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

Average Error: 1.2 → 0.5

Time: 19.4s

Precision: 64

Math

\[x + y \cdot \frac{z - t}{a - t}\]

↓

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

C

double f(double x, double y, double z, double t, double a) {
        double r30095261 = x;
        double r30095262 = y;
        double r30095263 = z;
        double r30095264 = t;
        double r30095265 = r30095263 - r30095264;
        double r30095266 = a;
        double r30095267 = r30095266 - r30095264;
        double r30095268 = r30095265 / r30095267;
        double r30095269 = r30095262 * r30095268;
        double r30095270 = r30095261 + r30095269;
        return r30095270;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r30095271 = z;
        double r30095272 = t;
        double r30095273 = r30095271 - r30095272;
        double r30095274 = cbrt(r30095273);
        double r30095275 = a;
        double r30095276 = r30095275 - r30095272;
        double r30095277 = cbrt(r30095276);
        double r30095278 = r30095274 / r30095277;
        double r30095279 = r30095278 * r30095278;
        double r30095280 = y;
        double r30095281 = r30095277 / r30095274;
        double r30095282 = r30095280 / r30095281;
        double r30095283 = r30095279 * r30095282;
        double r30095284 = x;
        double r30095285 = r30095283 + r30095284;
        return r30095285;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	1.2
Target	0.5
Herbie	0.5

\[\begin{array}{l} \mathbf{if}\;y \lt -8.508084860551241 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y \lt 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

1. Initial program 1.2

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

3. Applied add-cube-cbrt 1.7

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

4. Applied associate-*l* 1.7

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

6. Applied add-cube-cbrt 1.8

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

7. Applied add-cube-cbrt 1.8

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

8. Applied times-frac 1.8

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

9. Applied associate-*r* 1.2

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

10. Simplified 1.2

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

12. Applied pow1 1.2

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

13. Applied pow1 1.2

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

14. Applied pow-prod-down 1.2

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

15. Applied pow1 1.2

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

16. Applied pow1 1.2

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

17. Applied pow-prod-down 1.2

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

18. Applied pow-prod-down 1.2

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

19. Simplified 0.5

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

20. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019163 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"

  :herbie-target
  (if (< y -8.508084860551241e-17) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2.894426862792089e-49) (+ x (* (* y (- z t)) (/ 1 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

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