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

Average Error: 7.4 → 0.1

Time: 4.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -2.847567984179807 \cdot 10^{-300} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le 0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{y}{\left(x + y\right) \cdot z}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r630884 = x;
        double r630885 = y;
        double r630886 = r630884 + r630885;
        double r630887 = 1.0;
        double r630888 = z;
        double r630889 = r630885 / r630888;
        double r630890 = r630887 - r630889;
        double r630891 = r630886 / r630890;
        return r630891;
}

↓

double f(double x, double y, double z) {
        double r630892 = x;
        double r630893 = y;
        double r630894 = r630892 + r630893;
        double r630895 = 1.0;
        double r630896 = z;
        double r630897 = r630893 / r630896;
        double r630898 = r630895 - r630897;
        double r630899 = r630894 / r630898;
        double r630900 = -2.847567984179807e-300;
        bool r630901 = r630899 <= r630900;
        double r630902 = 0.0;
        bool r630903 = r630899 <= r630902;
        double r630904 = !r630903;
        bool r630905 = r630901 || r630904;
        double r630906 = 1.0;
        double r630907 = r630895 / r630894;
        double r630908 = r630894 * r630896;
        double r630909 = r630893 / r630908;
        double r630910 = r630907 - r630909;
        double r630911 = r630906 / r630910;
        double r630912 = r630905 ? r630899 : r630911;
        return r630912;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.4
Target	3.9
Herbie	0.1

\[\begin{array}{l} \mathbf{if}\;y \lt -3.74293107626898565 \cdot 10^{171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y \lt 3.55346624560867344 \cdot 10^{168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (/ (+ x y) (- 1.0 (/ y z))) < -2.847567984179807e-300 or 0.0 < (/ (+ x y) (- 1.0 (/ y z)))

   1. Initial program 0.1

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

   if -2.847567984179807e-300 < (/ (+ x y) (- 1.0 (/ y z))) < 0.0

   1. Initial program 59.8

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

   3. Applied clear-num 59.8

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

   5. Applied div-sub 59.8

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

   7. Applied div-inv 59.8

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

   8. Applied associate-/l* 0.4

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

   9. Simplified 0.4

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -2.847567984179807 \cdot 10^{-300} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le 0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{y}{\left(x + y\right) \cdot z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
  :precision binary64

  :herbie-target
  (if (< y -3.7429310762689856e+171) (* (/ (+ y x) (- y)) z) (if (< y 3.5534662456086734e+168) (/ (+ x y) (- 1 (/ y z))) (* (/ (+ y x) (- y)) z)))

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