Average Error: 5.9 → 1.0
Time: 14.0s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -2.717101417616620114043785351731635839082 \cdot 10^{48} \lor \neg \left(a \le 261096693021459858565601987977292223086600\right):\\ \;\;\;\;x + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\frac{\sqrt[3]{y}}{a} \cdot \left(t - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t - z\right) \cdot y\right) \cdot \frac{1}{a} + x\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;a \le -2.717101417616620114043785351731635839082 \cdot 10^{48} \lor \neg \left(a \le 261096693021459858565601987977292223086600\right):\\
\;\;\;\;x + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\frac{\sqrt[3]{y}}{a} \cdot \left(t - z\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\left(t - z\right) \cdot y\right) \cdot \frac{1}{a} + x\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r283235 = x;
        double r283236 = y;
        double r283237 = z;
        double r283238 = t;
        double r283239 = r283237 - r283238;
        double r283240 = r283236 * r283239;
        double r283241 = a;
        double r283242 = r283240 / r283241;
        double r283243 = r283235 - r283242;
        return r283243;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r283244 = a;
        double r283245 = -2.71710141761662e+48;
        bool r283246 = r283244 <= r283245;
        double r283247 = 2.6109669302145986e+41;
        bool r283248 = r283244 <= r283247;
        double r283249 = !r283248;
        bool r283250 = r283246 || r283249;
        double r283251 = x;
        double r283252 = y;
        double r283253 = cbrt(r283252);
        double r283254 = r283253 * r283253;
        double r283255 = r283253 / r283244;
        double r283256 = t;
        double r283257 = z;
        double r283258 = r283256 - r283257;
        double r283259 = r283255 * r283258;
        double r283260 = r283254 * r283259;
        double r283261 = r283251 + r283260;
        double r283262 = r283258 * r283252;
        double r283263 = 1.0;
        double r283264 = r283263 / r283244;
        double r283265 = r283262 * r283264;
        double r283266 = r283265 + r283251;
        double r283267 = r283250 ? r283261 : r283266;
        return r283267;
}

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
Target0.7
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. Split input into 2 regimes

  2. if a < -2.71710141761662e+48 or 2.6109669302145986e+41 < a

    1. Initial program 10.1

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

    2. Simplified1.9

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

    4. Applied fma-udef1.9

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

    5. Simplified0.5

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

    7. Applied *-un-lft-identity0.5

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

    8. Applied *-un-lft-identity0.5

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

    9. Applied times-frac0.5

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

    10. Applied add-cube-cbrt0.9

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

    11. Applied times-frac0.9

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

    12. Simplified0.9

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

    13. Simplified0.8

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

    if -2.71710141761662e+48 < a < 2.6109669302145986e+41

    1. Initial program 1.2

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

    2. Simplified3.5

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

    4. Applied fma-udef3.5

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

    5. Simplified10.8

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

    7. Applied div-inv10.8

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

    8. Applied *-un-lft-identity10.8

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

    9. Applied times-frac1.4

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

    10. Simplified1.3

      \[\leadsto \frac{1}{a} \cdot \color{blue}{\left(\left(t - z\right) \cdot y\right)} + x\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -2.717101417616620114043785351731635839082 \cdot 10^{48} \lor \neg \left(a \le 261096693021459858565601987977292223086600\right):\\ \;\;\;\;x + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\frac{\sqrt[3]{y}}{a} \cdot \left(t - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t - z\right) \cdot y\right) \cdot \frac{1}{a} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"

  :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)))