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

Average Error: 7.2 → 2.1

Time: 8.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} = -\infty \lor \neg \left(\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 1.5358838164040624209409443001876571781 \cdot 10^{277}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r417598 = x;
        double r417599 = y;
        double r417600 = z;
        double r417601 = r417599 * r417600;
        double r417602 = r417601 - r417598;
        double r417603 = t;
        double r417604 = r417603 * r417600;
        double r417605 = r417604 - r417598;
        double r417606 = r417602 / r417605;
        double r417607 = r417598 + r417606;
        double r417608 = 1.0;
        double r417609 = r417598 + r417608;
        double r417610 = r417607 / r417609;
        return r417610;
}

↓

double f(double x, double y, double z, double t) {
        double r417611 = x;
        double r417612 = y;
        double r417613 = z;
        double r417614 = r417612 * r417613;
        double r417615 = r417614 - r417611;
        double r417616 = t;
        double r417617 = r417616 * r417613;
        double r417618 = r417617 - r417611;
        double r417619 = r417615 / r417618;
        double r417620 = r417611 + r417619;
        double r417621 = 1.0;
        double r417622 = r417611 + r417621;
        double r417623 = r417620 / r417622;
        double r417624 = -inf.0;
        bool r417625 = r417623 <= r417624;
        double r417626 = 1.5358838164040624e+277;
        bool r417627 = r417623 <= r417626;
        double r417628 = !r417627;
        bool r417629 = r417625 || r417628;
        double r417630 = r417612 / r417616;
        double r417631 = r417611 + r417630;
        double r417632 = r417631 / r417622;
        double r417633 = r417629 ? r417632 : r417623;
        return r417633;
}

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	2.1

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

Derivation

1. Split input into 2 regimes

2. if (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)) < -inf.0 or 1.5358838164040624e+277 < (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))

   1. Initial program 61.5

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

   2. Taylor expanded around inf 14.4

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

   if -inf.0 < (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)) < 1.5358838164040624e+277

   1. Initial program 0.5

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

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} = -\infty \lor \neg \left(\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 1.5358838164040624209409443001876571781 \cdot 10^{277}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(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)))