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 r790076 = x;
        double r790077 = y;
        double r790078 = z;
        double r790079 = r790077 * r790078;
        double r790080 = r790079 - r790076;
        double r790081 = t;
        double r790082 = r790081 * r790078;
        double r790083 = r790082 - r790076;
        double r790084 = r790080 / r790083;
        double r790085 = r790076 + r790084;
        double r790086 = 1.0;
        double r790087 = r790076 + r790086;
        double r790088 = r790085 / r790087;
        return r790088;
}

↓

double f(double x, double y, double z, double t) {
        double r790089 = x;
        double r790090 = y;
        double r790091 = z;
        double r790092 = r790090 * r790091;
        double r790093 = r790092 - r790089;
        double r790094 = t;
        double r790095 = r790094 * r790091;
        double r790096 = r790095 - r790089;
        double r790097 = r790093 / r790096;
        double r790098 = r790089 + r790097;
        double r790099 = 1.0;
        double r790100 = r790089 + r790099;
        double r790101 = r790098 / r790100;
        double r790102 = -inf.0;
        bool r790103 = r790101 <= r790102;
        double r790104 = 5.304900116922104e+186;
        bool r790105 = r790101 <= r790104;
        double r790106 = !r790105;
        bool r790107 = r790103 || r790106;
        double r790108 = r790090 / r790094;
        double r790109 = r790089 + r790108;
        double r790110 = r790109 / r790100;
        double r790111 = r790107 ? r790110 : r790101;
        return r790111;
}

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)))