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

Average Error: 16.5 → 10.1

Time: 17.2s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -8.77698945948147 \cdot 10^{-231}:\\ \;\;\;\;\left(x + y\right) - \left(\left(\sqrt[3]{z - t} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \left(\sqrt[3]{1} \cdot \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\right)\\ \mathbf{elif}\;a \le 5.06495798850060336 \cdot 10^{-82}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \left(\left(\sqrt[3]{z - t} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \left(\left(\sqrt[3]{\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}}}\right) \cdot \frac{y}{\sqrt[3]{a - t}}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r739244 = x;
        double r739245 = y;
        double r739246 = r739244 + r739245;
        double r739247 = z;
        double r739248 = t;
        double r739249 = r739247 - r739248;
        double r739250 = r739249 * r739245;
        double r739251 = a;
        double r739252 = r739251 - r739248;
        double r739253 = r739250 / r739252;
        double r739254 = r739246 - r739253;
        return r739254;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r739255 = a;
        double r739256 = -8.77698945948147e-231;
        bool r739257 = r739255 <= r739256;
        double r739258 = x;
        double r739259 = y;
        double r739260 = r739258 + r739259;
        double r739261 = z;
        double r739262 = t;
        double r739263 = r739261 - r739262;
        double r739264 = cbrt(r739263);
        double r739265 = 1.0;
        double r739266 = r739255 - r739262;
        double r739267 = cbrt(r739266);
        double r739268 = r739267 * r739267;
        double r739269 = r739265 / r739268;
        double r739270 = cbrt(r739269);
        double r739271 = r739264 * r739270;
        double r739272 = r739263 / r739268;
        double r739273 = cbrt(r739272);
        double r739274 = r739271 * r739273;
        double r739275 = cbrt(r739265);
        double r739276 = r739259 / r739267;
        double r739277 = r739273 * r739276;
        double r739278 = r739275 * r739277;
        double r739279 = r739274 * r739278;
        double r739280 = r739260 - r739279;
        double r739281 = 5.0649579885006034e-82;
        bool r739282 = r739255 <= r739281;
        double r739283 = r739261 * r739259;
        double r739284 = r739283 / r739262;
        double r739285 = r739284 + r739258;
        double r739286 = r739264 * r739264;
        double r739287 = r739286 / r739267;
        double r739288 = cbrt(r739287);
        double r739289 = r739264 / r739267;
        double r739290 = cbrt(r739289);
        double r739291 = r739288 * r739290;
        double r739292 = r739291 * r739276;
        double r739293 = r739274 * r739292;
        double r739294 = r739260 - r739293;
        double r739295 = r739282 ? r739285 : r739294;
        double r739296 = r739257 ? r739280 : r739295;
        return r739296;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	16.5
Target	8.6
Herbie	10.1

\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.3664970889390727 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.47542934445772333 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if a < -8.77698945948147e-231

   1. Initial program 15.2

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

   3. Applied add-cube-cbrt 15.3

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

   4. Applied times-frac 10.2

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

   6. Applied add-cube-cbrt 10.3

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

   7. Applied associate-*l* 10.3

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

   9. Applied div-inv 10.3

      \[\leadsto \left(x + y\right) - \left(\sqrt[3]{\color{blue}{\left(z - t\right) \cdot \frac{1}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}} \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\]

   10. Applied cbrt-prod 10.3

       \[\leadsto \left(x + y\right) - \left(\color{blue}{\left(\sqrt[3]{z - t} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right)} \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\]
   11. Using strategy rm

   12. Applied *-un-lft-identity 10.3

       \[\leadsto \left(x + y\right) - \left(\left(\sqrt[3]{z - t} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \left(\sqrt[3]{\color{blue}{1 \cdot \frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\]

   13. Applied cbrt-prod 10.3

       \[\leadsto \left(x + y\right) - \left(\left(\sqrt[3]{z - t} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \left(\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right)} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\]

   14. Applied associate-*l* 10.3

       \[\leadsto \left(x + y\right) - \left(\left(\sqrt[3]{z - t} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \color{blue}{\left(\sqrt[3]{1} \cdot \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\right)}\]

   if -8.77698945948147e-231 < a < 5.0649579885006034e-82

   1. Initial program 20.2

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

   2. Taylor expanded around inf 11.2

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

   if 5.0649579885006034e-82 < a

   1. Initial program 15.8

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

   3. Applied add-cube-cbrt 15.9

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

   4. Applied times-frac 9.1

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

   6. Applied add-cube-cbrt 9.1

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

   7. Applied associate-*l* 9.1

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

   9. Applied div-inv 9.1

      \[\leadsto \left(x + y\right) - \left(\sqrt[3]{\color{blue}{\left(z - t\right) \cdot \frac{1}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}} \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\]

   10. Applied cbrt-prod 9.1

       \[\leadsto \left(x + y\right) - \left(\color{blue}{\left(\sqrt[3]{z - t} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right)} \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\]
   11. Using strategy rm

   12. Applied add-cube-cbrt 9.1

       \[\leadsto \left(x + y\right) - \left(\left(\sqrt[3]{z - t} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \left(\sqrt[3]{\frac{\color{blue}{\left(\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}\right) \cdot \sqrt[3]{z - t}}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\]

   13. Applied times-frac 9.1

       \[\leadsto \left(x + y\right) - \left(\left(\sqrt[3]{z - t} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \left(\sqrt[3]{\color{blue}{\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}}}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\]

   14. Applied cbrt-prod 9.1

       \[\leadsto \left(x + y\right) - \left(\left(\sqrt[3]{z - t} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \left(\color{blue}{\left(\sqrt[3]{\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}}}\right)} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 10.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -8.77698945948147 \cdot 10^{-231}:\\ \;\;\;\;\left(x + y\right) - \left(\left(\sqrt[3]{z - t} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \left(\sqrt[3]{1} \cdot \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\right)\\ \mathbf{elif}\;a \le 5.06495798850060336 \cdot 10^{-82}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \left(\left(\sqrt[3]{z - t} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \left(\left(\sqrt[3]{\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}}}\right) \cdot \frac{y}{\sqrt[3]{a - t}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) -1.3664970889390727e-07) (- (+ y x) (* (* (- z t) (/ 1 (- 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 (- a t))) y))))

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