Average Error: 3.9 → 0.3
Time: 13.8s
Precision: 64
\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot 3 \le -2.315641800986653731087017149548046290874:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + t \cdot \frac{1}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{elif}\;z \cdot 3 \le 2.181965382420920083950531709255992849268 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{t}{y}}{z \cdot 3} + \left(x - \frac{y}{z \cdot 3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{1}{z} \cdot \frac{y}{3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \end{array}\]
\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;z \cdot 3 \le -2.315641800986653731087017149548046290874:\\
\;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + t \cdot \frac{1}{\left(z \cdot 3\right) \cdot y}\\

\mathbf{elif}\;z \cdot 3 \le 2.181965382420920083950531709255992849268 \cdot 10^{-43}:\\
\;\;\;\;\frac{\frac{t}{y}}{z \cdot 3} + \left(x - \frac{y}{z \cdot 3}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(x - \frac{1}{z} \cdot \frac{y}{3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r671359 = x;
        double r671360 = y;
        double r671361 = z;
        double r671362 = 3.0;
        double r671363 = r671361 * r671362;
        double r671364 = r671360 / r671363;
        double r671365 = r671359 - r671364;
        double r671366 = t;
        double r671367 = r671363 * r671360;
        double r671368 = r671366 / r671367;
        double r671369 = r671365 + r671368;
        return r671369;
}
↓
double f(double x, double y, double z, double t) {
        double r671370 = z;
        double r671371 = 3.0;
        double r671372 = r671370 * r671371;
        double r671373 = -2.3156418009866537;
        bool r671374 = r671372 <= r671373;
        double r671375 = x;
        double r671376 = y;
        double r671377 = r671376 / r671372;
        double r671378 = r671375 - r671377;
        double r671379 = t;
        double r671380 = 1.0;
        double r671381 = r671372 * r671376;
        double r671382 = r671380 / r671381;
        double r671383 = r671379 * r671382;
        double r671384 = r671378 + r671383;
        double r671385 = 2.18196538242092e-43;
        bool r671386 = r671372 <= r671385;
        double r671387 = r671379 / r671376;
        double r671388 = r671387 / r671372;
        double r671389 = r671388 + r671378;
        double r671390 = r671380 / r671370;
        double r671391 = r671376 / r671371;
        double r671392 = r671390 * r671391;
        double r671393 = r671375 - r671392;
        double r671394 = r671379 / r671381;
        double r671395 = r671393 + r671394;
        double r671396 = r671386 ? r671389 : r671395;
        double r671397 = r671374 ? r671384 : r671396;
        return r671397;
}

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

Original3.9
Target1.9
Herbie0.3
\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y}\]

Derivation

  1. Split input into 3 regimes

  2. if (* z 3.0) < -2.3156418009866537

    1. Initial program 0.4

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

    3. Applied div-inv0.4

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

    if -2.3156418009866537 < (* z 3.0) < 2.18196538242092e-43

    1. Initial program 12.1

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

    3. Applied add-cube-cbrt12.3

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

    4. Applied times-frac1.1

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

    6. Applied div-inv1.1

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

    7. Applied associate-*r*3.8

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

    8. Simplified3.4

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

    10. Applied pow13.4

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

    11. Applied pow13.4

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

    12. Applied pow-prod-down3.4

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

    13. Simplified0.3

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

    if 2.18196538242092e-43 < (* z 3.0)

    1. Initial program 0.3

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

    3. Applied *-un-lft-identity0.3

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

    4. Applied times-frac0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \le -2.315641800986653731087017149548046290874:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + t \cdot \frac{1}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{elif}\;z \cdot 3 \le 2.181965382420920083950531709255992849268 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{t}{y}}{z \cdot 3} + \left(x - \frac{y}{z \cdot 3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{1}{z} \cdot \frac{y}{3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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))))