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

Average Error: 7.0 → 3.1

Time: 17.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -4.127986421304602107023968829046578659366 \cdot 10^{128} \lor \neg \left(z \le 1.778030437179678559841700639570196585266 \cdot 10^{125}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}{x + 1}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r564611 = x;
        double r564612 = y;
        double r564613 = z;
        double r564614 = r564612 * r564613;
        double r564615 = r564614 - r564611;
        double r564616 = t;
        double r564617 = r564616 * r564613;
        double r564618 = r564617 - r564611;
        double r564619 = r564615 / r564618;
        double r564620 = r564611 + r564619;
        double r564621 = 1.0;
        double r564622 = r564611 + r564621;
        double r564623 = r564620 / r564622;
        return r564623;
}

↓

double f(double x, double y, double z, double t) {
        double r564624 = z;
        double r564625 = -4.127986421304602e+128;
        bool r564626 = r564624 <= r564625;
        double r564627 = 1.7780304371796786e+125;
        bool r564628 = r564624 <= r564627;
        double r564629 = !r564628;
        bool r564630 = r564626 || r564629;
        double r564631 = x;
        double r564632 = y;
        double r564633 = t;
        double r564634 = r564632 / r564633;
        double r564635 = r564631 + r564634;
        double r564636 = 1.0;
        double r564637 = r564631 + r564636;
        double r564638 = r564635 / r564637;
        double r564639 = r564632 * r564624;
        double r564640 = r564639 - r564631;
        double r564641 = 1.0;
        double r564642 = r564633 * r564624;
        double r564643 = r564642 - r564631;
        double r564644 = r564641 / r564643;
        double r564645 = r564640 * r564644;
        double r564646 = r564631 + r564645;
        double r564647 = r564646 / r564637;
        double r564648 = r564630 ? r564638 : r564647;
        return r564648;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.0
Target	0.3
Herbie	3.1

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

Derivation

1. Split input into 2 regimes

2. if z < -4.127986421304602e+128 or 1.7780304371796786e+125 < z

   1. Initial program 20.5

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

   2. Taylor expanded around inf 6.6

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

   if -4.127986421304602e+128 < z < 1.7780304371796786e+125

   1. Initial program 1.7

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

   3. Applied div-inv 1.7

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

4. Final simplification 3.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.127986421304602107023968829046578659366 \cdot 10^{128} \lor \neg \left(z \le 1.778030437179678559841700639570196585266 \cdot 10^{125}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(y \cdot z - x\right) \cdot \frac{1}{t \cdot z - x}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 
(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)))