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

Average Error: 5.9 → 1.2

Time: 23.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i = -\infty \lor \neg \left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i \le 4.451860458118571560076905778940636290813 \cdot 10^{294}\right):\\ \;\;\;\;\left(\left(c \cdot b + \left(\left(y \cdot \left(\left(t \cdot x\right) \cdot z\right)\right) \cdot 18 - \left(a \cdot 4\right) \cdot t\right)\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(\sqrt[3]{k} \cdot \left(\left(j \cdot 27\right) \cdot \sqrt[3]{k}\right)\right) \cdot \sqrt[3]{k}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r589036 = x;
        double r589037 = 18.0;
        double r589038 = r589036 * r589037;
        double r589039 = y;
        double r589040 = r589038 * r589039;
        double r589041 = z;
        double r589042 = r589040 * r589041;
        double r589043 = t;
        double r589044 = r589042 * r589043;
        double r589045 = a;
        double r589046 = 4.0;
        double r589047 = r589045 * r589046;
        double r589048 = r589047 * r589043;
        double r589049 = r589044 - r589048;
        double r589050 = b;
        double r589051 = c;
        double r589052 = r589050 * r589051;
        double r589053 = r589049 + r589052;
        double r589054 = r589036 * r589046;
        double r589055 = i;
        double r589056 = r589054 * r589055;
        double r589057 = r589053 - r589056;
        double r589058 = j;
        double r589059 = 27.0;
        double r589060 = r589058 * r589059;
        double r589061 = k;
        double r589062 = r589060 * r589061;
        double r589063 = r589057 - r589062;
        return r589063;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r589064 = t;
        double r589065 = x;
        double r589066 = 18.0;
        double r589067 = r589065 * r589066;
        double r589068 = y;
        double r589069 = r589067 * r589068;
        double r589070 = z;
        double r589071 = r589069 * r589070;
        double r589072 = r589064 * r589071;
        double r589073 = a;
        double r589074 = 4.0;
        double r589075 = r589073 * r589074;
        double r589076 = r589075 * r589064;
        double r589077 = r589072 - r589076;
        double r589078 = c;
        double r589079 = b;
        double r589080 = r589078 * r589079;
        double r589081 = r589077 + r589080;
        double r589082 = r589065 * r589074;
        double r589083 = i;
        double r589084 = r589082 * r589083;
        double r589085 = r589081 - r589084;
        double r589086 = -inf.0;
        bool r589087 = r589085 <= r589086;
        double r589088 = 4.4518604581185716e+294;
        bool r589089 = r589085 <= r589088;
        double r589090 = !r589089;
        bool r589091 = r589087 || r589090;
        double r589092 = r589064 * r589065;
        double r589093 = r589092 * r589070;
        double r589094 = r589068 * r589093;
        double r589095 = r589094 * r589066;
        double r589096 = r589095 - r589076;
        double r589097 = r589080 + r589096;
        double r589098 = r589097 - r589084;
        double r589099 = j;
        double r589100 = 27.0;
        double r589101 = k;
        double r589102 = r589100 * r589101;
        double r589103 = r589099 * r589102;
        double r589104 = r589098 - r589103;
        double r589105 = cbrt(r589101);
        double r589106 = r589099 * r589100;
        double r589107 = r589106 * r589105;
        double r589108 = r589105 * r589107;
        double r589109 = r589108 * r589105;
        double r589110 = r589085 - r589109;
        double r589111 = r589091 ? r589104 : r589110;
        return r589111;
}

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	5.9
Target	1.7
Herbie	1.2

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

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 4.4518604581185716e+294 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i))

   1. Initial program 53.1

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

   2. Taylor expanded around inf 33.5

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

   3. Simplified 6.5

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

   5. Applied associate-*l* 6.6

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

   if -inf.0 < (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) < 4.4518604581185716e+294

   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

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

   4. Applied associate-*r* 0.6

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

   5. Simplified 0.6

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

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i = -\infty \lor \neg \left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i \le 4.451860458118571560076905778940636290813 \cdot 10^{294}\right):\\ \;\;\;\;\left(\left(c \cdot b + \left(\left(y \cdot \left(\left(t \cdot x\right) \cdot z\right)\right) \cdot 18 - \left(a \cdot 4\right) \cdot t\right)\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) - \left(a \cdot 4\right) \cdot t\right) + c \cdot b\right) - \left(x \cdot 4\right) \cdot i\right) - \left(\sqrt[3]{k} \cdot \left(\left(j \cdot 27\right) \cdot \sqrt[3]{k}\right)\right) \cdot \sqrt[3]{k}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(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)))