Average Error: 6.4 → 1.9
Time: 15.7s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[x - \left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}\right)\right) \cdot \left(\frac{\sqrt[3]{y}}{a} \cdot \sqrt[3]{z - t}\right)\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
x - \left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}\right)\right) \cdot \left(\frac{\sqrt[3]{y}}{a} \cdot \sqrt[3]{z - t}\right)
double f(double x, double y, double z, double t, double a) {
        double r184208 = x;
        double r184209 = y;
        double r184210 = z;
        double r184211 = t;
        double r184212 = r184210 - r184211;
        double r184213 = r184209 * r184212;
        double r184214 = a;
        double r184215 = r184213 / r184214;
        double r184216 = r184208 - r184215;
        return r184216;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r184217 = x;
        double r184218 = y;
        double r184219 = cbrt(r184218);
        double r184220 = r184219 * r184219;
        double r184221 = z;
        double r184222 = t;
        double r184223 = r184221 - r184222;
        double r184224 = cbrt(r184223);
        double r184225 = r184224 * r184224;
        double r184226 = r184220 * r184225;
        double r184227 = a;
        double r184228 = r184219 / r184227;
        double r184229 = r184228 * r184224;
        double r184230 = r184226 * r184229;
        double r184231 = r184217 - r184230;
        return r184231;
}

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

Original6.4
Target0.7
Herbie1.9
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Initial program 6.4

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

  3. Applied associate-/l*5.6

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

  5. Applied div-inv5.6

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

  6. Applied associate-/r*2.5

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

  8. Applied add-cube-cbrt3.0

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

  9. Applied add-cube-cbrt3.0

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

  10. Applied times-frac3.0

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

  11. Applied *-un-lft-identity3.0

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

  12. Applied add-cube-cbrt3.1

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

  13. Applied times-frac3.1

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

  14. Applied times-frac1.9

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

  15. Simplified1.9

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

  16. Simplified1.9

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

  17. Final simplification1.9

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

Reproduce

herbie shell --seed 2019323 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"
  :precision binary64

  :herbie-target
  (if (< y -1.0761266216389975e-10) (- x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (- x (/ (* y (- z t)) a)) (- x (/ y (/ a (- z t))))))

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