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

Average Error: 16.7 → 8.5

Time: 6.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -1.660989681485315977561284170805408884346 \cdot 10^{-107}:\\ \;\;\;\;x + \left(y - 1 \cdot \frac{\frac{{\left(\sqrt[3]{z - t}\right)}^{3}}{\sqrt[3]{a - t}}}{\frac{\sqrt[3]{a - t}}{\frac{y}{\sqrt[3]{a - t}}}}\right)\\ \mathbf{elif}\;a \le 4.71698568704097209317121327706243523854 \cdot 10^{-64}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r637367 = x;
        double r637368 = y;
        double r637369 = r637367 + r637368;
        double r637370 = z;
        double r637371 = t;
        double r637372 = r637370 - r637371;
        double r637373 = r637372 * r637368;
        double r637374 = a;
        double r637375 = r637374 - r637371;
        double r637376 = r637373 / r637375;
        double r637377 = r637369 - r637376;
        return r637377;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r637378 = a;
        double r637379 = -1.660989681485316e-107;
        bool r637380 = r637378 <= r637379;
        double r637381 = x;
        double r637382 = y;
        double r637383 = 1.0;
        double r637384 = z;
        double r637385 = t;
        double r637386 = r637384 - r637385;
        double r637387 = cbrt(r637386);
        double r637388 = 3.0;
        double r637389 = pow(r637387, r637388);
        double r637390 = r637378 - r637385;
        double r637391 = cbrt(r637390);
        double r637392 = r637389 / r637391;
        double r637393 = r637382 / r637391;
        double r637394 = r637391 / r637393;
        double r637395 = r637392 / r637394;
        double r637396 = r637383 * r637395;
        double r637397 = r637382 - r637396;
        double r637398 = r637381 + r637397;
        double r637399 = 4.716985687040972e-64;
        bool r637400 = r637378 <= r637399;
        double r637401 = r637384 * r637382;
        double r637402 = r637401 / r637385;
        double r637403 = r637402 + r637381;
        double r637404 = r637387 * r637387;
        double r637405 = r637404 / r637391;
        double r637406 = r637387 / r637391;
        double r637407 = r637406 * r637393;
        double r637408 = r637405 * r637407;
        double r637409 = r637382 - r637408;
        double r637410 = r637381 + r637409;
        double r637411 = r637400 ? r637403 : r637410;
        double r637412 = r637380 ? r637398 : r637411;
        return r637412;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	16.7
Target	8.3
Herbie	8.5

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

Derivation

1. Split input into 3 regimes

2. if a < -1.660989681485316e-107

   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 8.5

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

   6. Applied add-cube-cbrt 8.5

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

   7. Applied times-frac 8.5

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

   8. Applied associate-*l* 8.4

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

   10. Applied associate--l+ 6.3

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

   12. Applied *-un-lft-identity 6.3

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

   13. Applied associate-*l* 6.3

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

   14. Simplified 7.2

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

   if -1.660989681485316e-107 < a < 4.716985687040972e-64

   1. Initial program 20.5

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

   2. Taylor expanded around inf 12.6

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

   if 4.716985687040972e-64 < a

   1. Initial program 14.4

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

   3. Applied add-cube-cbrt 14.5

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

   4. Applied times-frac 6.9

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

   6. Applied add-cube-cbrt 6.9

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

   7. Applied times-frac 6.9

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

   8. Applied associate-*l* 6.9

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

   10. Applied associate--l+ 5.6

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

4. Final simplification 8.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.660989681485315977561284170805408884346 \cdot 10^{-107}:\\ \;\;\;\;x + \left(y - 1 \cdot \frac{\frac{{\left(\sqrt[3]{z - t}\right)}^{3}}{\sqrt[3]{a - t}}}{\frac{\sqrt[3]{a - t}}{\frac{y}{\sqrt[3]{a - t}}}}\right)\\ \mathbf{elif}\;a \le 4.71698568704097209317121327706243523854 \cdot 10^{-64}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(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))))