r698469;
        double r698471 = r698461 / r698470;
        double r698472 = r698463 / r698471;
        double r698473 = r698454 ? r698462 : r698472;
        return r698473;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.5
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 < -5.482999273352397e+83 or 1.420138958383812e+77 < z

   1. Initial program 18.8

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

   2. Taylor expanded around inf 8.1

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

   if -5.482999273352397e+83 < z < 1.420138958383812e+77

   1. Initial program 0.9

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

   3. Applied clear-num 0.9

      \[\leadsto \frac{x + \color{blue}{\frac{1}{\frac{t \cdot z - x}{y \cdot z - x}}}}{x + 1}\]
   4. Using strategy rm

   5. Applied clear-num 0.9

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

4. Final simplification 3.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.48299927335239672 \cdot 10^{83} \lor \neg \left(z \le 1.42013895838381203 \cdot 10^{77}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x + 1}{x + \frac{1}{\frac{t \cdot z - x}{y \cdot z - x}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 
(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)))