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

Average Error: 16.7 → 8.5

Time: 6.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -1.660989681485315977561284170805408884346 \cdot 10^{-107}:\\ \;\;\;\;x + \left(y - 1 \cdot \frac{\frac{{\left(\sqrt[3]{z - t}\right)}^{3}}{\sqrt[3]{a - t}}}{\frac{\sqrt[3]{a - t}}{\frac{y}{\sqrt[3]{a - t}}}}\right)\\ \mathbf{elif}\;a \le 4.71698568704097209317121327706243523854 \cdot 10^{-64}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r664989 = x;
        double r664990 = y;
        double r664991 = r664989 + r664990;
        double r664992 = z;
        double r664993 = t;
        double r664994 = r664992 - r664993;
        double r664995 = r664994 * r664990;
        double r664996 = a;
        double r664997 = r664996 - r664993;
        double r664998 = r664995 / r664997;
        double r664999 = r664991 - r664998;
        return r664999;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r665000 = a;
        double r665001 = -1.660989681485316e-107;
        bool r665002 = r665000 <= r665001;
        double r665003 = x;
        double r665004 = y;
        double r665005 = 1.0;
        double r665006 = z;
        double r665007 = t;
        double r665008 = r665006 - r665007;
        double r665009 = cbrt(r665008);
        double r665010 = 3.0;
        double r665011 = pow(r665009, r665010);
        double r665012 = r665000 - r665007;
        double r665013 = cbrt(r665012);
        double r665014 = r665011 / r665013;
        double r665015 = r665004 / r665013;
        double r665016 = r665013 / r665015;
        double r665017 = r665014 / r665016;
        double r665018 = r665005 * r665017;
        double r665019 = r665004 - r665018;
        double r665020 = r665003 + r665019;
        double r665021 = 4.716985687040972e-64;
        bool r665022 = r665000 <= r665021;
        double r665023 = r665006 * r665004;
        double r665024 = r665023 / r665007;
        double r665025 = r665024 + r665003;
        double r665026 = r665009 * r665009;
        double r665027 = r665026 / r665013;
        double r665028 = r665009 / r665013;
        double r665029 = r665028 * r665015;
        double r665030 = r665027 * r665029;
        double r665031 = r665004 - r665030;
        double r665032 = r665003 + r665031;
        double r665033 = r665022 ? r665025 : r665032;
        double r665034 = r665002 ? r665020 : r665033;
        return r665034;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	16.7
Target	8.3
Herbie	8.5

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

Derivation

1. Split input into 3 regimes

2. if a < -1.660989681485316e-107

   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 8.5

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

   6. Applied add-cube-cbrt 8.5

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

   7. Applied times-frac 8.5

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

   8. Applied associate-*l* 8.4

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

   10. Applied associate--l+ 6.3

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

   12. Applied *-un-lft-identity 6.3

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

   13. Applied associate-*l* 6.3

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

   14. Simplified 7.2

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

   if -1.660989681485316e-107 < a < 4.716985687040972e-64

   1. Initial program 20.5

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

   2. Taylor expanded around inf 12.6

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

   if 4.716985687040972e-64 < a

   1. Initial program 14.4

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

   3. Applied add-cube-cbrt 14.5

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

   4. Applied times-frac 6.9

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

   6. Applied add-cube-cbrt 6.9

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

   7. Applied times-frac 6.9

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

   8. Applied associate-*l* 6.9

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

   10. Applied associate--l+ 5.6

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

4. Final simplification 8.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.660989681485315977561284170805408884346 \cdot 10^{-107}:\\ \;\;\;\;x + \left(y - 1 \cdot \frac{\frac{{\left(\sqrt[3]{z - t}\right)}^{3}}{\sqrt[3]{a - t}}}{\frac{\sqrt[3]{a - t}}{\frac{y}{\sqrt[3]{a - t}}}}\right)\\ \mathbf{elif}\;a \le 4.71698568704097209317121327706243523854 \cdot 10^{-64}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(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))))