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

Average Error: 16.9 → 8.5

Time: 14.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le -4.5495847408092521 \cdot 10^{-213}:\\ \;\;\;\;x + \left(y - \left(\left(z - t\right) \cdot \frac{\frac{\frac{1}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \frac{\frac{\frac{y}{\sqrt[3]{\sqrt[3]{a - t}}}}{\sqrt[3]{\sqrt[3]{a - t}}}}{\sqrt[3]{\sqrt[3]{a - t}}}\right)\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 3.97803588627699 \cdot 10^{-275}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - \left(\left(z - t\right) \cdot \frac{\frac{\sqrt[3]{\frac{y}{\sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{y}{\sqrt[3]{a - t}}}}{\sqrt[3]{1}}}{\sqrt[3]{\sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}}\right) \cdot \frac{\frac{\sqrt[3]{\frac{y}{\sqrt[3]{a - t}}}}{\sqrt[3]{a - t}}}{\sqrt[3]{\sqrt[3]{a - t}}}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r686393 = x;
        double r686394 = y;
        double r686395 = r686393 + r686394;
        double r686396 = z;
        double r686397 = t;
        double r686398 = r686396 - r686397;
        double r686399 = r686398 * r686394;
        double r686400 = a;
        double r686401 = r686400 - r686397;
        double r686402 = r686399 / r686401;
        double r686403 = r686395 - r686402;
        return r686403;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r686404 = x;
        double r686405 = y;
        double r686406 = r686404 + r686405;
        double r686407 = z;
        double r686408 = t;
        double r686409 = r686407 - r686408;
        double r686410 = r686409 * r686405;
        double r686411 = a;
        double r686412 = r686411 - r686408;
        double r686413 = r686410 / r686412;
        double r686414 = r686406 - r686413;
        double r686415 = -4.549584740809252e-213;
        bool r686416 = r686414 <= r686415;
        double r686417 = 1.0;
        double r686418 = cbrt(r686412);
        double r686419 = r686418 * r686418;
        double r686420 = cbrt(r686419);
        double r686421 = r686417 / r686420;
        double r686422 = r686421 / r686420;
        double r686423 = r686422 / r686420;
        double r686424 = r686409 * r686423;
        double r686425 = cbrt(r686418);
        double r686426 = r686405 / r686425;
        double r686427 = r686426 / r686425;
        double r686428 = r686427 / r686425;
        double r686429 = r686424 * r686428;
        double r686430 = r686405 - r686429;
        double r686431 = r686404 + r686430;
        double r686432 = 3.978035886276994e-275;
        bool r686433 = r686414 <= r686432;
        double r686434 = r686407 * r686405;
        double r686435 = r686434 / r686408;
        double r686436 = r686435 + r686404;
        double r686437 = r686405 / r686418;
        double r686438 = cbrt(r686437);
        double r686439 = r686438 * r686438;
        double r686440 = cbrt(r686417);
        double r686441 = r686439 / r686440;
        double r686442 = r686425 * r686425;
        double r686443 = r686441 / r686442;
        double r686444 = r686409 * r686443;
        double r686445 = r686438 / r686418;
        double r686446 = r686445 / r686425;
        double r686447 = r686444 * r686446;
        double r686448 = r686405 - r686447;
        double r686449 = r686404 + r686448;
        double r686450 = r686433 ? r686436 : r686449;
        double r686451 = r686416 ? r686431 : r686450;
        return r686451;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	16.9
Target	8.4
Herbie	8.5

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

Derivation

1. Split input into 3 regimes

2. if (- (+ x y) (/ (* (- z t) y) (- a t))) < -4.549584740809252e-213

   1. Initial program 12.9

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

   3. Applied add-cube-cbrt 13.1

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

   4. Applied times-frac 7.6

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

   6. Applied associate--l+ 7.2

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

   8. Applied div-inv 7.3

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

   9. Applied associate-*l* 7.9

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

   10. Simplified 7.9

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

   12. Applied add-cube-cbrt 8.0

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

   13. Applied cbrt-prod 8.0

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

   14. Applied add-cube-cbrt 8.0

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

   15. Applied cbrt-prod 8.0

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

   16. Applied add-cube-cbrt 8.1

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

   17. Applied cbrt-prod 8.3

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

   18. Applied *-un-lft-identity 8.3

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

   19. Applied times-frac 8.2

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

   20. Applied times-frac 8.2

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

   21. Applied times-frac 8.2

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

   22. Applied associate-*r* 7.6

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

   if -4.549584740809252e-213 < (- (+ x y) (/ (* (- z t) y) (- a t))) < 3.978035886276994e-275

   1. Initial program 55.5

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

   2. Taylor expanded around inf 19.5

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

   if 3.978035886276994e-275 < (- (+ x y) (/ (* (- z t) y) (- a t)))

   1. Initial program 12.8

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

   3. Applied add-cube-cbrt 13.0

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

   4. Applied times-frac 7.3

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

   6. Applied associate--l+ 6.9

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

   8. Applied div-inv 6.9

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

   9. Applied associate-*l* 7.6

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

   10. Simplified 7.7

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

   12. Applied add-cube-cbrt 7.8

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

   13. Applied *-un-lft-identity 7.8

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

   14. Applied cbrt-prod 7.8

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

   15. Applied add-cube-cbrt 7.9

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

   16. Applied times-frac 7.9

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

   17. Applied times-frac 7.8

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

   18. Applied associate-*r* 7.1

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

4. Final simplification 8.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le -4.5495847408092521 \cdot 10^{-213}:\\ \;\;\;\;x + \left(y - \left(\left(z - t\right) \cdot \frac{\frac{\frac{1}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \frac{\frac{\frac{y}{\sqrt[3]{\sqrt[3]{a - t}}}}{\sqrt[3]{\sqrt[3]{a - t}}}}{\sqrt[3]{\sqrt[3]{a - t}}}\right)\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \le 3.97803588627699 \cdot 10^{-275}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - \left(\left(z - t\right) \cdot \frac{\frac{\sqrt[3]{\frac{y}{\sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{y}{\sqrt[3]{a - t}}}}{\sqrt[3]{1}}}{\sqrt[3]{\sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}}\right) \cdot \frac{\frac{\sqrt[3]{\frac{y}{\sqrt[3]{a - t}}}}{\sqrt[3]{a - t}}}{\sqrt[3]{\sqrt[3]{a - t}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 
(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))))