Average Error: 5.6 → 2.7
Time: 1.2m
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -7702228306921125838848:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(z \cdot \left(\left(y \cdot x\right) \cdot 18\right)\right) \cdot t - t \cdot \left(a \cdot 4\right)\right)\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{elif}\;t \le 1.806883127258244205731970851622029403045 \cdot 10^{-164}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(\left(18 \cdot t\right) \cdot x\right), z, b \cdot c - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, i \cdot x\right), \left(j \cdot k\right) \cdot 27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) - t \cdot \left(a \cdot 4\right)\right)\right) - \left(x \cdot 4\right) \cdot i\right) - k \cdot \left(j \cdot 27\right)\\ \end{array}\]
\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 -7702228306921125838848:\\
\;\;\;\;\left(\left(b \cdot c + \left(\left(z \cdot \left(\left(y \cdot x\right) \cdot 18\right)\right) \cdot t - t \cdot \left(a \cdot 4\right)\right)\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\

\mathbf{elif}\;t \le 1.806883127258244205731970851622029403045 \cdot 10^{-164}:\\
\;\;\;\;\mathsf{fma}\left(y \cdot \left(\left(18 \cdot t\right) \cdot x\right), z, b \cdot c - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, i \cdot x\right), \left(j \cdot k\right) \cdot 27\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\left(b \cdot c + \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) - t \cdot \left(a \cdot 4\right)\right)\right) - \left(x \cdot 4\right) \cdot i\right) - k \cdot \left(j \cdot 27\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 r34697969 = x;
        double r34697970 = 18.0;
        double r34697971 = r34697969 * r34697970;
        double r34697972 = y;
        double r34697973 = r34697971 * r34697972;
        double r34697974 = z;
        double r34697975 = r34697973 * r34697974;
        double r34697976 = t;
        double r34697977 = r34697975 * r34697976;
        double r34697978 = a;
        double r34697979 = 4.0;
        double r34697980 = r34697978 * r34697979;
        double r34697981 = r34697980 * r34697976;
        double r34697982 = r34697977 - r34697981;
        double r34697983 = b;
        double r34697984 = c;
        double r34697985 = r34697983 * r34697984;
        double r34697986 = r34697982 + r34697985;
        double r34697987 = r34697969 * r34697979;
        double r34697988 = i;
        double r34697989 = r34697987 * r34697988;
        double r34697990 = r34697986 - r34697989;
        double r34697991 = j;
        double r34697992 = 27.0;
        double r34697993 = r34697991 * r34697992;
        double r34697994 = k;
        double r34697995 = r34697993 * r34697994;
        double r34697996 = r34697990 - r34697995;
        return r34697996;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r34697997 = t;
        double r34697998 = -7.702228306921126e+21;
        bool r34697999 = r34697997 <= r34697998;
        double r34698000 = b;
        double r34698001 = c;
        double r34698002 = r34698000 * r34698001;
        double r34698003 = z;
        double r34698004 = y;
        double r34698005 = x;
        double r34698006 = r34698004 * r34698005;
        double r34698007 = 18.0;
        double r34698008 = r34698006 * r34698007;
        double r34698009 = r34698003 * r34698008;
        double r34698010 = r34698009 * r34697997;
        double r34698011 = a;
        double r34698012 = 4.0;
        double r34698013 = r34698011 * r34698012;
        double r34698014 = r34697997 * r34698013;
        double r34698015 = r34698010 - r34698014;
        double r34698016 = r34698002 + r34698015;
        double r34698017 = r34698005 * r34698012;
        double r34698018 = i;
        double r34698019 = r34698017 * r34698018;
        double r34698020 = r34698016 - r34698019;
        double r34698021 = j;
        double r34698022 = 27.0;
        double r34698023 = k;
        double r34698024 = r34698022 * r34698023;
        double r34698025 = r34698021 * r34698024;
        double r34698026 = r34698020 - r34698025;
        double r34698027 = 1.8068831272582442e-164;
        bool r34698028 = r34697997 <= r34698027;
        double r34698029 = r34698007 * r34697997;
        double r34698030 = r34698029 * r34698005;
        double r34698031 = r34698004 * r34698030;
        double r34698032 = r34698018 * r34698005;
        double r34698033 = fma(r34697997, r34698011, r34698032);
        double r34698034 = r34698021 * r34698023;
        double r34698035 = r34698034 * r34698022;
        double r34698036 = fma(r34698012, r34698033, r34698035);
        double r34698037 = r34698002 - r34698036;
        double r34698038 = fma(r34698031, r34698003, r34698037);
        double r34698039 = r34698004 * r34698003;
        double r34698040 = r34698005 * r34698039;
        double r34698041 = r34697997 * r34698040;
        double r34698042 = r34698007 * r34698041;
        double r34698043 = r34698042 - r34698014;
        double r34698044 = r34698002 + r34698043;
        double r34698045 = r34698044 - r34698019;
        double r34698046 = r34698021 * r34698022;
        double r34698047 = r34698023 * r34698046;
        double r34698048 = r34698045 - r34698047;
        double r34698049 = r34698028 ? r34698038 : r34698048;
        double r34698050 = r34697999 ? r34698026 : r34698049;
        return r34698050;
}

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
Target1.7
Herbie2.7
\[\begin{array}{l} \mathbf{if}\;t \lt -1.62108153975413982700795070153457058168 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \mathbf{elif}\;t \lt 165.6802794380522243500308832153677940369:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) + \left(c \cdot b - 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if t < -7.702228306921126e+21

    1. Initial program 2.1

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

    2. Taylor expanded around 0 2.1

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

    4. Applied associate-*l*2.1

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

    if -7.702228306921126e+21 < t < 1.8068831272582442e-164

    1. Initial program 8.2

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

    2. Simplified4.2

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

    4. Applied associate-*l*4.2

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

    6. Applied associate-*l*1.6

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

    if 1.8068831272582442e-164 < t

    1. Initial program 3.8

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

    2. Taylor expanded around inf 4.4

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

  4. Final simplification2.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -7702228306921125838848:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(z \cdot \left(\left(y \cdot x\right) \cdot 18\right)\right) \cdot t - t \cdot \left(a \cdot 4\right)\right)\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{elif}\;t \le 1.806883127258244205731970851622029403045 \cdot 10^{-164}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(\left(18 \cdot t\right) \cdot x\right), z, b \cdot c - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, i \cdot x\right), \left(j \cdot k\right) \cdot 27\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(18 \cdot \left(t \cdot \left(x \cdot \left(y \cdot z\right)\right)\right) - t \cdot \left(a \cdot 4\right)\right)\right) - \left(x \cdot 4\right) \cdot i\right) - k \cdot \left(j \cdot 27\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +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"

  :herbie-target
  (if (< t -1.6210815397541398e-69) (- (- (* (* 18.0 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4.0)) (- (* (* k j) 27.0) (* c b))) (if (< t 165.68027943805222) (+ (- (* (* 18.0 y) (* x (* z t))) (* (+ (* a t) (* i x)) 4.0)) (- (* c b) (* 27.0 (* k j)))) (- (- (* (* 18.0 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4.0)) (- (* (* k j) 27.0) (* c b)))))

  (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)))