Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B

Average Error: 16.6 → 10.5

Time: 19.8s

Precision: 64

Math

\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]

↓

\[\begin{array}{l} \mathbf{if}\;t \le -9.297768369592005284780782354057654742725 \cdot 10^{152}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \mathbf{elif}\;t \le 2.789813100340155218970434890375255818922 \cdot 10^{53}:\\ \;\;\;\;\left(x + y\right) - \frac{\sqrt[3]{z - t} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right)}{\sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}}\right)\\ \mathbf{elif}\;t \le 2.570599205910773841093883086425963460435 \cdot 10^{158}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\frac{\frac{a - t}{y}}{\sqrt[3]{z - t}}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r28787261 = x;
        double r28787262 = y;
        double r28787263 = r28787261 + r28787262;
        double r28787264 = z;
        double r28787265 = t;
        double r28787266 = r28787264 - r28787265;
        double r28787267 = r28787266 * r28787262;
        double r28787268 = a;
        double r28787269 = r28787268 - r28787265;
        double r28787270 = r28787267 / r28787269;
        double r28787271 = r28787263 - r28787270;
        return r28787271;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r28787272 = t;
        double r28787273 = -9.297768369592005e+152;
        bool r28787274 = r28787272 <= r28787273;
        double r28787275 = x;
        double r28787276 = z;
        double r28787277 = y;
        double r28787278 = r28787276 * r28787277;
        double r28787279 = r28787278 / r28787272;
        double r28787280 = r28787275 + r28787279;
        double r28787281 = 2.7898131003401552e+53;
        bool r28787282 = r28787272 <= r28787281;
        double r28787283 = r28787275 + r28787277;
        double r28787284 = r28787276 - r28787272;
        double r28787285 = cbrt(r28787284);
        double r28787286 = cbrt(r28787277);
        double r28787287 = cbrt(r28787286);
        double r28787288 = r28787286 * r28787286;
        double r28787289 = cbrt(r28787288);
        double r28787290 = r28787287 * r28787289;
        double r28787291 = r28787285 * r28787290;
        double r28787292 = a;
        double r28787293 = r28787292 - r28787272;
        double r28787294 = cbrt(r28787293);
        double r28787295 = r28787291 / r28787294;
        double r28787296 = r28787286 * r28787285;
        double r28787297 = r28787296 / r28787294;
        double r28787298 = r28787297 * r28787297;
        double r28787299 = r28787295 * r28787298;
        double r28787300 = r28787283 - r28787299;
        double r28787301 = 2.570599205910774e+158;
        bool r28787302 = r28787272 <= r28787301;
        double r28787303 = r28787285 * r28787285;
        double r28787304 = r28787293 / r28787277;
        double r28787305 = r28787304 / r28787285;
        double r28787306 = r28787303 / r28787305;
        double r28787307 = r28787283 - r28787306;
        double r28787308 = r28787302 ? r28787280 : r28787307;
        double r28787309 = r28787282 ? r28787300 : r28787308;
        double r28787310 = r28787274 ? r28787280 : r28787309;
        return r28787310;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	16.6
Target	8.4
Herbie	10.5

\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.366497088939072697550672266103566343531 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.475429344457723334351036314450840066235 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if t < -9.297768369592005e+152 or 2.7898131003401552e+53 < t < 2.570599205910774e+158

   1. Initial program 28.7

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]

   2. Taylor expanded around inf 17.5

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

   if -9.297768369592005e+152 < t < 2.7898131003401552e+53

   1. Initial program 8.8

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
   2. Using strategy rm

   3. Applied associate-/l* 6.3

      \[\leadsto \left(x + y\right) - \color{blue}{\frac{z - t}{\frac{a - t}{y}}}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 6.5

      \[\leadsto \left(x + y\right) - \frac{z - t}{\frac{a - t}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}}\]

   6. Applied add-cube-cbrt 6.6

      \[\leadsto \left(x + y\right) - \frac{z - t}{\frac{\color{blue}{\left(\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}\right) \cdot \sqrt[3]{a - t}}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}\]

   7. Applied times-frac 6.6

      \[\leadsto \left(x + y\right) - \frac{z - t}{\color{blue}{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\sqrt[3]{a - t}}{\sqrt[3]{y}}}}\]

   8. Applied add-cube-cbrt 6.6

      \[\leadsto \left(x + y\right) - \frac{\color{blue}{\left(\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}\right) \cdot \sqrt[3]{z - t}}}{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\sqrt[3]{a - t}}{\sqrt[3]{y}}}\]

   9. Applied times-frac 5.1

      \[\leadsto \left(x + y\right) - \color{blue}{\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot \frac{\sqrt[3]{z - t}}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{y}}}}\]

   10. Simplified 5.0

       \[\leadsto \left(x + y\right) - \color{blue}{\left(\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t}}\right)} \cdot \frac{\sqrt[3]{z - t}}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{y}}}\]

   11. Simplified 5.0

       \[\leadsto \left(x + y\right) - \left(\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t}}\right) \cdot \color{blue}{\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t}}}\]
   12. Using strategy rm

   13. Applied add-cube-cbrt 5.1

       \[\leadsto \left(x + y\right) - \left(\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t}}\right) \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}}{\sqrt[3]{a - t}}\]

   14. Applied cbrt-prod 5.1

       \[\leadsto \left(x + y\right) - \left(\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t}}\right) \cdot \frac{\sqrt[3]{z - t} \cdot \color{blue}{\left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)}}{\sqrt[3]{a - t}}\]

   if 2.570599205910774e+158 < t

   1. Initial program 35.0

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
   2. Using strategy rm

   3. Applied associate-/l* 25.6

      \[\leadsto \left(x + y\right) - \color{blue}{\frac{z - t}{\frac{a - t}{y}}}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 25.6

      \[\leadsto \left(x + y\right) - \frac{\color{blue}{\left(\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}\right) \cdot \sqrt[3]{z - t}}}{\frac{a - t}{y}}\]

   6. Applied associate-/l* 25.6

      \[\leadsto \left(x + y\right) - \color{blue}{\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\frac{\frac{a - t}{y}}{\sqrt[3]{z - t}}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 10.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -9.297768369592005284780782354057654742725 \cdot 10^{152}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \mathbf{elif}\;t \le 2.789813100340155218970434890375255818922 \cdot 10^{53}:\\ \;\;\;\;\left(x + y\right) - \frac{\sqrt[3]{z - t} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right)}{\sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}}\right)\\ \mathbf{elif}\;t \le 2.570599205910773841093883086425963460435 \cdot 10^{158}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\frac{\frac{a - t}{y}}{\sqrt[3]{z - t}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"

  :herbie-target
  (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) -1.3664970889390727e-07) (- (+ y x) (* (* (- z t) (/ 1.0 (- a t))) y)) (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) 1.4754293444577233e-239) (/ (- (* y (- a z)) (* x t)) (- a t)) (- (+ y x) (* (* (- z t) (/ 1.0 (- a t))) y))))

  (- (+ x y) (/ (* (- z t) y) (- a t))))