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

Average Error: 7.3 → 0.3

Time: 15.8s

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 r541246 = x;
        double r541247 = y;
        double r541248 = r541246 + r541247;
        double r541249 = 1.0;
        double r541250 = z;
        double r541251 = r541247 / r541250;
        double r541252 = r541249 - r541251;
        double r541253 = r541248 / r541252;
        return r541253;
}

↓

double f(double x, double y, double z) {
        double r541254 = y;
        double r541255 = -7.355065664441932e+31;
        bool r541256 = r541254 <= r541255;
        double r541257 = 9.808139144977446e+46;
        bool r541258 = r541254 <= r541257;
        double r541259 = !r541258;
        bool r541260 = r541256 || r541259;
        double r541261 = 1.0;
        double r541262 = 1.0;
        double r541263 = x;
        double r541264 = r541263 + r541254;
        double r541265 = r541262 / r541264;
        double r541266 = z;
        double r541267 = r541254 / r541264;
        double r541268 = r541266 / r541267;
        double r541269 = r541261 / r541268;
        double r541270 = r541265 - r541269;
        double r541271 = r541261 / r541270;
        double r541272 = r541254 / r541266;
        double r541273 = r541262 - r541272;
        double r541274 = r541264 / r541273;
        double r541275 = r541260 ? r541271 : r541274;
        return r541275;
}

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 clear-num 0.2

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

   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))))