Average Error: 3.8 → 0.7
Time: 13.7s
Precision: 64
\[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot 9 \le -3.338893372507656821369864426932132802203 \cdot 10^{-32}:\\ \;\;\;\;\left(x \cdot 2 - y \cdot \left(\sqrt{9} \cdot \left(\sqrt{9} \cdot \left(t \cdot z\right)\right)\right)\right) + 27 \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;y \cdot 9 \le 7.055620513394500257077151321056658993069 \cdot 10^{57}:\\ \;\;\;\;\left(x \cdot 2 - \left(y \cdot \left(z \cdot 9\right)\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(t \cdot z\right)\right) + \sqrt{27} \cdot \left(\sqrt{27} \cdot \left(a \cdot b\right)\right)\\ \end{array}\]
\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b
↓
\begin{array}{l}
\mathbf{if}\;y \cdot 9 \le -3.338893372507656821369864426932132802203 \cdot 10^{-32}:\\
\;\;\;\;\left(x \cdot 2 - y \cdot \left(\sqrt{9} \cdot \left(\sqrt{9} \cdot \left(t \cdot z\right)\right)\right)\right) + 27 \cdot \left(a \cdot b\right)\\

\mathbf{elif}\;y \cdot 9 \le 7.055620513394500257077151321056658993069 \cdot 10^{57}:\\
\;\;\;\;\left(x \cdot 2 - \left(y \cdot \left(z \cdot 9\right)\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\\

\mathbf{else}:\\
\;\;\;\;\left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(t \cdot z\right)\right) + \sqrt{27} \cdot \left(\sqrt{27} \cdot \left(a \cdot b\right)\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r487505 = x;
        double r487506 = 2.0;
        double r487507 = r487505 * r487506;
        double r487508 = y;
        double r487509 = 9.0;
        double r487510 = r487508 * r487509;
        double r487511 = z;
        double r487512 = r487510 * r487511;
        double r487513 = t;
        double r487514 = r487512 * r487513;
        double r487515 = r487507 - r487514;
        double r487516 = a;
        double r487517 = 27.0;
        double r487518 = r487516 * r487517;
        double r487519 = b;
        double r487520 = r487518 * r487519;
        double r487521 = r487515 + r487520;
        return r487521;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r487522 = y;
        double r487523 = 9.0;
        double r487524 = r487522 * r487523;
        double r487525 = -3.338893372507657e-32;
        bool r487526 = r487524 <= r487525;
        double r487527 = x;
        double r487528 = 2.0;
        double r487529 = r487527 * r487528;
        double r487530 = sqrt(r487523);
        double r487531 = t;
        double r487532 = z;
        double r487533 = r487531 * r487532;
        double r487534 = r487530 * r487533;
        double r487535 = r487530 * r487534;
        double r487536 = r487522 * r487535;
        double r487537 = r487529 - r487536;
        double r487538 = 27.0;
        double r487539 = a;
        double r487540 = b;
        double r487541 = r487539 * r487540;
        double r487542 = r487538 * r487541;
        double r487543 = r487537 + r487542;
        double r487544 = 7.0556205133945e+57;
        bool r487545 = r487524 <= r487544;
        double r487546 = r487532 * r487523;
        double r487547 = r487522 * r487546;
        double r487548 = r487547 * r487531;
        double r487549 = r487529 - r487548;
        double r487550 = r487539 * r487538;
        double r487551 = r487550 * r487540;
        double r487552 = r487549 + r487551;
        double r487553 = r487524 * r487533;
        double r487554 = r487529 - r487553;
        double r487555 = sqrt(r487538);
        double r487556 = r487555 * r487541;
        double r487557 = r487555 * r487556;
        double r487558 = r487554 + r487557;
        double r487559 = r487545 ? r487552 : r487558;
        double r487560 = r487526 ? r487543 : r487559;
        return r487560;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original3.8
Target2.6
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;y \lt 7.590524218811188954625810696587370427881 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - 9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* y 9.0) < -3.338893372507657e-32

    1. Initial program 7.1

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]
    2. Using strategy rm

    3. Applied associate-*l*0.8

      \[\leadsto \left(x \cdot 2 - \color{blue}{\left(y \cdot 9\right) \cdot \left(z \cdot t\right)}\right) + \left(a \cdot 27\right) \cdot b\]

