Average Error: 5.5 → 4.5
Time: 18.1s
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.16781444686311266 \cdot 10^{-150} \lor \neg \left(t \le 4.1450304107497156 \cdot 10^{-137}\right):\\ \;\;\;\;\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(\left(j \cdot 27\right) \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)\right) \cdot \sqrt[3]{k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, 0 - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \end{array}\]
\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}\;t \le -1.16781444686311266 \cdot 10^{-150} \lor \neg \left(t \le 4.1450304107497156 \cdot 10^{-137}\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, \left(\left(j \cdot 27\right) \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)\right) \cdot \sqrt[3]{k}\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 r1025017 = x;
        double r1025018 = 18.0;
        double r1025019 = r1025017 * r1025018;
        double r1025020 = y;
        double r1025021 = r1025019 * r1025020;
        double r1025022 = z;
        double r1025023 = r1025021 * r1025022;
        double r1025024 = t;
        double r1025025 = r1025023 * r1025024;
        double r1025026 = a;
        double r1025027 = 4.0;
        double r1025028 = r1025026 * r1025027;
        double r1025029 = r1025028 * r1025024;
        double r1025030 = r1025025 - r1025029;
        double r1025031 = b;
        double r1025032 = c;
        double r1025033 = r1025031 * r1025032;
        double r1025034 = r1025030 + r1025033;
        double r1025035 = r1025017 * r1025027;
        double r1025036 = i;
        double r1025037 = r1025035 * r1025036;
        double r1025038 = r1025034 - r1025037;
        double r1025039 = j;
        double r1025040 = 27.0;
        double r1025041 = r1025039 * r1025040;
        double r1025042 = k;
        double r1025043 = r1025041 * r1025042;
        double r1025044 = r1025038 - r1025043;
        return r1025044;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r1025045 = t;
        double r1025046 = -1.1678144468631127e-150;
        bool r1025047 = r1025045 <= r1025046;
        double r1025048 = 4.1450304107497156e-137;
        bool r1025049 = r1025045 <= r1025048;
        double r1025050 = !r1025049;
        bool r1025051 = r1025047 || r1025050;
        double r1025052 = x;
        double r1025053 = 18.0;
        double r1025054 = r1025052 * r1025053;
        double r1025055 = y;
        double r1025056 = r1025054 * r1025055;
        double r1025057 = z;
        double r1025058 = r1025056 * r1025057;
        double r1025059 = a;
        double r1025060 = 4.0;
        double r1025061 = r1025059 * r1025060;
        double r1025062 = r1025058 - r1025061;
        double r1025063 = b;
        double r1025064 = c;
        double r1025065 = r1025063 * r1025064;
        double r1025066 = i;
        double r1025067 = r1025060 * r1025066;
        double r1025068 = j;
        double r1025069 = 27.0;
        double r1025070 = r1025068 * r1025069;
        double r1025071 = k;
        double r1025072 = cbrt(r1025071);
        double r1025073 = r1025072 * r1025072;
        double r1025074 = r1025070 * r1025073;
        double r1025075 = r1025074 * r1025072;
        double r1025076 = fma(r1025052, r1025067, r1025075);
        double r1025077 = r1025065 - r1025076;
        double r1025078 = fma(r1025045, r1025062, r1025077);
        double r1025079 = 0.0;
        double r1025080 = r1025079 - r1025061;
        double r1025081 = r1025070 * r1025071;
        double r1025082 = fma(r1025052, r1025067, r1025081);
        double r1025083 = r1025065 - r1025082;
        double r1025084 = fma(r1025045, r1025080, r1025083);
        double r1025085 = r1025051 ? r1025078 : r1025084;
        return r1025085;
}

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
Target1.3
Herbie4.5
\[\begin{array}{l} \mathbf{if}\;t \lt -1.6210815397541398 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \mathbf{elif}\;t \lt 165.680279438052224:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) + \left(c \cdot b - 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -1.1678144468631127e-150 or 4.1450304107497156e-137 < t

    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. Simplified3.4

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

    4. Applied add-cube-cbrt3.6

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

    5. Applied associate-*r*3.6

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

    if -1.1678144468631127e-150 < t < 4.1450304107497156e-137

    1. Initial program 9.6

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

    2. Simplified9.6

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

    3. Taylor expanded around 0 6.2

      \[\leadsto \mathsf{fma}\left(t, \color{blue}{0} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification4.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.16781444686311266 \cdot 10^{-150} \lor \neg \left(t \le 4.1450304107497156 \cdot 10^{-137}\right):\\ \;\;\;\;\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(\left(j \cdot 27\right) \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)\right) \cdot \sqrt[3]{k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, 0 - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +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)))