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

Average Error: 11.1 → 1.5

Time: 55.5s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a) {
        double r27414565 = x;
        double r27414566 = y;
        double r27414567 = z;
        double r27414568 = t;
        double r27414569 = r27414567 - r27414568;
        double r27414570 = r27414566 * r27414569;
        double r27414571 = a;
        double r27414572 = r27414567 - r27414571;
        double r27414573 = r27414570 / r27414572;
        double r27414574 = r27414565 + r27414573;
        return r27414574;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r27414575 = x;
        double r27414576 = y;
        double r27414577 = cbrt(r27414576);
        double r27414578 = z;
        double r27414579 = a;
        double r27414580 = r27414578 - r27414579;
        double r27414581 = cbrt(r27414580);
        double r27414582 = cbrt(r27414581);
        double r27414583 = r27414577 / r27414582;
        double r27414584 = r27414583 / r27414582;
        double r27414585 = r27414577 * r27414577;
        double r27414586 = r27414585 / r27414581;
        double r27414587 = 1.0;
        double r27414588 = r27414581 * r27414581;
        double r27414589 = cbrt(r27414588);
        double r27414590 = r27414587 / r27414589;
        double r27414591 = r27414582 * r27414582;
        double r27414592 = r27414590 / r27414591;
        double r27414593 = r27414586 * r27414592;
        double r27414594 = t;
        double r27414595 = r27414578 - r27414594;
        double r27414596 = r27414593 * r27414595;
        double r27414597 = r27414584 * r27414596;
        double r27414598 = r27414575 + r27414597;
        return r27414598;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	11.1
Target	1.3
Herbie	1.5

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

Derivation

1. Initial program 11.1

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

2. Simplified 3.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(z - t, \frac{y}{z - a}, x\right)}\]
3. Using strategy rm

4. Applied fma-udef 3.0

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

6. Applied add-cube-cbrt 3.4

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

7. Applied associate-/r* 3.4

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

9. Applied *-un-lft-identity 3.4

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

10. Applied cbrt-prod 3.4

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

11. Applied add-cube-cbrt 3.5

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

12. Applied times-frac 3.5

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

13. Applied times-frac 3.5

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

14. Applied associate-*r* 3.5

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

15. Simplified 3.5

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

17. Applied add-cube-cbrt 3.6

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

18. Applied add-cube-cbrt 3.6

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

19. Applied cbrt-prod 3.7

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

20. Applied *-un-lft-identity 3.7

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

21. Applied cbrt-prod 3.7

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

22. Applied times-frac 3.7

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

23. Applied times-frac 3.7

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

24. Applied associate-*r* 3.7

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

25. Simplified 1.5

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

26. Final simplification 1.5

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A"

  :herbie-target
  (+ x (/ y (/ (- z a) (- z t))))

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