Average Error: 5.5 → 4.9
Time: 17.8s
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.4272481818337534 \cdot 10^{181}:\\ \;\;\;\;\mathsf{fma}\left(t, \left(x \cdot 18\right) \cdot \left(y \cdot z\right) - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{elif}\;x \le 1.2900037542469239 \cdot 10^{120}:\\ \;\;\;\;\mathsf{fma}\left(t, \left(x \cdot \left(18 \cdot y\right)\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, 0 - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \end{array}\]
\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}\;x \le -2.4272481818337534 \cdot 10^{181}:\\
\;\;\;\;\mathsf{fma}\left(t, \left(x \cdot 18\right) \cdot \left(y \cdot z\right) - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\

\mathbf{elif}\;x \le 1.2900037542469239 \cdot 10^{120}:\\
\;\;\;\;\mathsf{fma}\left(t, \left(x \cdot \left(18 \cdot y\right)\right) \cdot z - a \cdot 4, 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, 0 - 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 r719029 = x;
        double r719030 = 18.0;
        double r719031 = r719029 * r719030;
        double r719032 = y;
        double r719033 = r719031 * r719032;
        double r719034 = z;
        double r719035 = r719033 * r719034;
        double r719036 = t;
        double r719037 = r719035 * r719036;
        double r719038 = a;
        double r719039 = 4.0;
        double r719040 = r719038 * r719039;
        double r719041 = r719040 * r719036;
        double r719042 = r719037 - r719041;
        double r719043 = b;
        double r719044 = c;
        double r719045 = r719043 * r719044;
        double r719046 = r719042 + r719045;
        double r719047 = r719029 * r719039;
        double r719048 = i;
        double r719049 = r719047 * r719048;
        double r719050 = r719046 - r719049;
        double r719051 = j;
        double r719052 = 27.0;
        double r719053 = r719051 * r719052;
        double r719054 = k;
        double r719055 = r719053 * r719054;
        double r719056 = r719050 - r719055;
        return r719056;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r719057 = x;
        double r719058 = -2.4272481818337534e+181;
        bool r719059 = r719057 <= r719058;
        double r719060 = t;
        double r719061 = 18.0;
        double r719062 = r719057 * r719061;
        double r719063 = y;
        double r719064 = z;
        double r719065 = r719063 * r719064;
        double r719066 = r719062 * r719065;
        double r719067 = a;
        double r719068 = 4.0;
        double r719069 = r719067 * r719068;
        double r719070 = r719066 - r719069;
        double r719071 = b;
        double r719072 = c;
        double r719073 = r719071 * r719072;
        double r719074 = i;
        double r719075 = r719068 * r719074;
        double r719076 = j;
        double r719077 = 27.0;
        double r719078 = r719076 * r719077;
        double r719079 = k;
        double r719080 = r719078 * r719079;
        double r719081 = fma(r719057, r719075, r719080);
        double r719082 = r719073 - r719081;
        double r719083 = fma(r719060, r719070, r719082);
        double r719084 = 1.2900037542469239e+120;
        bool r719085 = r719057 <= r719084;
        double r719086 = r719061 * r719063;
        double r719087 = r719057 * r719086;
        double r719088 = r719087 * r719064;
        double r719089 = r719088 - r719069;
        double r719090 = r719077 * r719079;
        double r719091 = r719076 * r719090;
        double r719092 = fma(r719057, r719075, r719091);
        double r719093 = r719073 - r719092;
        double r719094 = fma(r719060, r719089, r719093);
        double r719095 = 0.0;
        double r719096 = r719095 - r719069;
        double r719097 = fma(r719060, r719096, r719082);
        double r719098 = r719085 ? r719094 : r719097;
        double r719099 = r719059 ? r719083 : r719098;
        return r719099;
}

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

Original5.5
Target1.5
Herbie4.9
\[\begin{array}{l} \mathbf{if}\;t \lt -1.6210815397541398 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \mathbf{elif}\;t \lt 165.680279438052224:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) + \left(c \cdot b - 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -2.4272481818337534e+181

    1. Initial program 18.7

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

    2. Simplified18.7

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

    4. Applied associate-*l*9.4

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

    if -2.4272481818337534e+181 < x < 1.2900037542469239e+120

    1. Initial program 3.4

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

    2. Simplified3.5

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

    4. Applied associate-*l*3.5

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

    6. Applied associate-*l*3.5

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

    if 1.2900037542469239e+120 < x

    1. Initial program 18.2

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

    2. Simplified18.2

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

    3. Taylor expanded around 0 15.4

      \[\leadsto \mathsf{fma}\left(t, \color{blue}{0} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification4.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.4272481818337534 \cdot 10^{181}:\\ \;\;\;\;\mathsf{fma}\left(t, \left(x \cdot 18\right) \cdot \left(y \cdot z\right) - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{elif}\;x \le 1.2900037542469239 \cdot 10^{120}:\\ \;\;\;\;\mathsf{fma}\left(t, \left(x \cdot \left(18 \cdot y\right)\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, 0 - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +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)))