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

Average Error: 7.3 → 2.1

Time: 4.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} = -\infty \lor \neg \left(\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 8.3183219701519654 \cdot 10^{263}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r860639 = x;
        double r860640 = y;
        double r860641 = z;
        double r860642 = r860640 * r860641;
        double r860643 = r860642 - r860639;
        double r860644 = t;
        double r860645 = r860644 * r860641;
        double r860646 = r860645 - r860639;
        double r860647 = r860643 / r860646;
        double r860648 = r860639 + r860647;
        double r860649 = 1.0;
        double r860650 = r860639 + r860649;
        double r860651 = r860648 / r860650;
        return r860651;
}

↓

double f(double x, double y, double z, double t) {
        double r860652 = x;
        double r860653 = y;
        double r860654 = z;
        double r860655 = r860653 * r860654;
        double r860656 = r860655 - r860652;
        double r860657 = t;
        double r860658 = r860657 * r860654;
        double r860659 = r860658 - r860652;
        double r860660 = r860656 / r860659;
        double r860661 = r860652 + r860660;
        double r860662 = 1.0;
        double r860663 = r860652 + r860662;
        double r860664 = r860661 / r860663;
        double r860665 = -inf.0;
        bool r860666 = r860664 <= r860665;
        double r860667 = 8.318321970151965e+263;
        bool r860668 = r860664 <= r860667;
        double r860669 = !r860668;
        bool r860670 = r860666 || r860669;
        double r860671 = r860653 / r860657;
        double r860672 = r860652 + r860671;
        double r860673 = r860672 / r860663;
        double r860674 = r860670 ? r860673 : r860664;
        return r860674;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.3
Target	0.4
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 8.318321970151965e+263 < (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))

   1. Initial program 61.3

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

   2. Taylor expanded around inf 14.0

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

   if -inf.0 < (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)) < 8.318321970151965e+263

   1. Initial program 0.7

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

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} = -\infty \lor \neg \left(\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 8.3183219701519654 \cdot 10^{263}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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)))