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

Average Error: 16.2 → 9.2

Time: 6.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -2.766011212036650170966031009633619175096 \cdot 10^{-223}:\\ \;\;\;\;x + \left(y - \frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\\ \mathbf{elif}\;a \le 1.34497367140487305797427523914802175895 \cdot 10^{-213}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - \left(\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{a - t}}}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r596215 = x;
        double r596216 = y;
        double r596217 = r596215 + r596216;
        double r596218 = z;
        double r596219 = t;
        double r596220 = r596218 - r596219;
        double r596221 = r596220 * r596216;
        double r596222 = a;
        double r596223 = r596222 - r596219;
        double r596224 = r596221 / r596223;
        double r596225 = r596217 - r596224;
        return r596225;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r596226 = a;
        double r596227 = -2.76601121203665e-223;
        bool r596228 = r596226 <= r596227;
        double r596229 = x;
        double r596230 = y;
        double r596231 = z;
        double r596232 = t;
        double r596233 = r596231 - r596232;
        double r596234 = r596226 - r596232;
        double r596235 = cbrt(r596234);
        double r596236 = r596235 * r596235;
        double r596237 = r596233 / r596236;
        double r596238 = r596230 / r596235;
        double r596239 = r596237 * r596238;
        double r596240 = r596230 - r596239;
        double r596241 = r596229 + r596240;
        double r596242 = 1.344973671404873e-213;
        bool r596243 = r596226 <= r596242;
        double r596244 = r596231 * r596230;
        double r596245 = r596244 / r596232;
        double r596246 = r596245 + r596229;
        double r596247 = cbrt(r596230);
        double r596248 = r596247 * r596247;
        double r596249 = cbrt(r596235);
        double r596250 = r596249 * r596249;
        double r596251 = r596248 / r596250;
        double r596252 = r596237 * r596251;
        double r596253 = r596247 / r596249;
        double r596254 = r596252 * r596253;
        double r596255 = r596230 - r596254;
        double r596256 = r596229 + r596255;
        double r596257 = r596243 ? r596246 : r596256;
        double r596258 = r596228 ? r596241 : r596257;
        return r596258;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	16.2
Target	8.0
Herbie	9.2

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

Derivation

1. Split input into 3 regimes

2. if a < -2.76601121203665e-223

   1. Initial program 15.7

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

   3. Applied add-cube-cbrt 15.8

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

   4. Applied times-frac 10.1

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

   6. Applied associate--l+ 9.7

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

   if -2.76601121203665e-223 < a < 1.344973671404873e-213

   1. Initial program 21.1

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

   2. Taylor expanded around inf 6.9

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

   if 1.344973671404873e-213 < a

   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 9.8

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

   6. Applied associate--l+ 9.2

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

   8. Applied add-cube-cbrt 9.5

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

   9. Applied add-cube-cbrt 9.6

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

   10. Applied times-frac 9.7

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

   11. Applied associate-*r* 9.4

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

4. Final simplification 9.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -2.766011212036650170966031009633619175096 \cdot 10^{-223}:\\ \;\;\;\;x + \left(y - \frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\\ \mathbf{elif}\;a \le 1.34497367140487305797427523914802175895 \cdot 10^{-213}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - \left(\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{a - t}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 
(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))))