Average Error: 4.1 → 1.1
Time: 4.4s
Precision: 64
\[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot 9 \le -3.68671894795848607 \cdot 10^{-77}:\\ \;\;\;\;\left(x \cdot 2 - y \cdot \left(9 \cdot \left(z \cdot t\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \mathbf{elif}\;y \cdot 9 \le 4.4323646665533492 \cdot 10^{-78}:\\ \;\;\;\;x \cdot 2 + \mathsf{fma}\left(a, 27 \cdot b, -\left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - y \cdot \left(\left(9 \cdot z\right) \cdot t\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array}\]
\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 -3.68671894795848607 \cdot 10^{-77}:\\
\;\;\;\;\left(x \cdot 2 - y \cdot \left(9 \cdot \left(z \cdot t\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\

\mathbf{elif}\;y \cdot 9 \le 4.4323646665533492 \cdot 10^{-78}:\\
\;\;\;\;x \cdot 2 + \mathsf{fma}\left(a, 27 \cdot b, -\left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r825512 = x;
        double r825513 = 2.0;
        double r825514 = r825512 * r825513;
        double r825515 = y;
        double r825516 = 9.0;
        double r825517 = r825515 * r825516;
        double r825518 = z;
        double r825519 = r825517 * r825518;
        double r825520 = t;
        double r825521 = r825519 * r825520;
        double r825522 = r825514 - r825521;
        double r825523 = a;
        double r825524 = 27.0;
        double r825525 = r825523 * r825524;
        double r825526 = b;
        double r825527 = r825525 * r825526;
        double r825528 = r825522 + r825527;
        return r825528;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r825529 = y;
        double r825530 = 9.0;
        double r825531 = r825529 * r825530;
        double r825532 = -3.686718947958486e-77;
        bool r825533 = r825531 <= r825532;
        double r825534 = x;
        double r825535 = 2.0;
        double r825536 = r825534 * r825535;
        double r825537 = z;
        double r825538 = t;
        double r825539 = r825537 * r825538;
        double r825540 = r825530 * r825539;
        double r825541 = r825529 * r825540;
        double r825542 = r825536 - r825541;
        double r825543 = a;
        double r825544 = 27.0;
        double r825545 = r825543 * r825544;
        double r825546 = b;
        double r825547 = r825545 * r825546;
        double r825548 = r825542 + r825547;
        double r825549 = 4.432364666553349e-78;
        bool r825550 = r825531 <= r825549;
        double r825551 = r825544 * r825546;
        double r825552 = r825531 * r825537;
        double r825553 = r825552 * r825538;
        double r825554 = -r825553;
        double r825555 = fma(r825543, r825551, r825554);
        double r825556 = r825536 + r825555;
        double r825557 = r825530 * r825537;
        double r825558 = r825557 * r825538;
        double r825559 = r825529 * r825558;
        double r825560 = r825536 - r825559;
        double r825561 = r825560 + r825547;
        double r825562 = r825550 ? r825556 : r825561;
        double r825563 = r825533 ? r825548 : r825562;
        return r825563;
}

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

Target

Original4.1
Target2.8
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;y \lt 7.590524218811189 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - 9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* y 9.0) < -3.686718947958486e-77

    1. Initial program 6.6

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

    3. Applied associate-*l*1.3

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

    5. Applied associate-*l*1.3

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

    if -3.686718947958486e-77 < (* y 9.0) < 4.432364666553349e-78

    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 sub-neg0.7

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

    4. Applied associate-+l+0.7

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

    5. Simplified0.7

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

    if 4.432364666553349e-78 < (* y 9.0)

    1. Initial program 7.0

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

    3. Applied associate-*l*1.4

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

    5. Applied associate-*l*1.3

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

    7. Applied associate-*r*1.4

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot 9 \le -3.68671894795848607 \cdot 10^{-77}:\\ \;\;\;\;\left(x \cdot 2 - y \cdot \left(9 \cdot \left(z \cdot t\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \mathbf{elif}\;y \cdot 9 \le 4.4323646665533492 \cdot 10^{-78}:\\ \;\;\;\;x \cdot 2 + \mathsf{fma}\left(a, 27 \cdot b, -\left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - y \cdot \left(\left(9 \cdot z\right) \cdot t\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +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)))