Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A

Average Error: 7.3 → 0.2

Time: 17.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -15383809086552125997056 \lor \neg \left(y \le 2.211156237826685988742868005409983379368 \cdot 10^{49}\right):\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{\frac{y}{x + y}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r457913 = x;
        double r457914 = y;
        double r457915 = r457913 + r457914;
        double r457916 = 1.0;
        double r457917 = z;
        double r457918 = r457914 / r457917;
        double r457919 = r457916 - r457918;
        double r457920 = r457915 / r457919;
        return r457920;
}

↓

double f(double x, double y, double z) {
        double r457921 = y;
        double r457922 = -1.5383809086552126e+22;
        bool r457923 = r457921 <= r457922;
        double r457924 = 2.211156237826686e+49;
        bool r457925 = r457921 <= r457924;
        double r457926 = !r457925;
        bool r457927 = r457923 || r457926;
        double r457928 = 1.0;
        double r457929 = 1.0;
        double r457930 = x;
        double r457931 = r457930 + r457921;
        double r457932 = r457929 / r457931;
        double r457933 = r457921 / r457931;
        double r457934 = z;
        double r457935 = r457933 / r457934;
        double r457936 = r457932 - r457935;
        double r457937 = r457928 / r457936;
        double r457938 = r457921 / r457934;
        double r457939 = r457929 - r457938;
        double r457940 = r457931 / r457939;
        double r457941 = r457927 ? r457937 : r457940;
        return r457941;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.3
Target	3.9
Herbie	0.2

\[\begin{array}{l} \mathbf{if}\;y \lt -3.742931076268985646434612946949172132145 \cdot 10^{171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y \lt 3.553466245608673435460441960303815115662 \cdot 10^{168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if y < -1.5383809086552126e+22 or 2.211156237826686e+49 < y

   1. Initial program 16.3

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

   3. Applied clear-num 16.4

      \[\leadsto \color{blue}{\frac{1}{\frac{1 - \frac{y}{z}}{x + y}}}\]
   4. Using strategy rm

   5. Applied div-sub 16.4

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

   6. Simplified 10.5

      \[\leadsto \frac{1}{\frac{1}{x + y} - \color{blue}{\frac{y}{\left(x + y\right) \cdot z}}}\]
   7. Using strategy rm

   8. Applied associate-/r* 0.2

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

   if -1.5383809086552126e+22 < y < 2.211156237826686e+49

   1. Initial program 0.2

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -15383809086552125997056 \lor \neg \left(y \le 2.211156237826685988742868005409983379368 \cdot 10^{49}\right):\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{\frac{y}{x + y}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
  :precision binary64

  :herbie-target
  (if (< y -3.74293107626898565e171) (* (/ (+ y x) (- y)) z) (if (< y 3.55346624560867344e168) (/ (+ x y) (- 1 (/ y z))) (* (/ (+ y x) (- y)) z)))

  (/ (+ x y) (- 1 (/ y z))))