Average Error: 6.3 → 1.5
Time: 18.0s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[x + \frac{\frac{y}{\frac{\sqrt[3]{a}}{\sqrt[3]{z - t}} \cdot \frac{\sqrt[3]{a}}{\sqrt[3]{z - t}}}}{\frac{\sqrt[3]{a}}{\sqrt[3]{z - t}}}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
x + \frac{\frac{y}{\frac{\sqrt[3]{a}}{\sqrt[3]{z - t}} \cdot \frac{\sqrt[3]{a}}{\sqrt[3]{z - t}}}}{\frac{\sqrt[3]{a}}{\sqrt[3]{z - t}}}
double f(double x, double y, double z, double t, double a) {
        double r25857441 = x;
        double r25857442 = y;
        double r25857443 = z;
        double r25857444 = t;
        double r25857445 = r25857443 - r25857444;
        double r25857446 = r25857442 * r25857445;
        double r25857447 = a;
        double r25857448 = r25857446 / r25857447;
        double r25857449 = r25857441 + r25857448;
        return r25857449;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r25857450 = x;
        double r25857451 = y;
        double r25857452 = a;
        double r25857453 = cbrt(r25857452);
        double r25857454 = z;
        double r25857455 = t;
        double r25857456 = r25857454 - r25857455;
        double r25857457 = cbrt(r25857456);
        double r25857458 = r25857453 / r25857457;
        double r25857459 = r25857458 * r25857458;
        double r25857460 = r25857451 / r25857459;
        double r25857461 = r25857460 / r25857458;
        double r25857462 = r25857450 + r25857461;
        return r25857462;
}

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

Original6.3
Target0.6
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Initial program 6.3

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

  3. Applied associate-/l*5.8

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

  5. Applied add-cube-cbrt6.2

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

  6. Applied add-cube-cbrt6.4

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

  7. Applied times-frac6.3

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

  8. Applied associate-/r*1.5

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

  9. Simplified1.5

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

  10. Final simplification1.5

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"

  :herbie-target
  (if (< y -1.0761266216389975e-10) (+ x (/ 1.0 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

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