Average Error: 5.9 → 4.7
Time: 18.4s
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -4.05720677656797186 \cdot 10^{-131} \lor \neg \left(t \le 1.1132970895009035 \cdot 10^{-169}\right):\\ \;\;\;\;\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, -4 \cdot a, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \end{array}\]
\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 -4.05720677656797186 \cdot 10^{-131} \lor \neg \left(t \le 1.1132970895009035 \cdot 10^{-169}\right):\\
\;\;\;\;\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, j \cdot \left(27 \cdot k\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(t, -4 \cdot a, 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 r2236102 = x;
        double r2236103 = 18.0;
        double r2236104 = r2236102 * r2236103;
        double r2236105 = y;
        double r2236106 = r2236104 * r2236105;
        double r2236107 = z;
        double r2236108 = r2236106 * r2236107;
        double r2236109 = t;
        double r2236110 = r2236108 * r2236109;
        double r2236111 = a;
        double r2236112 = 4.0;
        double r2236113 = r2236111 * r2236112;
        double r2236114 = r2236113 * r2236109;
        double r2236115 = r2236110 - r2236114;
        double r2236116 = b;
        double r2236117 = c;
        double r2236118 = r2236116 * r2236117;
        double r2236119 = r2236115 + r2236118;
        double r2236120 = r2236102 * r2236112;
        double r2236121 = i;
        double r2236122 = r2236120 * r2236121;
        double r2236123 = r2236119 - r2236122;
        double r2236124 = j;
        double r2236125 = 27.0;
        double r2236126 = r2236124 * r2236125;
        double r2236127 = k;
        double r2236128 = r2236126 * r2236127;
        double r2236129 = r2236123 - r2236128;
        return r2236129;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r2236130 = t;
        double r2236131 = -4.057206776567972e-131;
        bool r2236132 = r2236130 <= r2236131;
        double r2236133 = 1.1132970895009035e-169;
        bool r2236134 = r2236130 <= r2236133;
        double r2236135 = !r2236134;
        bool r2236136 = r2236132 || r2236135;
        double r2236137 = x;
        double r2236138 = 18.0;
        double r2236139 = r2236137 * r2236138;
        double r2236140 = y;
        double r2236141 = r2236139 * r2236140;
        double r2236142 = z;
        double r2236143 = r2236141 * r2236142;
        double r2236144 = a;
        double r2236145 = 4.0;
        double r2236146 = r2236144 * r2236145;
        double r2236147 = r2236143 - r2236146;
        double r2236148 = b;
        double r2236149 = c;
        double r2236150 = r2236148 * r2236149;
        double r2236151 = i;
        double r2236152 = r2236145 * r2236151;
        double r2236153 = j;
        double r2236154 = 27.0;
        double r2236155 = k;
        double r2236156 = r2236154 * r2236155;
        double r2236157 = r2236153 * r2236156;
        double r2236158 = fma(r2236137, r2236152, r2236157);
        double r2236159 = r2236150 - r2236158;
        double r2236160 = fma(r2236130, r2236147, r2236159);
        double r2236161 = -4.0;
        double r2236162 = r2236161 * r2236144;
        double r2236163 = r2236153 * r2236154;
        double r2236164 = r2236163 * r2236155;
        double r2236165 = fma(r2236137, r2236152, r2236164);
        double r2236166 = r2236150 - r2236165;
        double r2236167 = fma(r2236130, r2236162, r2236166);
        double r2236168 = r2236136 ? r2236160 : r2236167;
        return r2236168;
}

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

Original5.9
Target1.7
Herbie4.7
\[\begin{array}{l} \mathbf{if}\;t \lt -1.6210815397541398 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \mathbf{elif}\;t \lt 165.680279438052224:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) + \left(c \cdot b - 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -4.057206776567972e-131 or 1.1132970895009035e-169 < t

    1. Initial program 3.7

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

    2. Simplified3.7

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

    4. Applied associate-*l*3.7

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

    if -4.057206776567972e-131 < t < 1.1132970895009035e-169

    1. Initial program 10.4

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

    2. Simplified10.5

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

    4. Applied associate-*l*10.9

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

    5. Taylor expanded around 0 6.8

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

  4. Final simplification4.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4.05720677656797186 \cdot 10^{-131} \lor \neg \left(t \le 1.1132970895009035 \cdot 10^{-169}\right):\\ \;\;\;\;\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, -4 \cdot a, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +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)))