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

Average Error: 16.9 → 8.8

Time: 23.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -1.47322666539955977712838229113003878126 \cdot 10^{-157}:\\ \;\;\;\;\left(x + y\right) - \frac{\frac{z - t}{a - t}}{\frac{1}{y}}\\ \mathbf{elif}\;a \le 1.019767468767238742050870882472042955611 \cdot 10^{-151}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \left(\left(\frac{\sqrt[3]{y}}{\sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}}\right) \cdot \left(\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}\right)\right) \cdot \frac{\sqrt[3]{z - t}}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{y}}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r390792 = x;
        double r390793 = y;
        double r390794 = r390792 + r390793;
        double r390795 = z;
        double r390796 = t;
        double r390797 = r390795 - r390796;
        double r390798 = r390797 * r390793;
        double r390799 = a;
        double r390800 = r390799 - r390796;
        double r390801 = r390798 / r390800;
        double r390802 = r390794 - r390801;
        return r390802;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r390803 = a;
        double r390804 = -1.4732266653995598e-157;
        bool r390805 = r390803 <= r390804;
        double r390806 = x;
        double r390807 = y;
        double r390808 = r390806 + r390807;
        double r390809 = z;
        double r390810 = t;
        double r390811 = r390809 - r390810;
        double r390812 = r390803 - r390810;
        double r390813 = r390811 / r390812;
        double r390814 = 1.0;
        double r390815 = r390814 / r390807;
        double r390816 = r390813 / r390815;
        double r390817 = r390808 - r390816;
        double r390818 = 1.0197674687672387e-151;
        bool r390819 = r390803 <= r390818;
        double r390820 = r390809 * r390807;
        double r390821 = r390820 / r390810;
        double r390822 = r390821 + r390806;
        double r390823 = cbrt(r390807);
        double r390824 = cbrt(r390812);
        double r390825 = r390823 / r390824;
        double r390826 = r390825 * r390825;
        double r390827 = cbrt(r390811);
        double r390828 = r390827 * r390827;
        double r390829 = r390826 * r390828;
        double r390830 = r390824 / r390823;
        double r390831 = r390827 / r390830;
        double r390832 = r390829 * r390831;
        double r390833 = r390808 - r390832;
        double r390834 = r390819 ? r390822 : r390833;
        double r390835 = r390805 ? r390817 : r390834;
        return r390835;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	16.9
Target	8.5
Herbie	8.8

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

Derivation

1. Split input into 3 regimes

2. if a < -1.4732266653995598e-157

   1. Initial program 15.5

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

   3. Applied associate-/l* 9.4

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

   5. Applied div-inv 9.4

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

   6. Applied associate-/r* 8.5

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

   if -1.4732266653995598e-157 < a < 1.0197674687672387e-151

   1. Initial program 21.6

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

   2. Taylor expanded around inf 9.2

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

   if 1.0197674687672387e-151 < 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 associate-/l* 9.7

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

   5. Applied add-cube-cbrt 9.8

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

   6. Applied add-cube-cbrt 9.8

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

   7. Applied times-frac 9.9

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

   8. Applied *-un-lft-identity 9.9

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

   9. Applied times-frac 8.9

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

   10. Simplified 8.8

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

   12. Applied *-un-lft-identity 8.8

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

   13. Applied cbrt-prod 8.8

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

   14. Applied *-un-lft-identity 8.8

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

   15. Applied cbrt-prod 8.8

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

   16. Applied times-frac 8.8

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

   17. Applied add-cube-cbrt 8.9

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

   18. Applied times-frac 8.9

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

   19. Applied associate-*r* 8.9

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

   20. Simplified 8.9

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

4. Final simplification 8.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.47322666539955977712838229113003878126 \cdot 10^{-157}:\\ \;\;\;\;\left(x + y\right) - \frac{\frac{z - t}{a - t}}{\frac{1}{y}}\\ \mathbf{elif}\;a \le 1.019767468767238742050870882472042955611 \cdot 10^{-151}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \left(\left(\frac{\sqrt[3]{y}}{\sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a - t}}\right) \cdot \left(\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}\right)\right) \cdot \frac{\sqrt[3]{z - t}}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{y}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(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))))