Average Error: 5.5 → 4.8
Time: 7.5s
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -7.13446001620463910194711542650419294006 \cdot 10^{-127}:\\ \;\;\;\;t \cdot \left(18 \cdot \left(x \cdot \left(z \cdot y\right)\right) - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{elif}\;t \le 1.302442160720236031928431474235782745489 \cdot 10^{-107}:\\ \;\;\;\;t \cdot \left(0 - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(\sqrt[3]{j \cdot \left(27 \cdot k\right)} \cdot \left(\sqrt[3]{j} \cdot \sqrt[3]{27 \cdot k}\right)\right) \cdot \sqrt[3]{j \cdot \left(27 \cdot k\right)}\right)\right)\\ \end{array}\]
\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 -7.13446001620463910194711542650419294006 \cdot 10^{-127}:\\
\;\;\;\;t \cdot \left(18 \cdot \left(x \cdot \left(z \cdot y\right)\right) - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\right)\\

\mathbf{elif}\;t \le 1.302442160720236031928431474235782745489 \cdot 10^{-107}:\\
\;\;\;\;t \cdot \left(0 - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(\sqrt[3]{j \cdot \left(27 \cdot k\right)} \cdot \left(\sqrt[3]{j} \cdot \sqrt[3]{27 \cdot k}\right)\right) \cdot \sqrt[3]{j \cdot \left(27 \cdot k\right)}\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 r804171 = x;
        double r804172 = 18.0;
        double r804173 = r804171 * r804172;
        double r804174 = y;
        double r804175 = r804173 * r804174;
        double r804176 = z;
        double r804177 = r804175 * r804176;
        double r804178 = t;
        double r804179 = r804177 * r804178;
        double r804180 = a;
        double r804181 = 4.0;
        double r804182 = r804180 * r804181;
        double r804183 = r804182 * r804178;
        double r804184 = r804179 - r804183;
        double r804185 = b;
        double r804186 = c;
        double r804187 = r804185 * r804186;
        double r804188 = r804184 + r804187;
        double r804189 = r804171 * r804181;
        double r804190 = i;
        double r804191 = r804189 * r804190;
        double r804192 = r804188 - r804191;
        double r804193 = j;
        double r804194 = 27.0;
        double r804195 = r804193 * r804194;
        double r804196 = k;
        double r804197 = r804195 * r804196;
        double r804198 = r804192 - r804197;
        return r804198;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r804199 = t;
        double r804200 = -7.134460016204639e-127;
        bool r804201 = r804199 <= r804200;
        double r804202 = 18.0;
        double r804203 = x;
        double r804204 = z;
        double r804205 = y;
        double r804206 = r804204 * r804205;
        double r804207 = r804203 * r804206;
        double r804208 = r804202 * r804207;
        double r804209 = a;
        double r804210 = 4.0;
        double r804211 = r804209 * r804210;
        double r804212 = r804208 - r804211;
        double r804213 = r804199 * r804212;
        double r804214 = b;
        double r804215 = c;
        double r804216 = r804214 * r804215;
        double r804217 = r804203 * r804210;
        double r804218 = i;
        double r804219 = r804217 * r804218;
        double r804220 = j;
        double r804221 = 27.0;
        double r804222 = k;
        double r804223 = r804221 * r804222;
        double r804224 = r804220 * r804223;
        double r804225 = r804219 + r804224;
        double r804226 = r804216 - r804225;
        double r804227 = r804213 + r804226;
        double r804228 = 1.302442160720236e-107;
        bool r804229 = r804199 <= r804228;
        double r804230 = 0.0;
        double r804231 = r804230 - r804211;
        double r804232 = r804199 * r804231;
        double r804233 = r804232 + r804226;
        double r804234 = r804203 * r804202;
        double r804235 = r804234 * r804205;
        double r804236 = r804235 * r804204;
        double r804237 = r804236 - r804211;
        double r804238 = r804199 * r804237;
        double r804239 = cbrt(r804224);
        double r804240 = cbrt(r804220);
        double r804241 = cbrt(r804223);
        double r804242 = r804240 * r804241;
        double r804243 = r804239 * r804242;
        double r804244 = r804243 * r804239;
        double r804245 = r804219 + r804244;
        double r804246 = r804216 - r804245;
        double r804247 = r804238 + r804246;
        double r804248 = r804229 ? r804233 : r804247;
        double r804249 = r804201 ? r804227 : r804248;
        return r804249;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original5.5
Target1.6
Herbie4.8
\[\begin{array}{l} \mathbf{if}\;t \lt -1.62108153975413982700795070153457058168 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \mathbf{elif}\;t \lt 165.6802794380522243500308832153677940369:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) + \left(c \cdot b - 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if t < -7.134460016204639e-127

    1. Initial program 3.1

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

    2. Simplified3.1

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

    4. Applied associate-*l*3.1

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

    5. Taylor expanded around inf 3.9

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

    if -7.134460016204639e-127 < t < 1.302442160720236e-107

    1. Initial program 9.1

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

    2. Simplified9.1

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

    4. Applied associate-*l*9.2

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

    5. Taylor expanded around 0 6.6

      \[\leadsto t \cdot \left(\color{blue}{0} - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\right)\]

    if 1.302442160720236e-107 < t

    1. Initial program 3.0

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

    2. Simplified3.0

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

    4. Applied associate-*l*3.0

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

    6. Applied add-cube-cbrt3.2

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

    8. Applied cbrt-prod3.2

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

  4. Final simplification4.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -7.13446001620463910194711542650419294006 \cdot 10^{-127}:\\ \;\;\;\;t \cdot \left(18 \cdot \left(x \cdot \left(z \cdot y\right)\right) - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{elif}\;t \le 1.302442160720236031928431474235782745489 \cdot 10^{-107}:\\ \;\;\;\;t \cdot \left(0 - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(\sqrt[3]{j \cdot \left(27 \cdot k\right)} \cdot \left(\sqrt[3]{j} \cdot \sqrt[3]{27 \cdot k}\right)\right) \cdot \sqrt[3]{j \cdot \left(27 \cdot k\right)}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 
(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)))