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

Average Error: 7.3 → 2.2

Time: 5.5s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r670794 = x;
        double r670795 = y;
        double r670796 = z;
        double r670797 = r670795 * r670796;
        double r670798 = r670797 - r670794;
        double r670799 = t;
        double r670800 = r670799 * r670796;
        double r670801 = r670800 - r670794;
        double r670802 = r670798 / r670801;
        double r670803 = r670794 + r670802;
        double r670804 = 1.0;
        double r670805 = r670794 + r670804;
        double r670806 = r670803 / r670805;
        return r670806;
}

↓

double f(double x, double y, double z, double t) {
        double r670807 = x;
        double r670808 = y;
        double r670809 = z;
        double r670810 = r670808 * r670809;
        double r670811 = r670810 - r670807;
        double r670812 = t;
        double r670813 = r670812 * r670809;
        double r670814 = r670813 - r670807;
        double r670815 = r670811 / r670814;
        double r670816 = r670807 + r670815;
        double r670817 = 1.0;
        double r670818 = r670807 + r670817;
        double r670819 = r670816 / r670818;
        double r670820 = -inf.0;
        bool r670821 = r670819 <= r670820;
        double r670822 = 8.318321970151965e+263;
        bool r670823 = r670819 <= r670822;
        double r670824 = !r670823;
        bool r670825 = r670821 || r670824;
        double r670826 = r670808 / r670812;
        double r670827 = r670807 + r670826;
        double r670828 = r670827 / r670818;
        double r670829 = 1.0;
        double r670830 = r670814 / r670811;
        double r670831 = r670829 / r670830;
        double r670832 = r670807 + r670831;
        double r670833 = r670832 / r670818;
        double r670834 = r670825 ? r670828 : r670833;
        return r670834;
}

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	2.2

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

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

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

   3. Applied clear-num 0.7

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

4. Final simplification 2.2

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

Reproduce

herbie shell --seed 2020046 +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)))