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

Average Error: 1.2 → 0.5

Time: 20.5s

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 r28750938 = x;
        double r28750939 = y;
        double r28750940 = z;
        double r28750941 = t;
        double r28750942 = r28750940 - r28750941;
        double r28750943 = a;
        double r28750944 = r28750943 - r28750941;
        double r28750945 = r28750942 / r28750944;
        double r28750946 = r28750939 * r28750945;
        double r28750947 = r28750938 + r28750946;
        return r28750947;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r28750948 = z;
        double r28750949 = t;
        double r28750950 = r28750948 - r28750949;
        double r28750951 = cbrt(r28750950);
        double r28750952 = a;
        double r28750953 = r28750952 - r28750949;
        double r28750954 = cbrt(r28750953);
        double r28750955 = r28750951 / r28750954;
        double r28750956 = r28750955 * r28750955;
        double r28750957 = y;
        double r28750958 = r28750954 / r28750951;
        double r28750959 = r28750957 / r28750958;
        double r28750960 = r28750956 * r28750959;
        double r28750961 = x;
        double r28750962 = r28750960 + r28750961;
        return r28750962;
}

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)))))