Average Error: 3.7 → 1.3
Time: 18.5s
Precision: 64
\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le 1.018025094121265229501382308592741530739 \cdot 10^{-108}:\\ \;\;\;\;\left(x - \frac{\frac{y}{z}}{3}\right) + \frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{z} \cdot \frac{\frac{\sqrt[3]{t}}{3}}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{z \cdot \left(3 \cdot y\right)}\\ \end{array}\]
\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;t \le 1.018025094121265229501382308592741530739 \cdot 10^{-108}:\\
\;\;\;\;\left(x - \frac{\frac{y}{z}}{3}\right) + \frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{z} \cdot \frac{\frac{\sqrt[3]{t}}{3}}{y}\\

\mathbf{else}:\\
\;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{z \cdot \left(3 \cdot y\right)}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r621324 = x;
        double r621325 = y;
        double r621326 = z;
        double r621327 = 3.0;
        double r621328 = r621326 * r621327;
        double r621329 = r621325 / r621328;
        double r621330 = r621324 - r621329;
        double r621331 = t;
        double r621332 = r621328 * r621325;
        double r621333 = r621331 / r621332;
        double r621334 = r621330 + r621333;
        return r621334;
}
↓
double f(double x, double y, double z, double t) {
        double r621335 = t;
        double r621336 = 1.0180250941212652e-108;
        bool r621337 = r621335 <= r621336;
        double r621338 = x;
        double r621339 = y;
        double r621340 = z;
        double r621341 = r621339 / r621340;
        double r621342 = 3.0;
        double r621343 = r621341 / r621342;
        double r621344 = r621338 - r621343;
        double r621345 = cbrt(r621335);
        double r621346 = r621345 * r621345;
        double r621347 = r621346 / r621340;
        double r621348 = r621345 / r621342;
        double r621349 = r621348 / r621339;
        double r621350 = r621347 * r621349;
        double r621351 = r621344 + r621350;
        double r621352 = r621340 * r621342;
        double r621353 = r621339 / r621352;
        double r621354 = r621338 - r621353;
        double r621355 = r621342 * r621339;
        double r621356 = r621340 * r621355;
        double r621357 = r621335 / r621356;
        double r621358 = r621354 + r621357;
        double r621359 = r621337 ? r621351 : r621358;
        return r621359;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original3.7
Target1.8
Herbie1.3
\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y}\]

Derivation

  1. Split input into 2 regimes

  2. if t < 1.0180250941212652e-108

    1. Initial program 4.9

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\]
    2. Using strategy rm

    3. Applied associate-/r*1.8

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{\frac{t}{z \cdot 3}}{y}}\]
    4. Using strategy rm

    5. Applied associate-/r*1.8

      \[\leadsto \left(x - \color{blue}{\frac{\frac{y}{z}}{3}}\right) + \frac{\frac{t}{z \cdot 3}}{y}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity1.8

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

    8. Applied add-cube-cbrt2.1

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

    9. Applied times-frac2.0

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

    10. Applied times-frac1.4

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

    11. Simplified1.4

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

    if 1.0180250941212652e-108 < t

    1. Initial program 1.2

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\]
    2. Using strategy rm

    3. Applied associate-*l*1.2

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

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 1.018025094121265229501382308592741530739 \cdot 10^{-108}:\\ \;\;\;\;\left(x - \frac{\frac{y}{z}}{3}\right) + \frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{z} \cdot \frac{\frac{\sqrt[3]{t}}{3}}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{z \cdot \left(3 \cdot y\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, H"
  :precision binary64

  :herbie-target
  (+ (- x (/ y (* z 3))) (/ (/ t (* z 3)) y))

  (+ (- x (/ y (* z 3))) (/ t (* (* z 3) y))))