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

Average Error: 7.4 → 3.3

Time: 4.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -8.31331351955163841 \cdot 10^{144} \lor \neg \left(z \le 1.2516301622890457 \cdot 10^{149}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x + 1}{x + \frac{y \cdot z - x}{t \cdot z - x}}}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r683810 = x;
        double r683811 = y;
        double r683812 = z;
        double r683813 = r683811 * r683812;
        double r683814 = r683813 - r683810;
        double r683815 = t;
        double r683816 = r683815 * r683812;
        double r683817 = r683816 - r683810;
        double r683818 = r683814 / r683817;
        double r683819 = r683810 + r683818;
        double r683820 = 1.0;
        double r683821 = r683810 + r683820;
        double r683822 = r683819 / r683821;
        return r683822;
}

↓

double f(double x, double y, double z, double t) {
        double r683823 = z;
        double r683824 = -8.313313519551638e+144;
        bool r683825 = r683823 <= r683824;
        double r683826 = 1.2516301622890457e+149;
        bool r683827 = r683823 <= r683826;
        double r683828 = !r683827;
        bool r683829 = r683825 || r683828;
        double r683830 = x;
        double r683831 = y;
        double r683832 = t;
        double r683833 = r683831 / r683832;
        double r683834 = r683830 + r683833;
        double r683835 = 1.0;
        double r683836 = r683830 + r683835;
        double r683837 = r683834 / r683836;
        double r683838 = 1.0;
        double r683839 = r683831 * r683823;
        double r683840 = r683839 - r683830;
        double r683841 = r683832 * r683823;
        double r683842 = r683841 - r683830;
        double r683843 = r683840 / r683842;
        double r683844 = r683830 + r683843;
        double r683845 = r683836 / r683844;
        double r683846 = r683838 / r683845;
        double r683847 = r683829 ? r683837 : r683846;
        return r683847;
}

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	3.3

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

Derivation

1. Split input into 2 regimes

2. if z < -8.313313519551638e+144 or 1.2516301622890457e+149 < z

   1. Initial program 22.8

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

   2. Taylor expanded around inf 6.3

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

   if -8.313313519551638e+144 < z < 1.2516301622890457e+149

   1. Initial program 2.2

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

   3. Applied clear-num 2.2

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

4. Final simplification 3.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -8.31331351955163841 \cdot 10^{144} \lor \neg \left(z \le 1.2516301622890457 \cdot 10^{149}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x + 1}{x + \frac{y \cdot z - x}{t \cdot z - x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 
(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)))