Average Error: 5.9 → 1.0
Time: 16.4s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[x + \left(\left(z - t\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a}}\right) \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{a}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a}}\right)\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
x + \left(\left(z - t\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a}}\right) \cdot \left(\frac{\sqrt[3]{y}}{\sqrt[3]{a}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a}}\right)
double f(double x, double y, double z, double t, double a) {
        double r13637965 = x;
        double r13637966 = y;
        double r13637967 = z;
        double r13637968 = t;
        double r13637969 = r13637967 - r13637968;
        double r13637970 = r13637966 * r13637969;
        double r13637971 = a;
        double r13637972 = r13637970 / r13637971;
        double r13637973 = r13637965 + r13637972;
        return r13637973;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r13637974 = x;
        double r13637975 = z;
        double r13637976 = t;
        double r13637977 = r13637975 - r13637976;
        double r13637978 = y;
        double r13637979 = cbrt(r13637978);
        double r13637980 = a;
        double r13637981 = cbrt(r13637980);
        double r13637982 = r13637979 / r13637981;
        double r13637983 = r13637977 * r13637982;
        double r13637984 = r13637982 * r13637982;
        double r13637985 = r13637983 * r13637984;
        double r13637986 = r13637974 + r13637985;
        return r13637986;
}

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

Original5.9
Target0.8
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Initial program 5.9

    \[x + \frac{y \cdot \left(z - t\right)}{a}\]
  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 *-un-lft-identity5.7

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

  6. Applied add-cube-cbrt6.2

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

  7. Applied times-frac6.2

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

  8. Applied add-cube-cbrt6.3

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

  9. Applied times-frac2.1

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

  10. Simplified2.1

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

  11. Simplified1.0

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

  12. Final simplification1.0

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

Reproduce

herbie shell --seed 2019172 +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)))