    4. Simplified0.8

      \[\leadsto \left(x \cdot 2 - \left(y \cdot 9\right) \cdot \color{blue}{\left(t \cdot z\right)}\right) + \left(a \cdot 27\right) \cdot b\]

    5. Taylor expanded around 0 0.8

      \[\leadsto \left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(t \cdot z\right)\right) + \color{blue}{27 \cdot \left(a \cdot b\right)}\]
    6. Using strategy rm

    7. Applied associate-*l*0.7

      \[\leadsto \left(x \cdot 2 - \color{blue}{y \cdot \left(9 \cdot \left(t \cdot z\right)\right)}\right) + 27 \cdot \left(a \cdot b\right)\]
    8. Using strategy rm

    9. Applied add-sqr-sqrt0.7

      \[\leadsto \left(x \cdot 2 - y \cdot \left(\color{blue}{\left(\sqrt{9} \cdot \sqrt{9}\right)} \cdot \left(t \cdot z\right)\right)\right) + 27 \cdot \left(a \cdot b\right)\]

    10. Applied associate-*l*0.8

      \[\leadsto \left(x \cdot 2 - y \cdot \color{blue}{\left(\sqrt{9} \cdot \left(\sqrt{9} \cdot \left(t \cdot z\right)\right)\right)}\right) + 27 \cdot \left(a \cdot b\right)\]

    if -3.338893372507657e-32 < (* y 9.0) < 7.0556205133945e+57

    1. Initial program 0.7

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]
    2. Using strategy rm

    3. Applied associate-*l*0.7

      \[\leadsto \left(x \cdot 2 - \color{blue}{\left(y \cdot \left(9 \cdot z\right)\right)} \cdot t\right) + \left(a \cdot 27\right) \cdot b\]

    4. Simplified0.7

      \[\leadsto \left(x \cdot 2 - \left(y \cdot \color{blue}{\left(z \cdot 9\right)}\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]

    if 7.0556205133945e+57 < (* y 9.0)

    1. Initial program 10.4

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]
    2. Using strategy rm

    3. Applied associate-*l*0.9

      \[\leadsto \left(x \cdot 2 - \color{blue}{\left(y \cdot 9\right) \cdot \left(z \cdot t\right)}\right) + \left(a \cdot 27\right) \cdot b\]

    4. Simplified0.9

      \[\leadsto \left(x \cdot 2 - \left(y \cdot 9\right) \cdot \color{blue}{\left(t \cdot z\right)}\right) + \left(a \cdot 27\right) \cdot b\]

    5. Taylor expanded around 0 0.8

      \[\leadsto \left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(t \cdot z\right)\right) + \color{blue}{27 \cdot \left(a \cdot b\right)}\]
    6. Using strategy rm

    7. Applied add-sqr-sqrt0.8

      \[\leadsto \left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(t \cdot z\right)\right) + \color{blue}{\left(\sqrt{27} \cdot \sqrt{27}\right)} \cdot \left(a \cdot b\right)\]

    8. Applied associate-*l*0.9

      \[\leadsto \left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(t \cdot z\right)\right) + \color{blue}{\sqrt{27} \cdot \left(\sqrt{27} \cdot \left(a \cdot b\right)\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot 9 \le -3.338893372507656821369864426932132802203 \cdot 10^{-32}:\\ \;\;\;\;\left(x \cdot 2 - y \cdot \left(\sqrt{9} \cdot \left(\sqrt{9} \cdot \left(t \cdot z\right)\right)\right)\right) + 27 \cdot \left(a \cdot b\right)\\ \mathbf{elif}\;y \cdot 9 \le 7.055620513394500257077151321056658993069 \cdot 10^{57}:\\ \;\;\;\;\left(x \cdot 2 - \left(y \cdot \left(z \cdot 9\right)\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - \left(y \cdot 9\right) \cdot \left(t \cdot z\right)\right) + \sqrt{27} \cdot \left(\sqrt{27} \cdot \left(a \cdot b\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< y 7.590524218811189e-161) (+ (- (* x 2) (* (* (* y 9) z) t)) (* a (* 27 b))) (+ (- (* x 2) (* 9 (* y (* t z)))) (* (* a 27) b)))

  (+ (- (* x 2) (* (* (* y 9) z) t)) (* (* a 27) b)))