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

Average Error: 7.6 → 0.2

Time: 17.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -164807406.9940239489078521728515625 \lor \neg \left(y \le 21140046859011661824211317635153590227040000\right):\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{\frac{1}{\frac{x + y}{y}}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r400238 = x;
        double r400239 = y;
        double r400240 = r400238 + r400239;
        double r400241 = 1.0;
        double r400242 = z;
        double r400243 = r400239 / r400242;
        double r400244 = r400241 - r400243;
        double r400245 = r400240 / r400244;
        return r400245;
}

↓

double f(double x, double y, double z) {
        double r400246 = y;
        double r400247 = -164807406.99402395;
        bool r400248 = r400246 <= r400247;
        double r400249 = 2.1140046859011662e+43;
        bool r400250 = r400246 <= r400249;
        double r400251 = !r400250;
        bool r400252 = r400248 || r400251;
        double r400253 = 1.0;
        double r400254 = 1.0;
        double r400255 = x;
        double r400256 = r400255 + r400246;
        double r400257 = r400254 / r400256;
        double r400258 = r400256 / r400246;
        double r400259 = r400253 / r400258;
        double r400260 = z;
        double r400261 = r400259 / r400260;
        double r400262 = r400257 - r400261;
        double r400263 = r400253 / r400262;
        double r400264 = r400246 / r400260;
        double r400265 = r400254 - r400264;
        double r400266 = r400256 / r400265;
        double r400267 = r400252 ? r400263 : r400266;
        return r400267;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.6
Target	4.2
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 < -164807406.99402395 or 2.1140046859011662e+43 < 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

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

   8. Applied clear-num 0.2

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

   if -164807406.99402395 < y < 2.1140046859011662e+43

   1. Initial program 0.2

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -164807406.9940239489078521728515625 \lor \neg \left(y \le 21140046859011661824211317635153590227040000\right):\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{\frac{1}{\frac{x + y}{y}}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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))))