Average Error: 6.1 → 0.5
Time: 3.8s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -1.091631506502542157480520102888531476655 \cdot 10^{305}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 7.843751749471894793264472097780927032588 \cdot 10^{196}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + 1 \cdot \frac{z - t}{\frac{a}{y}}\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \le -1.091631506502542157480520102888531476655 \cdot 10^{305}:\\
\;\;\;\;x + y \cdot \frac{z - t}{a}\\

\mathbf{elif}\;y \cdot \left(z - t\right) \le 7.843751749471894793264472097780927032588 \cdot 10^{196}:\\
\;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r360213 = x;
        double r360214 = y;
        double r360215 = z;
        double r360216 = t;
        double r360217 = r360215 - r360216;
        double r360218 = r360214 * r360217;
        double r360219 = a;
        double r360220 = r360218 / r360219;
        double r360221 = r360213 + r360220;
        return r360221;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r360222 = y;
        double r360223 = z;
        double r360224 = t;
        double r360225 = r360223 - r360224;
        double r360226 = r360222 * r360225;
        double r360227 = -1.0916315065025422e+305;
        bool r360228 = r360226 <= r360227;
        double r360229 = x;
        double r360230 = a;
        double r360231 = r360225 / r360230;
        double r360232 = r360222 * r360231;
        double r360233 = r360229 + r360232;
        double r360234 = 7.843751749471895e+196;
        bool r360235 = r360226 <= r360234;
        double r360236 = r360226 / r360230;
        double r360237 = r360229 + r360236;
        double r360238 = 1.0;
        double r360239 = r360230 / r360222;
        double r360240 = r360225 / r360239;
        double r360241 = r360238 * r360240;
        double r360242 = r360229 + r360241;
        double r360243 = r360235 ? r360237 : r360242;
        double r360244 = r360228 ? r360233 : r360243;
        return r360244;
}

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

Original6.1
Target0.7
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* y (- z t)) < -1.0916315065025422e+305

    1. Initial program 61.4

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

    3. Applied *-un-lft-identity61.4

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

    4. Applied times-frac0.3

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

    5. Simplified0.3

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

    if -1.0916315065025422e+305 < (* y (- z t)) < 7.843751749471895e+196

    1. Initial program 0.4

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

    if 7.843751749471895e+196 < (* y (- z t))

    1. Initial program 27.8

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

    3. Applied clear-num27.8

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

    5. Applied associate-/r*0.8

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

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

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

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

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

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

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

    10. Applied times-frac0.8

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

    11. Applied times-frac0.8

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

    12. Applied add-cube-cbrt0.8

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

    13. Applied times-frac0.8

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

    14. Simplified0.8

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

    15. Simplified0.8

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -1.091631506502542157480520102888531476655 \cdot 10^{305}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 7.843751749471894793264472097780927032588 \cdot 10^{196}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + 1 \cdot \frac{z - t}{\frac{a}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :precision binary64

  :herbie-target
  (if (< y -1.0761266216389975e-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)))