Average Error: 5.6 → 6.7
Time: 33.1s
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(j \cdot 27\right) \cdot k \le -2.954442886009252531198288738982951568249 \cdot 10^{-102}:\\ \;\;\;\;\left(\left(-a \cdot 4\right) \cdot t + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \sqrt{27} \cdot \left(\sqrt{27} \cdot \left(k \cdot j\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\\ \end{array}\]
\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}\;\left(j \cdot 27\right) \cdot k \le -2.954442886009252531198288738982951568249 \cdot 10^{-102}:\\
\;\;\;\;\left(\left(-a \cdot 4\right) \cdot t + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \sqrt{27} \cdot \left(\sqrt{27} \cdot \left(k \cdot j\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \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 r52911 = x;
        double r52912 = 18.0;
        double r52913 = r52911 * r52912;
        double r52914 = y;
        double r52915 = r52913 * r52914;
        double r52916 = z;
        double r52917 = r52915 * r52916;
        double r52918 = t;
        double r52919 = r52917 * r52918;
        double r52920 = a;
        double r52921 = 4.0;
        double r52922 = r52920 * r52921;
        double r52923 = r52922 * r52918;
        double r52924 = r52919 - r52923;
        double r52925 = b;
        double r52926 = c;
        double r52927 = r52925 * r52926;
        double r52928 = r52924 + r52927;
        double r52929 = r52911 * r52921;
        double r52930 = i;
        double r52931 = r52929 * r52930;
        double r52932 = r52928 - r52931;
        double r52933 = j;
        double r52934 = 27.0;
        double r52935 = r52933 * r52934;
        double r52936 = k;
        double r52937 = r52935 * r52936;
        double r52938 = r52932 - r52937;
        return r52938;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r52939 = j;
        double r52940 = 27.0;
        double r52941 = r52939 * r52940;
        double r52942 = k;
        double r52943 = r52941 * r52942;
        double r52944 = -2.9544428860092525e-102;
        bool r52945 = r52943 <= r52944;
        double r52946 = a;
        double r52947 = 4.0;
        double r52948 = r52946 * r52947;
        double r52949 = -r52948;
        double r52950 = t;
        double r52951 = r52949 * r52950;
        double r52952 = b;
        double r52953 = c;
        double r52954 = r52952 * r52953;
        double r52955 = r52951 + r52954;
        double r52956 = x;
        double r52957 = r52956 * r52947;
        double r52958 = i;
        double r52959 = r52957 * r52958;
        double r52960 = sqrt(r52940);
        double r52961 = r52942 * r52939;
        double r52962 = r52960 * r52961;
        double r52963 = r52960 * r52962;
        double r52964 = r52959 + r52963;
        double r52965 = r52955 - r52964;
        double r52966 = 18.0;
        double r52967 = r52956 * r52966;
        double r52968 = y;
        double r52969 = r52967 * r52968;
        double r52970 = z;
        double r52971 = r52969 * r52970;
        double r52972 = r52971 - r52948;
        double r52973 = r52950 * r52972;
        double r52974 = r52973 + r52954;
        double r52975 = r52940 * r52942;
        double r52976 = r52939 * r52975;
        double r52977 = r52959 + r52976;
        double r52978 = r52974 - r52977;
        double r52979 = r52945 ? r52965 : r52978;
        return r52979;
}

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

Original5.6
Target1.6
Herbie6.7
\[\begin{array}{l} \mathbf{if}\;t \lt -1.62108153975413982700795070153457058168 \cdot 10^{-69}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \mathbf{elif}\;t \lt 165.6802794380522243500308832153677940369:\\ \;\;\;\;\left(\left(18 \cdot y\right) \cdot \left(x \cdot \left(z \cdot t\right)\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) + \left(c \cdot b - 27 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(18 \cdot t\right) \cdot \left(\left(x \cdot y\right) \cdot z\right) - \left(a \cdot t + i \cdot x\right) \cdot 4\right) - \left(\left(k \cdot j\right) \cdot 27 - c \cdot b\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* (* j 27.0) k) < -2.9544428860092525e-102

    1. Initial program 5.1

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

    2. Simplified5.1

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

    4. Applied associate-*l*5.1

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

    5. Taylor expanded around 0 5.0

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

    7. Applied add-sqr-sqrt5.0

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

    8. Applied associate-*l*5.1

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

    9. Taylor expanded around 0 8.7

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

    if -2.9544428860092525e-102 < (* (* j 27.0) k)

    1. Initial program 5.8

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

    2. Simplified5.8

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

    4. Applied associate-*l*5.8

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

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(j \cdot 27\right) \cdot k \le -2.954442886009252531198288738982951568249 \cdot 10^{-102}:\\ \;\;\;\;\left(\left(-a \cdot 4\right) \cdot t + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + \sqrt{27} \cdot \left(\sqrt{27} \cdot \left(k \cdot j\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right) + b \cdot c\right) - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019315 
(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 165.680279438052224) (+ (- (* (* 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)))