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

Average Error: 7.5 → 0.2

Time: 11.4s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -8.7834791853338754 \cdot 10^{-11} \lor \neg \left(y \le 3.1266093299729342 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{\frac{y}{x + y}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r320528 = x;
        double r320529 = y;
        double r320530 = r320528 + r320529;
        double r320531 = 1.0;
        double r320532 = z;
        double r320533 = r320529 / r320532;
        double r320534 = r320531 - r320533;
        double r320535 = r320530 / r320534;
        return r320535;
}

↓

double f(double x, double y, double z) {
        double r320536 = y;
        double r320537 = -8.783479185333875e-11;
        bool r320538 = r320536 <= r320537;
        double r320539 = 3.126609329972934e-31;
        bool r320540 = r320536 <= r320539;
        double r320541 = !r320540;
        bool r320542 = r320538 || r320541;
        double r320543 = 1.0;
        double r320544 = 1.0;
        double r320545 = x;
        double r320546 = r320545 + r320536;
        double r320547 = r320544 / r320546;
        double r320548 = r320536 / r320546;
        double r320549 = z;
        double r320550 = r320548 / r320549;
        double r320551 = r320547 - r320550;
        double r320552 = r320543 / r320551;
        double r320553 = r320536 / r320549;
        double r320554 = r320544 - r320553;
        double r320555 = r320546 / r320554;
        double r320556 = r320542 ? r320552 : r320555;
        return r320556;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.5
Target	3.9
Herbie	0.2

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

Derivation

1. Split input into 2 regimes

2. if y < -8.783479185333875e-11 or 3.126609329972934e-31 < y

   1. Initial program 14.0

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

   3. Applied clear-num 14.1

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

   5. Applied div-sub 14.1

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

   6. Simplified 0.3

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

   if -8.783479185333875e-11 < y < 3.126609329972934e-31

   1. Initial program 0.1

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -8.7834791853338754 \cdot 10^{-11} \lor \neg \left(y \le 3.1266093299729342 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{\frac{y}{x + y}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(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))))