Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4

Average Error: 2.2 → 2.4

Time: 5.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le 3.29194830615707638 \cdot 10^{-308} \lor \neg \left(z \le 2.88733508478686524 \cdot 10^{205}\right):\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\sqrt{z} \cdot \left(y - x\right)\right) \cdot \frac{\sqrt{z}}{t}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r612720 = x;
        double r612721 = y;
        double r612722 = r612721 - r612720;
        double r612723 = z;
        double r612724 = t;
        double r612725 = r612723 / r612724;
        double r612726 = r612722 * r612725;
        double r612727 = r612720 + r612726;
        return r612727;
}

↓

double f(double x, double y, double z, double t) {
        double r612728 = z;
        double r612729 = 3.2919483061570764e-308;
        bool r612730 = r612728 <= r612729;
        double r612731 = 2.8873350847868652e+205;
        bool r612732 = r612728 <= r612731;
        double r612733 = !r612732;
        bool r612734 = r612730 || r612733;
        double r612735 = x;
        double r612736 = y;
        double r612737 = r612736 - r612735;
        double r612738 = t;
        double r612739 = r612728 / r612738;
        double r612740 = r612737 * r612739;
        double r612741 = r612735 + r612740;
        double r612742 = sqrt(r612728);
        double r612743 = r612742 * r612737;
        double r612744 = r612742 / r612738;
        double r612745 = r612743 * r612744;
        double r612746 = r612735 + r612745;
        double r612747 = r612734 ? r612741 : r612746;
        return r612747;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	2.2
Target	2.5
Herbie	2.4

\[\begin{array}{l} \mathbf{if}\;\left(y - x\right) \cdot \frac{z}{t} \lt -1013646692435.887:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;\left(y - x\right) \cdot \frac{z}{t} \lt -0.0:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < 3.2919483061570764e-308 or 2.8873350847868652e+205 < z

   1. Initial program 2.6

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

   if 3.2919483061570764e-308 < z < 2.8873350847868652e+205

   1. Initial program 1.8

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

   3. Applied *-un-lft-identity 1.8

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

   4. Applied add-sqr-sqrt 2.0

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

   5. Applied times-frac 2.0

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

   6. Applied associate-*r* 2.3

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

   7. Simplified 2.3

      \[\leadsto x + \color{blue}{\left(\sqrt{z} \cdot \left(y - x\right)\right)} \cdot \frac{\sqrt{z}}{t}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le 3.29194830615707638 \cdot 10^{-308} \lor \neg \left(z \le 2.88733508478686524 \cdot 10^{205}\right):\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\sqrt{z} \cdot \left(y - x\right)\right) \cdot \frac{\sqrt{z}}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4"
  :precision binary64

  :herbie-target
  (if (< (* (- y x) (/ z t)) -1013646692435.887) (+ x (/ (- y x) (/ t z))) (if (< (* (- y x) (/ z t)) -0.0) (+ x (/ (* (- y x) z) t)) (+ x (/ (- y x) (/ t z)))))

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