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

Average Error: 7.3 → 0.3

Time: 15.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -73550656644419316191634323931136 \lor \neg \left(y \le 9.808139144977446004678209925944464050109 \cdot 10^{46}\right):\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{1}{\frac{z}{\frac{y}{x + y}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r422322 = x;
        double r422323 = y;
        double r422324 = r422322 + r422323;
        double r422325 = 1.0;
        double r422326 = z;
        double r422327 = r422323 / r422326;
        double r422328 = r422325 - r422327;
        double r422329 = r422324 / r422328;
        return r422329;
}

↓

double f(double x, double y, double z) {
        double r422330 = y;
        double r422331 = -7.355065664441932e+31;
        bool r422332 = r422330 <= r422331;
        double r422333 = 9.808139144977446e+46;
        bool r422334 = r422330 <= r422333;
        double r422335 = !r422334;
        bool r422336 = r422332 || r422335;
        double r422337 = 1.0;
        double r422338 = 1.0;
        double r422339 = x;
        double r422340 = r422339 + r422330;
        double r422341 = r422338 / r422340;
        double r422342 = z;
        double r422343 = r422330 / r422340;
        double r422344 = r422342 / r422343;
        double r422345 = r422337 / r422344;
        double r422346 = r422341 - r422345;
        double r422347 = r422337 / r422346;
        double r422348 = r422330 / r422342;
        double r422349 = r422338 - r422348;
        double r422350 = r422340 / r422349;
        double r422351 = r422336 ? r422347 : r422350;
        return r422351;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.3
Target	4.1
Herbie	0.3

\[\begin{array}{l} \mathbf{if}\;y \lt -3.742931076268985646434612946949172132145 \cdot 10^{171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y \lt 3.553466245608673435460441960303815115662 \cdot 10^{168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if y < -7.355065664441932e+31 or 9.808139144977446e+46 < y

   1. Initial program 16.3

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

   3. Applied clear-num 16.5

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

   5. Applied div-sub 16.5

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

   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 *-un-lft-identity 0.2

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

   9. Applied *-un-lft-identity 0.2

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

   10. Applied times-frac 0.2

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

   11. Applied associate-/l* 0.2

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

   if -7.355065664441932e+31 < y < 9.808139144977446e+46

   1. Initial program 0.3

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -73550656644419316191634323931136 \lor \neg \left(y \le 9.808139144977446004678209925944464050109 \cdot 10^{46}\right):\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{1}{\frac{z}{\frac{y}{x + y}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(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))))