Average Error: 5.6 → 3.7
Time: 13.7s
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -943409284927925125000 \lor \neg \left(z \le 2.0209759102865675 \cdot 10^{25}\right):\\ \;\;\;\;t \cdot \left(\left(x \cdot \left(18 \cdot y\right)\right) \cdot z - a \cdot 4\right) + \left(b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, {\left(18 \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \end{array}\]
\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, {\left(18 \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)}^{1} - 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 r710165 = x;
        double r710166 = 18.0;
        double r710167 = r710165 * r710166;
        double r710168 = y;
        double r710169 = r710167 * r710168;
        double r710170 = z;
        double r710171 = r710169 * r710170;
        double r710172 = t;
        double r710173 = r710171 * r710172;
        double r710174 = a;
        double r710175 = 4.0;
        double r710176 = r710174 * r710175;
        double r710177 = r710176 * r710172;
        double r710178 = r710173 - r710177;
        double r710179 = b;
        double r710180 = c;
        double r710181 = r710179 * r710180;
        double r710182 = r710178 + r710181;
        double r710183 = r710165 * r710175;
        double r710184 = i;
        double r710185 = r710183 * r710184;
        double r710186 = r710182 - r710185;
        double r710187 = j;
        double r710188 = 27.0;
        double r710189 = r710187 * r710188;
        double r710190 = k;
        double r710191 = r710189 * r710190;
        double r710192 = r710186 - r710191;
        return r710192;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r710193 = z;
        double r710194 = -9.434092849279251e+20;
        bool r710195 = r710193 <= r710194;
        double r710196 = 2.0209759102865675e+25;
        bool r710197 = r710193 <= r710196;
        double r710198 = !r710197;
        bool r710199 = r710195 || r710198;
        double r710200 = t;
        double r710201 = x;
        double r710202 = 18.0;
        double r710203 = y;
        double r710204 = r710202 * r710203;
        double r710205 = r710201 * r710204;
        double r710206 = r710205 * r710193;
        double r710207 = a;
        double r710208 = 4.0;
        double r710209 = r710207 * r710208;
        double r710210 = r710206 - r710209;
        double r710211 = r710200 * r710210;
        double r710212 = b;
        double r710213 = c;
        double r710214 = r710212 * r710213;
        double r710215 = i;
        double r710216 = r710208 * r710215;
        double r710217 = j;
        double r710218 = 27.0;
        double r710219 = k;
        double r710220 = r710218 * r710219;
        double r710221 = r710217 * r710220;
        double r710222 = fma(r710201, r710216, r710221);
        double r710223 = r710214 - r710222;
        double r710224 = r710211 + r710223;
        double r710225 = r710193 * r710203;
        double r710226 = r710201 * r710225;
        double r710227 = r710202 * r710226;
        double r710228 = 1.0;
        double r710229 = pow(r710227, r710228);
        double r710230 = r710229 - r710209;
        double r710231 = r710217 * r710218;
        double r710232 = r710231 * r710219;
        double r710233 = fma(r710201, r710216, r710232);
        double r710234 = r710214 - r710233;
        double r710235 = fma(r710200, r710230, r710234);
        double r710236 = r710199 ? r710224 : r710235;
        return r710236;
}

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
Herbie3.7
\[\begin{array}{l} \mathbf{if}\;t \lt -1.6210815397541398 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \mathbf{elif}\;t \lt 165.680279438052224:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) + \left(c \cdot b - 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if 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. Using strategy rm

    4. Applied pow14.3

      \[\leadsto \mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot \color{blue}{{z}^{1}} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\]

    5. Applied pow14.3

      \[\leadsto \mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot \color{blue}{{y}^{1}}\right) \cdot {z}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\]

    6. Applied pow14.3

      \[\leadsto \mathsf{fma}\left(t, \left(\left(x \cdot \color{blue}{{18}^{1}}\right) \cdot {y}^{1}\right) \cdot {z}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\]

    7. Applied pow14.3

      \[\leadsto \mathsf{fma}\left(t, \left(\left(\color{blue}{{x}^{1}} \cdot {18}^{1}\right) \cdot {y}^{1}\right) \cdot {z}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\]

    8. Applied pow-prod-down4.3

      \[\leadsto \mathsf{fma}\left(t, \left(\color{blue}{{\left(x \cdot 18\right)}^{1}} \cdot {y}^{1}\right) \cdot {z}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\]

    9. Applied pow-prod-down4.3

      \[\leadsto \mathsf{fma}\left(t, \color{blue}{{\left(\left(x \cdot 18\right) \cdot y\right)}^{1}} \cdot {z}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\]

    10. Applied pow-prod-down4.3

      \[\leadsto \mathsf{fma}\left(t, \color{blue}{{\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right)}^{1}} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\]

    11. Simplified1.2

      \[\leadsto \mathsf{fma}\left(t, {\color{blue}{\left(18 \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)}}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -943409284927925125000 \lor \neg \left(z \le 2.0209759102865675 \cdot 10^{25}\right):\\ \;\;\;\;t \cdot \left(\left(x \cdot \left(18 \cdot y\right)\right) \cdot z - a \cdot 4\right) + \left(b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, {\left(18 \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)}^{1} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \end{array}\]

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, 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)))