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

Average Error: 7.4 → 4.0

Time: 23.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 4.2451363091712777 \cdot 10^{183}:\\ \;\;\;\;\frac{x + \left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r486408 = x;
        double r486409 = y;
        double r486410 = z;
        double r486411 = r486409 * r486410;
        double r486412 = r486411 - r486408;
        double r486413 = t;
        double r486414 = r486413 * r486410;
        double r486415 = r486414 - r486408;
        double r486416 = r486412 / r486415;
        double r486417 = r486408 + r486416;
        double r486418 = 1.0;
        double r486419 = r486408 + r486418;
        double r486420 = r486417 / r486419;
        return r486420;
}

↓

double f(double x, double y, double z, double t) {
        double r486421 = x;
        double r486422 = y;
        double r486423 = z;
        double r486424 = r486422 * r486423;
        double r486425 = r486424 - r486421;
        double r486426 = t;
        double r486427 = r486426 * r486423;
        double r486428 = r486427 - r486421;
        double r486429 = r486425 / r486428;
        double r486430 = r486421 + r486429;
        double r486431 = 1.0;
        double r486432 = r486421 + r486431;
        double r486433 = r486430 / r486432;
        double r486434 = 4.2451363091712777e+183;
        bool r486435 = r486433 <= r486434;
        double r486436 = 1.0;
        double r486437 = r486436 / r486428;
        double r486438 = r486425 * r486437;
        double r486439 = r486421 + r486438;
        double r486440 = r486439 / r486432;
        double r486441 = r486422 / r486426;
        double r486442 = r486421 + r486441;
        double r486443 = r486442 / r486432;
        double r486444 = r486435 ? r486440 : r486443;
        return r486444;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.4
Target	0.3
Herbie	4.0

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

Derivation

1. Split input into 2 regimes

2. if (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)) < 4.2451363091712777e+183

   1. Initial program 2.9

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

   3. Applied div-inv 2.9

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

   if 4.2451363091712777e+183 < (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))

   1. Initial program 50.1

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

   2. Taylor expanded around inf 13.9

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

4. Final simplification 4.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 4.2451363091712777 \cdot 10^{183}:\\ \;\;\;\;\frac{x + \left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 
(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)))