Average Error: 6.2 → 0.9
Time: 35.0s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[x + \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}} \cdot \frac{\sqrt[3]{1}}{\frac{\frac{\sqrt[3]{1}}{\sqrt[3]{z - t}}}{\frac{\sqrt[3]{y}}{\sqrt[3]{a}}}}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
x + \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}} \cdot \frac{\sqrt[3]{1}}{\frac{\frac{\sqrt[3]{1}}{\sqrt[3]{z - t}}}{\frac{\sqrt[3]{y}}{\sqrt[3]{a}}}}
double f(double x, double y, double z, double t, double a) {
        double r254456 = x;
        double r254457 = y;
        double r254458 = z;
        double r254459 = t;
        double r254460 = r254458 - r254459;
        double r254461 = r254457 * r254460;
        double r254462 = a;
        double r254463 = r254461 / r254462;
        double r254464 = r254456 + r254463;
        return r254464;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r254465 = x;
        double r254466 = 1.0;
        double r254467 = cbrt(r254466);
        double r254468 = r254467 * r254467;
        double r254469 = z;
        double r254470 = t;
        double r254471 = r254469 - r254470;
        double r254472 = cbrt(r254471);
        double r254473 = r254472 * r254472;
        double r254474 = r254468 / r254473;
        double r254475 = y;
        double r254476 = cbrt(r254475);
        double r254477 = r254476 * r254476;
        double r254478 = a;
        double r254479 = cbrt(r254478);
        double r254480 = r254479 * r254479;
        double r254481 = r254477 / r254480;
        double r254482 = r254474 / r254481;
        double r254483 = r254468 / r254482;
        double r254484 = r254467 / r254472;
        double r254485 = r254476 / r254479;
        double r254486 = r254484 / r254485;
        double r254487 = r254467 / r254486;
        double r254488 = r254483 * r254487;
        double r254489 = r254465 + r254488;
        return r254489;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.2
Target0.5
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Initial program 6.2

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

  3. Applied associate-/l*5.7

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

  5. Applied div-inv5.7

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

  6. Applied associate-/r*2.4

    \[\leadsto x + \color{blue}{\frac{\frac{y}{a}}{\frac{1}{z - t}}}\]
  7. Using strategy rm

  8. Applied *-un-lft-identity2.4

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

  9. Applied *-un-lft-identity2.4

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

  10. Applied times-frac2.4

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

  11. Applied associate-/l*2.4

    \[\leadsto x + \color{blue}{\frac{\frac{1}{1}}{\frac{\frac{1}{z - t}}{\frac{y}{a}}}}\]
  12. Using strategy rm

  13. Applied add-cube-cbrt2.8

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

  14. Applied add-cube-cbrt3.0

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

  15. Applied times-frac3.0

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

  16. Applied add-cube-cbrt3.0

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

  17. Applied add-cube-cbrt3.0

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

  18. Applied times-frac3.1

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

  19. Applied times-frac0.9

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

  20. Applied add-cube-cbrt0.9

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

  21. Applied times-frac0.9

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

  22. Simplified0.9

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

  23. Simplified0.9

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

  24. Final simplification0.9

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

Reproduce

herbie shell --seed 2019294 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :precision binary64

  :herbie-target
  (if (< y -1.07612662163899753e-10) (+ x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2.8944268627920891e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

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