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

Average Error: 16.6 → 9.7

Time: 28.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -9.69352787633815922929071472670459335417 \cdot 10^{-114}:\\ \;\;\;\;\left(x + y\right) - \left(\sqrt[3]{\left(\frac{\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{a - t}}}\right) \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}}} \cdot \sqrt[3]{\left(\frac{\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{a - t}}}\right) \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}}}\right) \cdot \sqrt[3]{\left(\frac{\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{a - t}}}\right) \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}}}\\ \mathbf{elif}\;a \le 1.889084039169875659727922857236046312161 \cdot 10^{-110}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{elif}\;a \le 5.919178498880237805604501360892005847147 \cdot 10^{-44}:\\ \;\;\;\;\left(x + y\right) - \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\\ \mathbf{elif}\;a \le 9.934875400147095918607310666158699856074 \cdot 10^{-12}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \frac{z - t}{\frac{a - t}{y}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r444467 = x;
        double r444468 = y;
        double r444469 = r444467 + r444468;
        double r444470 = z;
        double r444471 = t;
        double r444472 = r444470 - r444471;
        double r444473 = r444472 * r444468;
        double r444474 = a;
        double r444475 = r444474 - r444471;
        double r444476 = r444473 / r444475;
        double r444477 = r444469 - r444476;
        return r444477;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r444478 = a;
        double r444479 = -9.693527876338159e-114;
        bool r444480 = r444478 <= r444479;
        double r444481 = x;
        double r444482 = y;
        double r444483 = r444481 + r444482;
        double r444484 = z;
        double r444485 = t;
        double r444486 = r444484 - r444485;
        double r444487 = cbrt(r444486);
        double r444488 = r444478 - r444485;
        double r444489 = cbrt(r444488);
        double r444490 = r444487 / r444489;
        double r444491 = r444489 * r444489;
        double r444492 = cbrt(r444491);
        double r444493 = r444490 / r444492;
        double r444494 = cbrt(r444489);
        double r444495 = r444482 / r444494;
        double r444496 = r444493 * r444495;
        double r444497 = r444487 * r444487;
        double r444498 = r444497 / r444489;
        double r444499 = r444496 * r444498;
        double r444500 = cbrt(r444499);
        double r444501 = r444500 * r444500;
        double r444502 = r444501 * r444500;
        double r444503 = r444483 - r444502;
        double r444504 = 1.8890840391698757e-110;
        bool r444505 = r444478 <= r444504;
        double r444506 = r444484 * r444482;
        double r444507 = r444506 / r444485;
        double r444508 = r444507 + r444481;
        double r444509 = 5.919178498880238e-44;
        bool r444510 = r444478 <= r444509;
        double r444511 = r444486 / r444491;
        double r444512 = cbrt(r444511);
        double r444513 = r444512 * r444512;
        double r444514 = r444482 / r444489;
        double r444515 = r444512 * r444514;
        double r444516 = r444513 * r444515;
        double r444517 = r444483 - r444516;
        double r444518 = 9.934875400147096e-12;
        bool r444519 = r444478 <= r444518;
        double r444520 = r444488 / r444482;
        double r444521 = r444486 / r444520;
        double r444522 = r444483 - r444521;
        double r444523 = r444519 ? r444508 : r444522;
        double r444524 = r444510 ? r444517 : r444523;
        double r444525 = r444505 ? r444508 : r444524;
        double r444526 = r444480 ? r444503 : r444525;
        return r444526;
}

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	9.7

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

Derivation

1. Split input into 4 regimes

2. if a < -9.693527876338159e-114

   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 8.4

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

   6. Applied add-cube-cbrt 8.4

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

   7. Applied cbrt-prod 8.5

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

   8. Applied *-un-lft-identity 8.5

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

   9. Applied times-frac 8.5

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

   10. Applied associate-*r* 8.5

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

   11. Simplified 8.4

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

   13. Applied *-un-lft-identity 8.4

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

   14. Applied add-cube-cbrt 8.5

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

   15. Applied times-frac 8.5

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

   16. Applied times-frac 8.5

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

   17. Applied associate-*l* 8.3

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

   19. Applied add-cube-cbrt 8.3

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

   20. Simplified 8.3

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

   21. Simplified 8.3

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

   if -9.693527876338159e-114 < a < 1.8890840391698757e-110 or 5.919178498880238e-44 < a < 9.934875400147096e-12

   1. Initial program 20.0

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

   2. Taylor expanded around inf 12.5

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

   if 1.8890840391698757e-110 < a < 5.919178498880238e-44

   1. Initial program 15.6

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
   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 14.9

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

   6. Applied add-cube-cbrt 14.9

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

   7. Applied associate-*l* 14.9

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

   if 9.934875400147096e-12 < a

   1. Initial program 14.3

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

   3. Applied associate-/l* 6.8

      \[\leadsto \left(x + y\right) - \color{blue}{\frac{z - t}{\frac{a - t}{y}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 9.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -9.69352787633815922929071472670459335417 \cdot 10^{-114}:\\ \;\;\;\;\left(x + y\right) - \left(\sqrt[3]{\left(\frac{\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{a - t}}}\right) \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}}} \cdot \sqrt[3]{\left(\frac{\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{a - t}}}\right) \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}}}\right) \cdot \sqrt[3]{\left(\frac{\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{\sqrt[3]{a - t}}}\right) \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}}}\\ \mathbf{elif}\;a \le 1.889084039169875659727922857236046312161 \cdot 10^{-110}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{elif}\;a \le 5.919178498880237805604501360892005847147 \cdot 10^{-44}:\\ \;\;\;\;\left(x + y\right) - \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}}\right) \cdot \left(\sqrt[3]{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\\ \mathbf{elif}\;a \le 9.934875400147095918607310666158699856074 \cdot 10^{-12}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \frac{z - t}{\frac{a - t}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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-7) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y)) (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) 1.47542934445772333e-239) (/ (- (* y (- a z)) (* x t)) (- a t)) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y))))

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