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

Average Error: 16.5 → 8.4

Time: 25.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -2.100455726661580592884813505585211088656 \cdot 10^{-151}:\\ \;\;\;\;x + \left(y - \frac{\sqrt[3]{z - t}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \left(\left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right) \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{\sqrt[3]{a - t}}}\right)\right)\\ \mathbf{elif}\;a \le 7.050287584713671167902036548554306331866 \cdot 10^{-215}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - \frac{\sqrt[3]{z - t}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \left(\left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right) \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{\sqrt[3]{a - t}}}\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r25149819 = x;
        double r25149820 = y;
        double r25149821 = r25149819 + r25149820;
        double r25149822 = z;
        double r25149823 = t;
        double r25149824 = r25149822 - r25149823;
        double r25149825 = r25149824 * r25149820;
        double r25149826 = a;
        double r25149827 = r25149826 - r25149823;
        double r25149828 = r25149825 / r25149827;
        double r25149829 = r25149821 - r25149828;
        return r25149829;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r25149830 = a;
        double r25149831 = -2.1004557266615806e-151;
        bool r25149832 = r25149830 <= r25149831;
        double r25149833 = x;
        double r25149834 = y;
        double r25149835 = z;
        double r25149836 = t;
        double r25149837 = r25149835 - r25149836;
        double r25149838 = cbrt(r25149837);
        double r25149839 = r25149830 - r25149836;
        double r25149840 = cbrt(r25149839);
        double r25149841 = r25149840 * r25149840;
        double r25149842 = cbrt(r25149841);
        double r25149843 = r25149838 / r25149842;
        double r25149844 = r25149838 / r25149840;
        double r25149845 = r25149834 / r25149840;
        double r25149846 = r25149844 * r25149845;
        double r25149847 = cbrt(r25149840);
        double r25149848 = r25149838 / r25149847;
        double r25149849 = r25149846 * r25149848;
        double r25149850 = r25149843 * r25149849;
        double r25149851 = r25149834 - r25149850;
        double r25149852 = r25149833 + r25149851;
        double r25149853 = 7.050287584713671e-215;
        bool r25149854 = r25149830 <= r25149853;
        double r25149855 = r25149834 * r25149835;
        double r25149856 = r25149855 / r25149836;
        double r25149857 = r25149833 + r25149856;
        double r25149858 = r25149854 ? r25149857 : r25149852;
        double r25149859 = r25149832 ? r25149852 : r25149858;
        return r25149859;
}

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	8.1
Herbie	8.4

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

Derivation

1. Split input into 2 regimes

2. if a < -2.1004557266615806e-151 or 7.050287584713671e-215 < a

   1. Initial program 15.6

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

   3. Applied add-cube-cbrt 15.7

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

   4. Applied times-frac 9.7

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

   6. Applied add-cube-cbrt 9.8

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

   7. Applied times-frac 9.8

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

   8. Applied associate-*l* 9.4

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

   10. Applied add-cube-cbrt 9.4

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

   11. Applied cbrt-prod 9.4

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

   12. Applied times-frac 9.4

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

   13. Applied associate-*l* 9.4

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

   15. Applied associate--l+ 8.4

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

   if -2.1004557266615806e-151 < a < 7.050287584713671e-215

   1. Initial program 20.6

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

   2. Taylor expanded around inf 8.4

      \[\leadsto \color{blue}{\frac{z \cdot y}{t} + x}\]
3. Recombined 2 regimes into one program.

4. Final simplification 8.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -2.100455726661580592884813505585211088656 \cdot 10^{-151}:\\ \;\;\;\;x + \left(y - \frac{\sqrt[3]{z - t}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \left(\left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right) \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{\sqrt[3]{a - t}}}\right)\right)\\ \mathbf{elif}\;a \le 7.050287584713671167902036548554306331866 \cdot 10^{-215}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - \frac{\sqrt[3]{z - t}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \left(\left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right) \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{\sqrt[3]{a - t}}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"

  :herbie-target
  (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) -1.3664970889390727e-07) (- (+ y x) (* (* (- z t) (/ 1.0 (- 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) (/ 1.0 (- a t))) y))))

  (- (+ x y) (/ (* (- z t) y) (- a t))))