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

Average Error: 16.8 → 9.7

Time: 6.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -5.0892061953826007 \cdot 10^{-142}:\\ \;\;\;\;x + \left(y - \left(\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{a - t}}}\right)\\ \mathbf{elif}\;a \le 1.6745009116298949 \cdot 10^{-272}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \mathbf{elif}\;a \le 2.475987323717541 \cdot 10^{-180}:\\ \;\;\;\;\left(x + y\right) - \left(z - t\right) \cdot \frac{y}{a - t}\\ \mathbf{elif}\;a \le 2.96691100945856406 \cdot 10^{-48}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - \left(\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}}\right) \cdot \frac{\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{y}}}{\sqrt[3]{\sqrt[3]{a - t}}}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r655868 = x;
        double r655869 = y;
        double r655870 = r655868 + r655869;
        double r655871 = z;
        double r655872 = t;
        double r655873 = r655871 - r655872;
        double r655874 = r655873 * r655869;
        double r655875 = a;
        double r655876 = r655875 - r655872;
        double r655877 = r655874 / r655876;
        double r655878 = r655870 - r655877;
        return r655878;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r655879 = a;
        double r655880 = -5.089206195382601e-142;
        bool r655881 = r655879 <= r655880;
        double r655882 = x;
        double r655883 = y;
        double r655884 = z;
        double r655885 = t;
        double r655886 = r655884 - r655885;
        double r655887 = r655879 - r655885;
        double r655888 = cbrt(r655887);
        double r655889 = r655888 * r655888;
        double r655890 = r655886 / r655889;
        double r655891 = cbrt(r655883);
        double r655892 = r655891 * r655891;
        double r655893 = cbrt(r655888);
        double r655894 = r655893 * r655893;
        double r655895 = r655892 / r655894;
        double r655896 = r655890 * r655895;
        double r655897 = r655891 / r655893;
        double r655898 = r655896 * r655897;
        double r655899 = r655883 - r655898;
        double r655900 = r655882 + r655899;
        double r655901 = 1.674500911629895e-272;
        bool r655902 = r655879 <= r655901;
        double r655903 = r655884 * r655883;
        double r655904 = r655903 / r655885;
        double r655905 = r655882 + r655904;
        double r655906 = 2.475987323717541e-180;
        bool r655907 = r655879 <= r655906;
        double r655908 = r655882 + r655883;
        double r655909 = r655883 / r655887;
        double r655910 = r655886 * r655909;
        double r655911 = r655908 - r655910;
        double r655912 = 2.966911009458564e-48;
        bool r655913 = r655879 <= r655912;
        double r655914 = cbrt(r655891);
        double r655915 = r655914 * r655914;
        double r655916 = r655915 * r655914;
        double r655917 = r655916 / r655893;
        double r655918 = r655896 * r655917;
        double r655919 = r655883 - r655918;
        double r655920 = r655882 + r655919;
        double r655921 = r655913 ? r655905 : r655920;
        double r655922 = r655907 ? r655911 : r655921;
        double r655923 = r655902 ? r655905 : r655922;
        double r655924 = r655881 ? r655900 : r655923;
        return r655924;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	16.8
Target	8.6
Herbie	9.7

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

Derivation

1. Split input into 4 regimes

2. if a < -5.089206195382601e-142

   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 8.6

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

   6. Applied associate--l+ 8.1

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

   8. Applied add-cube-cbrt 8.3

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

   9. Applied add-cube-cbrt 8.3

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

   10. Applied times-frac 8.3

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

   11. Applied associate-*r* 8.0

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

   if -5.089206195382601e-142 < a < 1.674500911629895e-272 or 2.475987323717541e-180 < a < 2.966911009458564e-48

   1. Initial program 20.8

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

   3. Applied add-cube-cbrt 21.1

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

   4. Applied times-frac 20.4

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

   6. Applied associate--l+ 19.0

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

   7. Taylor expanded around inf 13.1

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

   if 1.674500911629895e-272 < a < 2.475987323717541e-180

   1. Initial program 20.5

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

   3. Applied *-un-lft-identity 20.5

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

   4. Applied times-frac 17.5

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

   5. Simplified 17.5

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

   if 2.966911009458564e-48 < 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 add-cube-cbrt 14.9

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

   4. Applied times-frac 7.4

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

   6. Applied associate--l+ 7.0

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

   8. Applied add-cube-cbrt 7.2

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

   9. Applied add-cube-cbrt 7.3

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

   10. Applied times-frac 7.3

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

   11. Applied associate-*r* 7.2

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

   13. Applied add-cube-cbrt 7.3

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

4. Final simplification 9.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -5.0892061953826007 \cdot 10^{-142}:\\ \;\;\;\;x + \left(y - \left(\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{a - t}}}\right)\\ \mathbf{elif}\;a \le 1.6745009116298949 \cdot 10^{-272}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \mathbf{elif}\;a \le 2.475987323717541 \cdot 10^{-180}:\\ \;\;\;\;\left(x + y\right) - \left(z - t\right) \cdot \frac{y}{a - t}\\ \mathbf{elif}\;a \le 2.96691100945856406 \cdot 10^{-48}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - \left(\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}}\right) \cdot \frac{\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{y}}}{\sqrt[3]{\sqrt[3]{a - t}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 
(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))))