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

Average Error: 10.7 → 1.0

Time: 3.7s

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 r620880 = x;
        double r620881 = y;
        double r620882 = z;
        double r620883 = r620881 - r620882;
        double r620884 = t;
        double r620885 = r620883 * r620884;
        double r620886 = a;
        double r620887 = r620886 - r620882;
        double r620888 = r620885 / r620887;
        double r620889 = r620880 + r620888;
        return r620889;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r620890 = x;
        double r620891 = y;
        double r620892 = z;
        double r620893 = r620891 - r620892;
        double r620894 = t;
        double r620895 = cbrt(r620894);
        double r620896 = r620895 * r620895;
        double r620897 = a;
        double r620898 = r620897 - r620892;
        double r620899 = cbrt(r620898);
        double r620900 = r620899 * r620899;
        double r620901 = r620896 / r620900;
        double r620902 = r620893 * r620901;
        double r620903 = r620895 / r620899;
        double r620904 = r620902 * r620903;
        double r620905 = r620890 + r620904;
        return r620905;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	10.7
Target	0.6
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 10.7

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

3. Applied *-un-lft-identity 10.7

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

4. Applied times-frac 3.0

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

5. Simplified 3.0

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

7. Applied add-cube-cbrt 3.5

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

8. Applied add-cube-cbrt 3.6

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

9. Applied times-frac 3.6

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

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 2019356 
(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))))