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

Average Error: 11.3 → 0.9

Time: 5.6s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a) {
        double r690235 = x;
        double r690236 = y;
        double r690237 = z;
        double r690238 = r690236 - r690237;
        double r690239 = t;
        double r690240 = r690238 * r690239;
        double r690241 = a;
        double r690242 = r690241 - r690237;
        double r690243 = r690240 / r690242;
        double r690244 = r690235 + r690243;
        return r690244;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r690245 = x;
        double r690246 = y;
        double r690247 = z;
        double r690248 = r690246 - r690247;
        double r690249 = cbrt(r690248);
        double r690250 = a;
        double r690251 = r690250 - r690247;
        double r690252 = cbrt(r690251);
        double r690253 = r690252 * r690252;
        double r690254 = r690253 / r690249;
        double r690255 = r690249 / r690254;
        double r690256 = cbrt(r690249);
        double r690257 = r690256 * r690256;
        double r690258 = t;
        double r690259 = cbrt(r690258);
        double r690260 = r690259 * r690259;
        double r690261 = r690257 * r690260;
        double r690262 = r690252 / r690259;
        double r690263 = r690256 / r690262;
        double r690264 = r690261 * r690263;
        double r690265 = r690255 * r690264;
        double r690266 = r690245 + r690265;
        return r690266;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	11.3
Target	0.5
Herbie	0.9

\[\begin{array}{l} \mathbf{if}\;t \lt -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t \lt 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array}\]

Derivation

1. Initial program 11.3

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

3. Applied associate-/l* 3.1

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

5. Applied *-un-lft-identity 3.1

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

6. Applied add-cube-cbrt 3.5

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

7. Applied times-frac 3.5

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

8. Applied add-cube-cbrt 3.5

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

9. Applied times-frac 1.1

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

10. Simplified 1.1

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

12. Applied add-cube-cbrt 1.3

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

13. Applied *-un-lft-identity 1.3

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

14. Applied cbrt-prod 1.3

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

15. Applied times-frac 1.3

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

16. Applied add-cube-cbrt 1.4

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

17. Applied times-frac 0.9

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

18. Simplified 0.9

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

19. Final simplification 0.9

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

Reproduce

herbie shell --seed 2020025 
(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))))