Average Error: 5.6 → 3.5
Time: 17.2s
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -10735.131202477718:\\ \;\;\;\;\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\left(\left(\sqrt[3]{\sqrt[3]{j}} \cdot \sqrt[3]{\sqrt[3]{j}}\right) \cdot \sqrt[3]{\sqrt[3]{j}}\right) \cdot \left(27 \cdot k\right)\right)\right)\right)\\ \mathbf{elif}\;t \le 3.5577987813979318 \cdot 10^{43}:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(z \cdot t\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, 18 \cdot \left(x \cdot \left(z \cdot y\right)\right) - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \end{array}\]
\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k
↓
\begin{array}{l}
\mathbf{if}\;t \le -10735.131202477718:\\
\;\;\;\;\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\left(\left(\sqrt[3]{\sqrt[3]{j}} \cdot \sqrt[3]{\sqrt[3]{j}}\right) \cdot \sqrt[3]{\sqrt[3]{j}}\right) \cdot \left(27 \cdot k\right)\right)\right)\right)\\

\mathbf{elif}\;t \le 3.5577987813979318 \cdot 10^{43}:\\
\;\;\;\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(z \cdot t\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(t, 18 \cdot \left(x \cdot \left(z \cdot y\right)\right) - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r777630 = x;
        double r777631 = 18.0;
        double r777632 = r777630 * r777631;
        double r777633 = y;
        double r777634 = r777632 * r777633;
        double r777635 = z;
        double r777636 = r777634 * r777635;
        double r777637 = t;
        double r777638 = r777636 * r777637;
        double r777639 = a;
        double r777640 = 4.0;
        double r777641 = r777639 * r777640;
        double r777642 = r777641 * r777637;
        double r777643 = r777638 - r777642;
        double r777644 = b;
        double r777645 = c;
        double r777646 = r777644 * r777645;
        double r777647 = r777643 + r777646;
        double r777648 = r777630 * r777640;
        double r777649 = i;
        double r777650 = r777648 * r777649;
        double r777651 = r777647 - r777650;
        double r777652 = j;
        double r777653 = 27.0;
        double r777654 = r777652 * r777653;
        double r777655 = k;
        double r777656 = r777654 * r777655;
        double r777657 = r777651 - r777656;
        return r777657;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r777658 = t;
        double r777659 = -10735.131202477718;
        bool r777660 = r777658 <= r777659;
        double r777661 = x;
        double r777662 = 18.0;
        double r777663 = r777661 * r777662;
        double r777664 = y;
        double r777665 = r777663 * r777664;
        double r777666 = z;
        double r777667 = r777665 * r777666;
        double r777668 = a;
        double r777669 = 4.0;
        double r777670 = r777668 * r777669;
        double r777671 = r777667 - r777670;
        double r777672 = b;
        double r777673 = c;
        double r777674 = r777672 * r777673;
        double r777675 = i;
        double r777676 = r777669 * r777675;
        double r777677 = j;
        double r777678 = cbrt(r777677);
        double r777679 = r777678 * r777678;
        double r777680 = cbrt(r777678);
        double r777681 = r777680 * r777680;
        double r777682 = r777681 * r777680;
        double r777683 = 27.0;
        double r777684 = k;
        double r777685 = r777683 * r777684;
        double r777686 = r777682 * r777685;
        double r777687 = r777679 * r777686;
        double r777688 = fma(r777661, r777676, r777687);
        double r777689 = r777674 - r777688;
        double r777690 = fma(r777658, r777671, r777689);
        double r777691 = 3.557798781397932e+43;
        bool r777692 = r777658 <= r777691;
        double r777693 = r777666 * r777658;
        double r777694 = r777665 * r777693;
        double r777695 = r777670 * r777658;
        double r777696 = r777694 - r777695;
        double r777697 = r777696 + r777674;
        double r777698 = r777661 * r777669;
        double r777699 = r777698 * r777675;
        double r777700 = r777697 - r777699;
        double r777701 = r777677 * r777683;
        double r777702 = r777701 * r777684;
        double r777703 = r777700 - r777702;
        double r777704 = r777666 * r777664;
        double r777705 = r777661 * r777704;
        double r777706 = r777662 * r777705;
        double r777707 = r777706 - r777670;
        double r777708 = fma(r777661, r777676, r777702);
        double r777709 = r777674 - r777708;
        double r777710 = fma(r777658, r777707, r777709);
        double r777711 = r777692 ? r777703 : r777710;
        double r777712 = r777660 ? r777690 : r777711;
        return r777712;
}

