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

Average Error: 5.6 → 3.7

Time: 8.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -5.5038978783466573 \cdot 10^{-5}:\\ \;\;\;\;t \cdot \left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \sqrt[3]{z} - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{elif}\;z \le 1.12710979643317163 \cdot 10^{-39}:\\ \;\;\;\;\left(18 \cdot \left(t \cdot \left(x \cdot \left(z \cdot y\right)\right)\right) + t \cdot \left(-a \cdot 4\right)\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \sqrt{z}\right) \cdot \sqrt{z} - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r738966 = x;
        double r738967 = 18.0;
        double r738968 = r738966 * r738967;
        double r738969 = y;
        double r738970 = r738968 * r738969;
        double r738971 = z;
        double r738972 = r738970 * r738971;
        double r738973 = t;
        double r738974 = r738972 * r738973;
        double r738975 = a;
        double r738976 = 4.0;
        double r738977 = r738975 * r738976;
        double r738978 = r738977 * r738973;
        double r738979 = r738974 - r738978;
        double r738980 = b;
        double r738981 = c;
        double r738982 = r738980 * r738981;
        double r738983 = r738979 + r738982;
        double r738984 = r738966 * r738976;
        double r738985 = i;
        double r738986 = r738984 * r738985;
        double r738987 = r738983 - r738986;
        double r738988 = j;
        double r738989 = 27.0;
        double r738990 = r738988 * r738989;
        double r738991 = k;
        double r738992 = r738990 * r738991;
        double r738993 = r738987 - r738992;
        return r738993;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r738994 = z;
        double r738995 = -5.503897878346657e-05;
        bool r738996 = r738994 <= r738995;
        double r738997 = t;
        double r738998 = x;
        double r738999 = 18.0;
        double r739000 = r738998 * r738999;
        double r739001 = y;
        double r739002 = r739000 * r739001;
        double r739003 = cbrt(r738994);
        double r739004 = r739003 * r739003;
        double r739005 = r739002 * r739004;
        double r739006 = r739005 * r739003;
        double r739007 = a;
        double r739008 = 4.0;
        double r739009 = r739007 * r739008;
        double r739010 = r739006 - r739009;
        double r739011 = r738997 * r739010;
        double r739012 = b;
        double r739013 = c;
        double r739014 = r739012 * r739013;
        double r739015 = r738998 * r739008;
        double r739016 = i;
        double r739017 = r739015 * r739016;
        double r739018 = j;
        double r739019 = 27.0;
        double r739020 = k;
        double r739021 = r739019 * r739020;
        double r739022 = r739018 * r739021;
        double r739023 = r739017 + r739022;
        double r739024 = r739014 - r739023;
        double r739025 = r739011 + r739024;
        double r739026 = 1.1271097964331716e-39;
        bool r739027 = r738994 <= r739026;
        double r739028 = r738994 * r739001;
        double r739029 = r738998 * r739028;
        double r739030 = r738997 * r739029;
        double r739031 = r738999 * r739030;
        double r739032 = -r739009;
        double r739033 = r738997 * r739032;
        double r739034 = r739031 + r739033;
        double r739035 = r739034 + r739024;
        double r739036 = sqrt(r738994);
        double r739037 = r739002 * r739036;
        double r739038 = r739037 * r739036;
        double r739039 = r739038 - r739009;
        double r739040 = r738997 * r739039;
        double r739041 = r739018 * r739019;
        double r739042 = r739041 * r739020;
        double r739043 = r739017 + r739042;
        double r739044 = r739014 - r739043;
        double r739045 = r739040 + r739044;
        double r739046 = r739027 ? r739035 : r739045;
        double r739047 = r738996 ? r739025 : r739046;
        return r739047;
}

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	5.6
Target	1.6
Herbie	3.7

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

Derivation

1. Split input into 3 regimes

2. if z < -5.503897878346657e-05

   1. Initial program 6.0

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

   2. Simplified 6.0

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

   4. Applied associate-*l* 6.0

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

   6. Applied add-cube-cbrt 6.2

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

   7. Applied associate-*r* 6.2

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

   if -5.503897878346657e-05 < z < 1.1271097964331716e-39

   1. Initial program 5.0

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

   2. Simplified 5.0

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

   4. Applied associate-*l* 4.9

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

   6. Applied add-cube-cbrt 4.9

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

   7. Applied associate-*r* 4.9

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

   9. Applied sub-neg 4.9

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

   10. Applied distribute-lft-in 4.9

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

   11. Simplified 1.2

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

   if 1.1271097964331716e-39 < z

   1. Initial program 6.5

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

   2. Simplified 6.5

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

   4. Applied add-sqr-sqrt 6.5

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

   5. Applied associate-*r* 6.5

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

4. Final simplification 3.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.5038978783466573 \cdot 10^{-5}:\\ \;\;\;\;t \cdot \left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \sqrt[3]{z} - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{elif}\;z \le 1.12710979643317163 \cdot 10^{-39}:\\ \;\;\;\;\left(18 \cdot \left(t \cdot \left(x \cdot \left(z \cdot y\right)\right)\right) + t \cdot \left(-a \cdot 4\right)\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \sqrt{z}\right) \cdot \sqrt{z} - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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)))