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

Average Error: 7.2 → 3.4

Time: 23.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -7.150582175885458915277202527443364197416 \cdot 10^{54} \lor \neg \left(z \le 6.043915376694697440989411808244092375242 \cdot 10^{86}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{1}{\frac{t \cdot z - x}{y \cdot z - x}}}{x + 1}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r446448 = x;
        double r446449 = y;
        double r446450 = z;
        double r446451 = r446449 * r446450;
        double r446452 = r446451 - r446448;
        double r446453 = t;
        double r446454 = r446453 * r446450;
        double r446455 = r446454 - r446448;
        double r446456 = r446452 / r446455;
        double r446457 = r446448 + r446456;
        double r446458 = 1.0;
        double r446459 = r446448 + r446458;
        double r446460 = r446457 / r446459;
        return r446460;
}

↓

double f(double x, double y, double z, double t) {
        double r446461 = z;
        double r446462 = -7.150582175885459e+54;
        bool r446463 = r446461 <= r446462;
        double r446464 = 6.043915376694697e+86;
        bool r446465 = r446461 <= r446464;
        double r446466 = !r446465;
        bool r446467 = r446463 || r446466;
        double r446468 = x;
        double r446469 = y;
        double r446470 = t;
        double r446471 = r446469 / r446470;
        double r446472 = r446468 + r446471;
        double r446473 = 1.0;
        double r446474 = r446468 + r446473;
        double r446475 = r446472 / r446474;
        double r446476 = 1.0;
        double r446477 = r446470 * r446461;
        double r446478 = r446477 - r446468;
        double r446479 = r446469 * r446461;
        double r446480 = r446479 - r446468;
        double r446481 = r446478 / r446480;
        double r446482 = r446476 / r446481;
        double r446483 = r446468 + r446482;
        double r446484 = r446483 / r446474;
        double r446485 = r446467 ? r446475 : r446484;
        return r446485;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.2
Target	0.3
Herbie	3.4

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

Derivation

1. Split input into 2 regimes

2. if z < -7.150582175885459e+54 or 6.043915376694697e+86 < z

   1. Initial program 18.2

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

   2. Taylor expanded around inf 7.8

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

   if -7.150582175885459e+54 < z < 6.043915376694697e+86

   1. Initial program 0.7

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
   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 3.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -7.150582175885458915277202527443364197416 \cdot 10^{54} \lor \neg \left(z \le 6.043915376694697440989411808244092375242 \cdot 10^{86}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{1}{\frac{t \cdot z - x}{y \cdot z - x}}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +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)))