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

Average Error: 16.5 → 9.2

Time: 4.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -6.2755159521377276 \cdot 10^{-71}:\\ \;\;\;\;x + \left(y - \left(\left(z - t\right) \cdot \frac{\sqrt[3]{y} \cdot \left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}}\right)\\ \mathbf{elif}\;a \le 1.2029862383154317 \cdot 10^{-78}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - \left(z - t\right) \cdot \frac{y}{a - t}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r601214 = x;
        double r601215 = y;
        double r601216 = r601214 + r601215;
        double r601217 = z;
        double r601218 = t;
        double r601219 = r601217 - r601218;
        double r601220 = r601219 * r601215;
        double r601221 = a;
        double r601222 = r601221 - r601218;
        double r601223 = r601220 / r601222;
        double r601224 = r601216 - r601223;
        return r601224;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r601225 = a;
        double r601226 = -6.2755159521377276e-71;
        bool r601227 = r601225 <= r601226;
        double r601228 = x;
        double r601229 = y;
        double r601230 = z;
        double r601231 = t;
        double r601232 = r601230 - r601231;
        double r601233 = cbrt(r601229);
        double r601234 = r601233 * r601233;
        double r601235 = cbrt(r601234);
        double r601236 = cbrt(r601233);
        double r601237 = r601235 * r601236;
        double r601238 = r601233 * r601237;
        double r601239 = r601225 - r601231;
        double r601240 = cbrt(r601239);
        double r601241 = r601240 * r601240;
        double r601242 = r601238 / r601241;
        double r601243 = r601232 * r601242;
        double r601244 = r601233 / r601240;
        double r601245 = r601243 * r601244;
        double r601246 = r601229 - r601245;
        double r601247 = r601228 + r601246;
        double r601248 = 1.2029862383154317e-78;
        bool r601249 = r601225 <= r601248;
        double r601250 = r601230 * r601229;
        double r601251 = r601250 / r601231;
        double r601252 = r601228 + r601251;
        double r601253 = r601229 / r601239;
        double r601254 = r601232 * r601253;
        double r601255 = r601229 - r601254;
        double r601256 = r601228 + r601255;
        double r601257 = r601249 ? r601252 : r601256;
        double r601258 = r601227 ? r601247 : r601257;
        return r601258;
}

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	9.2

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

Derivation

1. Split input into 3 regimes

2. if a < -6.2755159521377276e-71

   1. Initial program 15.1

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

   3. Applied *-un-lft-identity 15.1

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

   4. Applied times-frac 8.3

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

   5. Simplified 8.3

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

   7. Applied associate--l+ 6.9

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

   9. Applied add-cube-cbrt 8.0

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

   10. Applied add-cube-cbrt 8.2

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

   11. Applied times-frac 8.2

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

   12. Applied associate-*r* 7.4

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

   14. Applied add-cube-cbrt 7.4

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

   15. Applied cbrt-prod 7.4

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

   if -6.2755159521377276e-71 < a < 1.2029862383154317e-78

   1. Initial program 19.2

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

   3. Applied *-un-lft-identity 19.2

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

   4. Applied times-frac 18.2

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

   5. Simplified 18.2

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

   7. Applied associate--l+ 13.0

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

   9. Applied add-cube-cbrt 17.3

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

   10. Applied add-cube-cbrt 18.1

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

   11. Applied times-frac 18.1

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

   12. Applied associate-*r* 15.7

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

   13. Taylor expanded around inf 12.7

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

   if 1.2029862383154317e-78 < a

   1. Initial program 14.8

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

   3. Applied *-un-lft-identity 14.8

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

   4. Applied times-frac 8.9

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

   5. Simplified 8.9

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

   7. Applied associate--l+ 7.1

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

4. Final simplification 9.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -6.2755159521377276 \cdot 10^{-71}:\\ \;\;\;\;x + \left(y - \left(\left(z - t\right) \cdot \frac{\sqrt[3]{y} \cdot \left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}}\right)\\ \mathbf{elif}\;a \le 1.2029862383154317 \cdot 10^{-78}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - \left(z - t\right) \cdot \frac{y}{a - t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 
(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))))