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

Average Error: 16.1 → 9.8

Time: 4.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -7.228876828185814212701565578115473709341 \cdot 10^{-183}:\\ \;\;\;\;x + \left(y - \left(\left(z - t\right) \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}}\right)\\ \mathbf{elif}\;a \le 3.206826751751319370953297038795426487923:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - \frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r471756 = x;
        double r471757 = y;
        double r471758 = r471756 + r471757;
        double r471759 = z;
        double r471760 = t;
        double r471761 = r471759 - r471760;
        double r471762 = r471761 * r471757;
        double r471763 = a;
        double r471764 = r471763 - r471760;
        double r471765 = r471762 / r471764;
        double r471766 = r471758 - r471765;
        return r471766;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r471767 = a;
        double r471768 = -7.228876828185814e-183;
        bool r471769 = r471767 <= r471768;
        double r471770 = x;
        double r471771 = y;
        double r471772 = z;
        double r471773 = t;
        double r471774 = r471772 - r471773;
        double r471775 = cbrt(r471771);
        double r471776 = r471775 * r471775;
        double r471777 = r471767 - r471773;
        double r471778 = cbrt(r471777);
        double r471779 = r471778 * r471778;
        double r471780 = r471776 / r471779;
        double r471781 = r471774 * r471780;
        double r471782 = r471775 / r471778;
        double r471783 = r471781 * r471782;
        double r471784 = r471771 - r471783;
        double r471785 = r471770 + r471784;
        double r471786 = 3.2068267517513194;
        bool r471787 = r471767 <= r471786;
        double r471788 = r471772 * r471771;
        double r471789 = r471788 / r471773;
        double r471790 = r471789 + r471770;
        double r471791 = r471774 / r471779;
        double r471792 = r471771 / r471778;
        double r471793 = r471791 * r471792;
        double r471794 = r471771 - r471793;
        double r471795 = r471770 + r471794;
        double r471796 = r471787 ? r471790 : r471795;
        double r471797 = r471769 ? r471785 : r471796;
        return r471797;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	16.1
Target	7.8
Herbie	9.8

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

Derivation

1. Split input into 3 regimes

2. if a < -7.228876828185814e-183

   1. Initial program 15.2

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

   3. Applied *-un-lft-identity 15.2

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

   4. Applied times-frac 10.5

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

   5. Simplified 10.5

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

   7. Applied associate--l+ 8.7

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

   9. Applied add-cube-cbrt 10.3

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

   10. Applied add-cube-cbrt 10.5

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

   11. Applied times-frac 10.5

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

   12. Applied associate-*r* 9.2

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

   if -7.228876828185814e-183 < a < 3.2068267517513194

   1. Initial program 19.1

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

   2. Taylor expanded around inf 13.7

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

   if 3.2068267517513194 < a

   1. Initial program 13.8

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

   3. Applied *-un-lft-identity 13.8

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

   4. Applied times-frac 6.6

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

   5. Simplified 6.6

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

   7. Applied associate--l+ 5.7

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

   9. Applied add-cube-cbrt 6.6

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

   10. Applied *-un-lft-identity 6.6

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

   11. Applied times-frac 6.6

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

   12. Applied associate-*r* 5.9

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

   13. Simplified 5.9

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

4. Final simplification 9.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -7.228876828185814212701565578115473709341 \cdot 10^{-183}:\\ \;\;\;\;x + \left(y - \left(\left(z - t\right) \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}}\right)\\ \mathbf{elif}\;a \le 3.206826751751319370953297038795426487923:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - \frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 
(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))))