Average Error: 3.3 → 2.0
Time: 7.9s
Precision: 64
\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -9.4052837436924809 \cdot 10^{37}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + 0.333333333333333315 \cdot \frac{t}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - {\left(\frac{\frac{y}{3}}{z}\right)}^{1}\right) + \frac{\frac{1}{z}}{\frac{y}{\frac{t}{3}}}\\ \end{array}\]
\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;t \le -9.4052837436924809 \cdot 10^{37}:\\
\;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + 0.333333333333333315 \cdot \frac{t}{z \cdot y}\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r826410 = x;
        double r826411 = y;
        double r826412 = z;
        double r826413 = 3.0;
        double r826414 = r826412 * r826413;
        double r826415 = r826411 / r826414;
        double r826416 = r826410 - r826415;
        double r826417 = t;
        double r826418 = r826414 * r826411;
        double r826419 = r826417 / r826418;
        double r826420 = r826416 + r826419;
        return r826420;
}
↓
double f(double x, double y, double z, double t) {
        double r826421 = t;
        double r826422 = -9.40528374369248e+37;
        bool r826423 = r826421 <= r826422;
        double r826424 = x;
        double r826425 = y;
        double r826426 = z;
        double r826427 = 3.0;
        double r826428 = r826426 * r826427;
        double r826429 = r826425 / r826428;
        double r826430 = r826424 - r826429;
        double r826431 = 0.3333333333333333;
        double r826432 = r826426 * r826425;
        double r826433 = r826421 / r826432;
        double r826434 = r826431 * r826433;
        double r826435 = r826430 + r826434;
        double r826436 = r826425 / r826427;
        double r826437 = r826436 / r826426;
        double r826438 = 1.0;
        double r826439 = pow(r826437, r826438);
        double r826440 = r826424 - r826439;
        double r826441 = r826438 / r826426;
        double r826442 = r826421 / r826427;
        double r826443 = r826425 / r826442;
        double r826444 = r826441 / r826443;
        double r826445 = r826440 + r826444;
        double r826446 = r826423 ? r826435 : r826445;
        return r826446;
}

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

Original3.3
Target1.8
Herbie2.0
\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -9.40528374369248e+37

    1. Initial program 0.7

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

    2. Taylor expanded around 0 0.8

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{0.333333333333333315 \cdot \frac{t}{z \cdot y}}\]

    if -9.40528374369248e+37 < t

    1. Initial program 3.9

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

    3. Applied associate-/r*1.5

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

    5. Applied *-un-lft-identity1.5

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

    6. Applied times-frac1.5

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

    7. Applied associate-/l*2.3

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

    9. Applied *-un-lft-identity2.3

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

    10. Applied times-frac2.3

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

    12. Applied pow12.3

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

    13. Applied pow12.3

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

    14. Applied pow-prod-down2.3

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

    15. Simplified2.3

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

  4. Final simplification2.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -9.4052837436924809 \cdot 10^{37}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + 0.333333333333333315 \cdot \frac{t}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - {\left(\frac{\frac{y}{3}}{z}\right)}^{1}\right) + \frac{\frac{1}{z}}{\frac{y}{\frac{t}{3}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +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))))