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

Average Error: 7.5 → 0.2

Time: 8.5s

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 r783395 = x;
        double r783396 = y;
        double r783397 = r783395 + r783396;
        double r783398 = 1.0;
        double r783399 = z;
        double r783400 = r783396 / r783399;
        double r783401 = r783398 - r783400;
        double r783402 = r783397 / r783401;
        return r783402;
}

↓

double f(double x, double y, double z) {
        double r783403 = y;
        double r783404 = -8.783479185333875e-11;
        bool r783405 = r783403 <= r783404;
        double r783406 = 3.126609329972934e-31;
        bool r783407 = r783403 <= r783406;
        double r783408 = !r783407;
        bool r783409 = r783405 || r783408;
        double r783410 = 1.0;
        double r783411 = 1.0;
        double r783412 = x;
        double r783413 = r783412 + r783403;
        double r783414 = r783411 / r783413;
        double r783415 = r783403 / r783413;
        double r783416 = z;
        double r783417 = r783415 / r783416;
        double r783418 = r783414 - r783417;
        double r783419 = r783410 / r783418;
        double r783420 = r783403 / r783416;
        double r783421 = r783411 - r783420;
        double r783422 = r783413 / r783421;
        double r783423 = r783409 ? r783419 : r783422;
        return r783423;
}

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 +o rules:numerics
(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))))