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

Average Error: 10.5 → 1.0

Time: 12.1s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a) {
        double r437707 = x;
        double r437708 = y;
        double r437709 = z;
        double r437710 = r437708 - r437709;
        double r437711 = t;
        double r437712 = r437710 * r437711;
        double r437713 = a;
        double r437714 = r437713 - r437709;
        double r437715 = r437712 / r437714;
        double r437716 = r437707 + r437715;
        return r437716;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r437717 = x;
        double r437718 = y;
        double r437719 = z;
        double r437720 = r437718 - r437719;
        double r437721 = t;
        double r437722 = cbrt(r437721);
        double r437723 = r437722 * r437722;
        double r437724 = a;
        double r437725 = r437724 - r437719;
        double r437726 = cbrt(r437725);
        double r437727 = r437726 * r437726;
        double r437728 = r437723 / r437727;
        double r437729 = r437720 * r437728;
        double r437730 = r437722 / r437726;
        double r437731 = r437729 * r437730;
        double r437732 = r437717 + r437731;
        return r437732;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	10.5
Target	0.7
Herbie	1.0

\[\begin{array}{l} \mathbf{if}\;t \lt -1.068297449017406694366747246993994850729 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t \lt 3.911094988758637497591020599238553861375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array}\]

Derivation

1. Initial program 10.5

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

3. Applied *-un-lft-identity 10.5

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

4. Applied times-frac 3.3

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

5. Simplified 3.3

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

7. Applied add-cube-cbrt 3.7

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

8. Applied add-cube-cbrt 3.8

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

9. Applied times-frac 3.8

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

10. Applied associate-*r* 1.0

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

11. Final simplification 1.0

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

Reproduce

herbie shell --seed 2019323 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (if (< t -1.0682974490174067e-39) (+ x (* (/ (- y z) (- a z)) t)) (if (< t 3.9110949887586375e-141) (+ x (/ (* (- y z) t) (- a z))) (+ x (* (/ (- y z) (- a z)) t))))

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