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

Average Error: 5.7 → 0.8

Time: 47.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i = -\infty \lor \neg \left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i \le 1.332226899097801573653352800247938704487 \cdot 10^{292}\right):\\ \;\;\;\;\left(\left(\left(18 \cdot \left(y \cdot \left(x \cdot \left(t \cdot z\right)\right)\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot j\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot k\right) \cdot j\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r613952 = x;
        double r613953 = 18.0;
        double r613954 = r613952 * r613953;
        double r613955 = y;
        double r613956 = r613954 * r613955;
        double r613957 = z;
        double r613958 = r613956 * r613957;
        double r613959 = t;
        double r613960 = r613958 * r613959;
        double r613961 = a;
        double r613962 = 4.0;
        double r613963 = r613961 * r613962;
        double r613964 = r613963 * r613959;
        double r613965 = r613960 - r613964;
        double r613966 = b;
        double r613967 = c;
        double r613968 = r613966 * r613967;
        double r613969 = r613965 + r613968;
        double r613970 = r613952 * r613962;
        double r613971 = i;
        double r613972 = r613970 * r613971;
        double r613973 = r613969 - r613972;
        double r613974 = j;
        double r613975 = 27.0;
        double r613976 = r613974 * r613975;
        double r613977 = k;
        double r613978 = r613976 * r613977;
        double r613979 = r613973 - r613978;
        return r613979;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r613980 = t;
        double r613981 = x;
        double r613982 = 18.0;
        double r613983 = r613981 * r613982;
        double r613984 = y;
        double r613985 = r613983 * r613984;
        double r613986 = z;
        double r613987 = r613985 * r613986;
        double r613988 = r613980 * r613987;
        double r613989 = a;
        double r613990 = 4.0;
        double r613991 = r613989 * r613990;
        double r613992 = r613991 * r613980;
        double r613993 = r613988 - r613992;
        double r613994 = c;
        double r613995 = b;
        double r613996 = r613994 * r613995;
        double r613997 = r613993 + r613996;
        double r613998 = r613981 * r613990;
        double r613999 = i;
        double r614000 = r613998 * r613999;
        double r614001 = r613997 - r614000;
        double r614002 = -inf.0;
        bool r614003 = r614001 <= r614002;
        double r614004 = 1.3322268990978016e+292;
        bool r614005 = r614001 <= r614004;
        double r614006 = !r614005;
        bool r614007 = r614003 || r614006;
        double r614008 = r613980 * r613986;
        double r614009 = r613981 * r614008;
        double r614010 = r613984 * r614009;
        double r614011 = r613982 * r614010;
        double r614012 = r614011 - r613992;
        double r614013 = r614012 + r613996;
        double r614014 = r614013 - r614000;
        double r614015 = 27.0;
        double r614016 = j;
        double r614017 = r614015 * r614016;
        double r614018 = k;
        double r614019 = r614017 * r614018;
        double r614020 = r614014 - r614019;
        double r614021 = r614015 * r614018;
        double r614022 = r614021 * r614016;
        double r614023 = r614001 - r614022;
        double r614024 = r614007 ? r614020 : r614023;
        return r614024;
}

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	5.7
Target	1.8
Herbie	0.8

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

Derivation

1. Split input into 2 regimes

2. if (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < -inf.0 or 1.3322268990978016e+292 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i))

   1. Initial program 50.4

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

   2. Taylor expanded around inf 32.1

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

   3. Simplified 7.7

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

   5. Applied pow1 7.7

      \[\leadsto \left(\left(\left(\left(\left(\left(t \cdot x\right) \cdot \color{blue}{{z}^{1}}\right) \cdot y\right) \cdot 18 - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   6. Applied pow1 7.7

      \[\leadsto \left(\left(\left(\left(\left(\left(t \cdot \color{blue}{{x}^{1}}\right) \cdot {z}^{1}\right) \cdot y\right) \cdot 18 - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   7. Applied pow1 7.7

      \[\leadsto \left(\left(\left(\left(\left(\left(\color{blue}{{t}^{1}} \cdot {x}^{1}\right) \cdot {z}^{1}\right) \cdot y\right) \cdot 18 - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   8. Applied pow-prod-down 7.7

      \[\leadsto \left(\left(\left(\left(\left(\color{blue}{{\left(t \cdot x\right)}^{1}} \cdot {z}^{1}\right) \cdot y\right) \cdot 18 - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   9. Applied pow-prod-down 7.7

      \[\leadsto \left(\left(\left(\left(\color{blue}{{\left(\left(t \cdot x\right) \cdot z\right)}^{1}} \cdot y\right) \cdot 18 - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   10. Simplified 5.1

       \[\leadsto \left(\left(\left(\left({\color{blue}{\left(x \cdot \left(z \cdot t\right)\right)}}^{1} \cdot y\right) \cdot 18 - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

   if -inf.0 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < 1.3322268990978016e+292

   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 associate-*l* 0.3

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

   4. Simplified 0.3

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i = -\infty \lor \neg \left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i \le 1.332226899097801573653352800247938704487 \cdot 10^{292}\right):\\ \;\;\;\;\left(\left(\left(18 \cdot \left(y \cdot \left(x \cdot \left(t \cdot z\right)\right)\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot j\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(27 \cdot k\right) \cdot j\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(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)))