Average Error: 5.6 → 3.7
Time: 13.3s
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -943409284927925125000 \lor \neg \left(z \le 2.0209759102865675 \cdot 10^{25}\right):\\ \;\;\;\;t \cdot \left(\left(x \cdot \left(18 \cdot y\right)\right) \cdot z - a \cdot 4\right) + \left(b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, 18 \cdot \left(x \cdot \left(z \cdot y\right)\right) - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \end{array}\]
\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}\;z \le -943409284927925125000 \lor \neg \left(z \le 2.0209759102865675 \cdot 10^{25}\right):\\
\;\;\;\;t \cdot \left(\left(x \cdot \left(18 \cdot y\right)\right) \cdot z - a \cdot 4\right) + \left(b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\

\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 r112010 = x;
        double r112011 = 18.0;
        double r112012 = r112010 * r112011;
        double r112013 = y;
        double r112014 = r112012 * r112013;
        double r112015 = z;
        double r112016 = r112014 * r112015;
        double r112017 = t;
        double r112018 = r112016 * r112017;
        double r112019 = a;
        double r112020 = 4.0;
        double r112021 = r112019 * r112020;
        double r112022 = r112021 * r112017;
        double r112023 = r112018 - r112022;
        double r112024 = b;
        double r112025 = c;
        double r112026 = r112024 * r112025;
        double r112027 = r112023 + r112026;
        double r112028 = r112010 * r112020;
        double r112029 = i;
        double r112030 = r112028 * r112029;
        double r112031 = r112027 - r112030;
        double r112032 = j;
        double r112033 = 27.0;
        double r112034 = r112032 * r112033;
        double r112035 = k;
        double r112036 = r112034 * r112035;
        double r112037 = r112031 - r112036;
        return r112037;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r112038 = z;
        double r112039 = -9.434092849279251e+20;
        bool r112040 = r112038 <= r112039;
        double r112041 = 2.0209759102865675e+25;
        bool r112042 = r112038 <= r112041;
        double r112043 = !r112042;
        bool r112044 = r112040 || r112043;
        double r112045 = t;
        double r112046 = x;
        double r112047 = 18.0;
        double r112048 = y;
        double r112049 = r112047 * r112048;
        double r112050 = r112046 * r112049;
        double r112051 = r112050 * r112038;
        double r112052 = a;
        double r112053 = 4.0;
        double r112054 = r112052 * r112053;
        double r112055 = r112051 - r112054;
        double r112056 = r112045 * r112055;
        double r112057 = b;
        double r112058 = c;
        double r112059 = r112057 * r112058;
        double r112060 = i;
        double r112061 = r112053 * r112060;
        double r112062 = j;
        double r112063 = 27.0;
        double r112064 = k;
        double r112065 = r112063 * r112064;
        double r112066 = r112062 * r112065;
        double r112067 = fma(r112046, r112061, r112066);
        double r112068 = r112059 - r112067;
        double r112069 = r112056 + r112068;
        double r112070 = r112038 * r112048;
        double r112071 = r112046 * r112070;
        double r112072 = r112047 * r112071;
        double r112073 = r112072 - r112054;
        double r112074 = r112062 * r112063;
        double r112075 = r112074 * r112064;
        double r112076 = fma(r112046, r112061, r112075);
        double r112077 = r112059 - r112076;
        double r112078 = fma(r112045, r112073, r112077);
        double r112079 = r112044 ? r112069 : r112078;
        return r112079;
}

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

Derivation

  1. Split input into 2 regimes

  2. if z < -9.434092849279251e+20 or 2.0209759102865675e+25 < z

    1. Initial program 7.4

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

    2. Simplified7.4

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

    4. Applied associate-*l*7.4

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

    6. Applied fma-udef7.4

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

    8. Applied associate-*l*7.4

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

    if -9.434092849279251e+20 < z < 2.0209759102865675e+25

    1. Initial program 4.3

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

    2. Simplified4.3

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

    3. Taylor expanded around inf 1.2

      \[\leadsto \mathsf{fma}\left(t, \color{blue}{18 \cdot \left(x \cdot \left(z \cdot y\right)\right)} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -943409284927925125000 \lor \neg \left(z \le 2.0209759102865675 \cdot 10^{25}\right):\\ \;\;\;\;t \cdot \left(\left(x \cdot \left(18 \cdot y\right)\right) \cdot z - a \cdot 4\right) + \left(b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, 18 \cdot \left(x \cdot \left(z \cdot y\right)\right) - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1"
  :precision binary64
  (- (- (+ (- (* (* (* (* x 18) y) z) t) (* (* a 4) t)) (* b c)) (* (* x 4) i)) (* (* j 27) k)))