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

Bits error versus c

Bits error versus i

Bits error versus j

Bits error versus k

Target

Original5.6
Target1.6
Herbie3.5
\[\begin{array}{l} \mathbf{if}\;t \lt -1.6210815397541398 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \mathbf{elif}\;t \lt 165.680279438052224:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) + \left(c \cdot b - 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if t < -10735.131202477718

    1. Initial program 1.8

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

    2. Simplified1.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)}\]
    3. Using strategy rm

    4. Applied associate-*l*1.9

      \[\leadsto \mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \color{blue}{j \cdot \left(27 \cdot k\right)}\right)\right)\]
    5. Using strategy rm

    6. Applied add-cube-cbrt2.1

      \[\leadsto \mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \color{blue}{\left(\left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \sqrt[3]{j}\right)} \cdot \left(27 \cdot k\right)\right)\right)\]

    7. Applied associate-*l*2.1

      \[\leadsto \mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \color{blue}{\left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{j} \cdot \left(27 \cdot k\right)\right)}\right)\right)\]
    8. Using strategy rm

    9. Applied add-cube-cbrt2.1

      \[\leadsto \mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{j}} \cdot \sqrt[3]{\sqrt[3]{j}}\right) \cdot \sqrt[3]{\sqrt[3]{j}}\right)} \cdot \left(27 \cdot k\right)\right)\right)\right)\]

    if -10735.131202477718 < t < 3.557798781397932e+43

    1. Initial program 7.5

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
    2. Using strategy rm

    3. Applied associate-*l*4.2

      \[\leadsto \left(\left(\left(\color{blue}{\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(z \cdot t\right)} - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

    if 3.557798781397932e+43 < t

    1. Initial program 1.6

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

    2. Simplified1.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)}\]

    3. Taylor expanded around inf 2.0

      \[\leadsto \mathsf{fma}\left(t, \color{blue}{18 \cdot \left(x \cdot \left(z \cdot y\right)\right)} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification3.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -10735.131202477718:\\ \;\;\;\;\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(\sqrt[3]{j} \cdot \sqrt[3]{j}\right) \cdot \left(\left(\left(\sqrt[3]{\sqrt[3]{j}} \cdot \sqrt[3]{\sqrt[3]{j}}\right) \cdot \sqrt[3]{\sqrt[3]{j}}\right) \cdot \left(27 \cdot k\right)\right)\right)\right)\\ \mathbf{elif}\;t \le 3.5577987813979318 \cdot 10^{43}:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(z \cdot t\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, 18 \cdot \left(x \cdot \left(z \cdot y\right)\right) - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, E"
  :precision binary64

  :herbie-target
  (if (< t -1.6210815397541398e-69) (- (- (* (* 18 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4)) (- (* (* k j) 27) (* c b))) (if (< t 165.68027943805222) (+ (- (* (* 18 y) (* x (* z t))) (* (+ (* a t) (* i x)) 4)) (- (* c b) (* 27 (* k j)))) (- (- (* (* 18 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4)) (- (* (* k j) 27) (* c b)))))

  (- (- (+ (- (* (* (* (* x 18) y) z) t) (* (* a 4) t)) (* b c)) (* (* x 4) i)) (* (* j 27) k)))