Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, E

Average Error: 5.8 → 1.1

Time: 27.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i = -\infty \lor \neg \left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i \le 4.448119603122321026114316643374029554495 \cdot 10^{305}\right):\\ \;\;\;\;\left(\left(c \cdot b + \left(\left(y \cdot \left(\left(t \cdot x\right) \cdot z\right)\right) \cdot 18 - \left(a \cdot 4\right) \cdot t\right)\right) - \left(x \cdot 4\right) \cdot i\right) - k \cdot \left(j \cdot 27\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(\sqrt[3]{k \cdot 27} \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{\left(k \cdot 27\right) \cdot j} \cdot \sqrt[3]{\left(k \cdot 27\right) \cdot j}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r532953 = x;
        double r532954 = 18.0;
        double r532955 = r532953 * r532954;
        double r532956 = y;
        double r532957 = r532955 * r532956;
        double r532958 = z;
        double r532959 = r532957 * r532958;
        double r532960 = t;
        double r532961 = r532959 * r532960;
        double r532962 = a;
        double r532963 = 4.0;
        double r532964 = r532962 * r532963;
        double r532965 = r532964 * r532960;
        double r532966 = r532961 - r532965;
        double r532967 = b;
        double r532968 = c;
        double r532969 = r532967 * r532968;
        double r532970 = r532966 + r532969;
        double r532971 = r532953 * r532963;
        double r532972 = i;
        double r532973 = r532971 * r532972;
        double r532974 = r532970 - r532973;
        double r532975 = j;
        double r532976 = 27.0;
        double r532977 = r532975 * r532976;
        double r532978 = k;
        double r532979 = r532977 * r532978;
        double r532980 = r532974 - r532979;
        return r532980;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r532981 = t;
        double r532982 = x;
        double r532983 = 18.0;
        double r532984 = r532982 * r532983;
        double r532985 = y;
        double r532986 = r532984 * r532985;
        double r532987 = z;
        double r532988 = r532986 * r532987;
        double r532989 = r532981 * r532988;
        double r532990 = a;
        double r532991 = 4.0;
        double r532992 = r532990 * r532991;
        double r532993 = r532992 * r532981;
        double r532994 = r532989 - r532993;
        double r532995 = c;
        double r532996 = b;
        double r532997 = r532995 * r532996;
        double r532998 = r532994 + r532997;
        double r532999 = r532982 * r532991;
        double r533000 = i;
        double r533001 = r532999 * r533000;
        double r533002 = r532998 - r533001;
        double r533003 = -inf.0;
        bool r533004 = r533002 <= r533003;
        double r533005 = 4.448119603122321e+305;
        bool r533006 = r533002 <= r533005;
        double r533007 = !r533006;
        bool r533008 = r533004 || r533007;
        double r533009 = r532981 * r532982;
        double r533010 = r533009 * r532987;
        double r533011 = r532985 * r533010;
        double r533012 = r533011 * r532983;
        double r533013 = r533012 - r532993;
        double r533014 = r532997 + r533013;
        double r533015 = r533014 - r533001;
        double r533016 = k;
        double r533017 = j;
        double r533018 = 27.0;
        double r533019 = r533017 * r533018;
        double r533020 = r533016 * r533019;
        double r533021 = r533015 - r533020;
        double r533022 = r533016 * r533018;
        double r533023 = cbrt(r533022);
        double r533024 = cbrt(r533017);
        double r533025 = r533023 * r533024;
        double r533026 = r533022 * r533017;
        double r533027 = cbrt(r533026);
        double r533028 = r533027 * r533027;
        double r533029 = r533025 * r533028;
        double r533030 = r533002 - r533029;
        double r533031 = r533008 ? r533021 : r533030;
        return r533031;
}

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

Original	5.8
Target	1.4
Herbie	1.1

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

Derivation

1. Split input into 2 regimes

2. if (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < -inf.0 or 4.448119603122321e+305 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i))

   1. Initial program 61.8

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

   2. Taylor expanded around inf 40.6

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

   3. Simplified 6.2

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

   if -inf.0 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < 4.448119603122321e+305

   1. Initial program 0.3

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

   3. Applied add-cube-cbrt 0.6

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

   4. Simplified 0.7

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

   5. Simplified 0.6

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

   7. Applied cbrt-prod 0.6

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

   8. Simplified 0.6

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

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i = -\infty \lor \neg \left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i \le 4.448119603122321026114316643374029554495 \cdot 10^{305}\right):\\ \;\;\;\;\left(\left(c \cdot b + \left(\left(y \cdot \left(\left(t \cdot x\right) \cdot z\right)\right) \cdot 18 - \left(a \cdot 4\right) \cdot t\right)\right) - \left(x \cdot 4\right) \cdot i\right) - k \cdot \left(j \cdot 27\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(\sqrt[3]{k \cdot 27} \cdot \sqrt[3]{j}\right) \cdot \left(\sqrt[3]{\left(k \cdot 27\right) \cdot j} \cdot \sqrt[3]{\left(k \cdot 27\right) \cdot j}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, E"

  :herbie-target
  (if (< t -1.6210815397541398e-69) (- (- (* (* 18.0 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4.0)) (- (* (* k j) 27.0) (* c b))) (if (< t 165.68027943805222) (+ (- (* (* 18.0 y) (* x (* z t))) (* (+ (* a t) (* i x)) 4.0)) (- (* c b) (* 27.0 (* k j)))) (- (- (* (* 18.0 t) (* (* x y) z)) (* (+ (* a t) (* i x)) 4.0)) (- (* (* k j) 27.0) (* c b)))))

  (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)))