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

Average Error: 7.4 → 0.3

Time: 11.3s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -412474744.9286561 \lor \neg \left(y \le 2.5532616039319462 \cdot 10^{62}\right):\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{\frac{y}{x + y}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r660658 = x;
        double r660659 = y;
        double r660660 = r660658 + r660659;
        double r660661 = 1.0;
        double r660662 = z;
        double r660663 = r660659 / r660662;
        double r660664 = r660661 - r660663;
        double r660665 = r660660 / r660664;
        return r660665;
}

↓

double f(double x, double y, double z) {
        double r660666 = y;
        double r660667 = -412474744.9286561;
        bool r660668 = r660666 <= r660667;
        double r660669 = 2.553261603931946e+62;
        bool r660670 = r660666 <= r660669;
        double r660671 = !r660670;
        bool r660672 = r660668 || r660671;
        double r660673 = 1.0;
        double r660674 = 1.0;
        double r660675 = x;
        double r660676 = r660675 + r660666;
        double r660677 = r660674 / r660676;
        double r660678 = r660666 / r660676;
        double r660679 = z;
        double r660680 = r660678 / r660679;
        double r660681 = r660677 - r660680;
        double r660682 = r660673 / r660681;
        double r660683 = r660666 / r660679;
        double r660684 = r660674 - r660683;
        double r660685 = r660676 / r660684;
        double r660686 = r660672 ? r660682 : r660685;
        return r660686;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.4
Target	3.8
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if y < -412474744.9286561 or 2.553261603931946e+62 < y

   1. Initial program 16.5

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

   3. Applied clear-num 16.6

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

   5. Applied div-sub 16.6

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

   6. Simplified 0.2

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

   if -412474744.9286561 < y < 2.553261603931946e+62

   1. Initial program 0.3

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -412474744.9286561 \lor \neg \left(y \le 2.5532616039319462 \cdot 10^{62}\right):\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{\frac{y}{x + y}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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))))