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

Average Error: 7.2 → 2.6

Time: 4.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} = -\infty \lor \neg \left(\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 5.304900116922103618735525591865166312009 \cdot 10^{186}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r649778 = x;
        double r649779 = y;
        double r649780 = z;
        double r649781 = r649779 * r649780;
        double r649782 = r649781 - r649778;
        double r649783 = t;
        double r649784 = r649783 * r649780;
        double r649785 = r649784 - r649778;
        double r649786 = r649782 / r649785;
        double r649787 = r649778 + r649786;
        double r649788 = 1.0;
        double r649789 = r649778 + r649788;
        double r649790 = r649787 / r649789;
        return r649790;
}

↓

double f(double x, double y, double z, double t) {
        double r649791 = x;
        double r649792 = y;
        double r649793 = z;
        double r649794 = r649792 * r649793;
        double r649795 = r649794 - r649791;
        double r649796 = t;
        double r649797 = r649796 * r649793;
        double r649798 = r649797 - r649791;
        double r649799 = r649795 / r649798;
        double r649800 = r649791 + r649799;
        double r649801 = 1.0;
        double r649802 = r649791 + r649801;
        double r649803 = r649800 / r649802;
        double r649804 = -inf.0;
        bool r649805 = r649803 <= r649804;
        double r649806 = 5.304900116922104e+186;
        bool r649807 = r649803 <= r649806;
        double r649808 = !r649807;
        bool r649809 = r649805 || r649808;
        double r649810 = r649792 / r649796;
        double r649811 = r649791 + r649810;
        double r649812 = r649811 / r649802;
        double r649813 = r649809 ? r649812 : r649803;
        return r649813;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.2
Target	0.4
Herbie	2.6

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

Derivation

1. Split input into 2 regimes

2. if (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)) < -inf.0 or 5.304900116922104e+186 < (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))

   1. Initial program 54.0

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

   2. Taylor expanded around inf 16.5

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

   if -inf.0 < (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)) < 5.304900116922104e+186

   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 2.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} = -\infty \lor \neg \left(\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 5.304900116922103618735525591865166312009 \cdot 10^{186}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(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)))