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

Average Error: 5.4 → 4.2

Time: 8.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -7.5200119166880343 \cdot 10^{-93} \lor \neg \left(t \le 8.24455457144902905 \cdot 10^{-121}\right):\\ \;\;\;\;t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(0 - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r569094 = x;
        double r569095 = 18.0;
        double r569096 = r569094 * r569095;
        double r569097 = y;
        double r569098 = r569096 * r569097;
        double r569099 = z;
        double r569100 = r569098 * r569099;
        double r569101 = t;
        double r569102 = r569100 * r569101;
        double r569103 = a;
        double r569104 = 4.0;
        double r569105 = r569103 * r569104;
        double r569106 = r569105 * r569101;
        double r569107 = r569102 - r569106;
        double r569108 = b;
        double r569109 = c;
        double r569110 = r569108 * r569109;
        double r569111 = r569107 + r569110;
        double r569112 = r569094 * r569104;
        double r569113 = i;
        double r569114 = r569112 * r569113;
        double r569115 = r569111 - r569114;
        double r569116 = j;
        double r569117 = 27.0;
        double r569118 = r569116 * r569117;
        double r569119 = k;
        double r569120 = r569118 * r569119;
        double r569121 = r569115 - r569120;
        return r569121;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r569122 = t;
        double r569123 = -7.520011916688034e-93;
        bool r569124 = r569122 <= r569123;
        double r569125 = 8.244554571449029e-121;
        bool r569126 = r569122 <= r569125;
        double r569127 = !r569126;
        bool r569128 = r569124 || r569127;
        double r569129 = x;
        double r569130 = 18.0;
        double r569131 = r569129 * r569130;
        double r569132 = y;
        double r569133 = r569131 * r569132;
        double r569134 = z;
        double r569135 = r569133 * r569134;
        double r569136 = a;
        double r569137 = 4.0;
        double r569138 = r569136 * r569137;
        double r569139 = r569135 - r569138;
        double r569140 = r569122 * r569139;
        double r569141 = b;
        double r569142 = c;
        double r569143 = r569141 * r569142;
        double r569144 = r569129 * r569137;
        double r569145 = i;
        double r569146 = r569144 * r569145;
        double r569147 = j;
        double r569148 = 27.0;
        double r569149 = k;
        double r569150 = r569148 * r569149;
        double r569151 = r569147 * r569150;
        double r569152 = r569146 + r569151;
        double r569153 = r569143 - r569152;
        double r569154 = r569140 + r569153;
        double r569155 = 0.0;
        double r569156 = r569155 - r569138;
        double r569157 = r569122 * r569156;
        double r569158 = r569147 * r569148;
        double r569159 = r569158 * r569149;
        double r569160 = r569146 + r569159;
        double r569161 = r569143 - r569160;
        double r569162 = r569157 + r569161;
        double r569163 = r569128 ? r569154 : r569162;
        return r569163;
}

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	5.4
Target	1.6
Herbie	4.2

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

Derivation

1. Split input into 2 regimes

2. if t < -7.520011916688034e-93 or 8.244554571449029e-121 < t

   1. Initial program 2.9

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

   2. Simplified 2.9

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

   4. Applied associate-*l* 3.0

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

   if -7.520011916688034e-93 < t < 8.244554571449029e-121

   1. Initial program 8.9

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

   2. Simplified 8.9

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

   3. Taylor expanded around 0 5.9

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

4. Final simplification 4.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -7.5200119166880343 \cdot 10^{-93} \lor \neg \left(t \le 8.24455457144902905 \cdot 10^{-121}\right):\\ \;\;\;\;t \cdot \left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + j \cdot \left(27 \cdot k\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(0 - a \cdot 4\right) + \left(b \cdot c - \left(\left(x \cdot 4\right) \cdot i + \left(j \cdot 27\right) \cdot k\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020003 
(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)))