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

Average Error: 5.3 → 6.2

Time: 7.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \le 7.0669981561857078 \cdot 10^{-218} \lor \neg \left(k \le 4.00595738363528446 \cdot 10^{-73}\right):\\ \;\;\;\;t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(0 - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r854009 = x;
        double r854010 = 18.0;
        double r854011 = r854009 * r854010;
        double r854012 = y;
        double r854013 = r854011 * r854012;
        double r854014 = z;
        double r854015 = r854013 * r854014;
        double r854016 = t;
        double r854017 = r854015 * r854016;
        double r854018 = a;
        double r854019 = 4.0;
        double r854020 = r854018 * r854019;
        double r854021 = r854020 * r854016;
        double r854022 = r854017 - r854021;
        double r854023 = b;
        double r854024 = c;
        double r854025 = r854023 * r854024;
        double r854026 = r854022 + r854025;
        double r854027 = r854009 * r854019;
        double r854028 = i;
        double r854029 = r854027 * r854028;
        double r854030 = r854026 - r854029;
        double r854031 = j;
        double r854032 = 27.0;
        double r854033 = r854031 * r854032;
        double r854034 = k;
        double r854035 = r854033 * r854034;
        double r854036 = r854030 - r854035;
        return r854036;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r854037 = k;
        double r854038 = 7.066998156185708e-218;
        bool r854039 = r854037 <= r854038;
        double r854040 = 4.0059573836352845e-73;
        bool r854041 = r854037 <= r854040;
        double r854042 = !r854041;
        bool r854043 = r854039 || r854042;
        double r854044 = t;
        double r854045 = x;
        double r854046 = 18.0;
        double r854047 = r854045 * r854046;
        double r854048 = y;
        double r854049 = r854047 * r854048;
        double r854050 = z;
        double r854051 = r854049 * r854050;
        double r854052 = a;
        double r854053 = 4.0;
        double r854054 = r854052 * r854053;
        double r854055 = r854051 - r854054;
        double r854056 = r854044 * r854055;
        double r854057 = b;
        double r854058 = c;
        double r854059 = r854057 * r854058;
        double r854060 = r854045 * r854053;
        double r854061 = i;
        double r854062 = r854060 * r854061;
        double r854063 = j;
        double r854064 = 27.0;
        double r854065 = r854064 * r854037;
        double r854066 = r854063 * r854065;
        double r854067 = r854062 + r854066;
        double r854068 = r854059 - r854067;
        double r854069 = r854056 + r854068;
        double r854070 = 0.0;
        double r854071 = r854070 - r854054;
        double r854072 = r854044 * r854071;
        double r854073 = r854063 * r854064;
        double r854074 = r854073 * r854037;
        double r854075 = r854062 + r854074;
        double r854076 = r854059 - r854075;
        double r854077 = r854072 + r854076;
        double r854078 = r854043 ? r854069 : r854077;
        return r854078;
}

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	5.3
Target	1.5
Herbie	6.2

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

Derivation

1. Split input into 2 regimes

2. if k < 7.066998156185708e-218 or 4.0059573836352845e-73 < k

   1. Initial program 5.4

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

   2. Simplified 5.4

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

   4. Applied associate-*l* 5.4

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

   if 7.066998156185708e-218 < k < 4.0059573836352845e-73

   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. Taylor expanded around 0 11.1

      \[\leadsto t \cdot \left(\color{blue}{0} - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 6.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 7.0669981561857078 \cdot 10^{-218} \lor \neg \left(k \le 4.00595738363528446 \cdot 10^{-73}\right):\\ \;\;\;\;t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(0 - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(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)))