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

Average Error: 5.6 → 2.7

Time: 49.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -9.65545760621799240184121524041608806323 \cdot 10^{111}:\\ \;\;\;\;\left(\left(\left(\left(x \cdot 18\right) \cdot \left(\left(y \cdot z\right) \cdot t\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - 27 \cdot \left(k \cdot j\right)\\ \mathbf{elif}\;x \le 3.101862889713283380041023072579951440529 \cdot 10^{79}:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot \left(y \cdot 18\right)\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(x \cdot 18\right) \cdot \left(y \cdot \left(z \cdot t\right)\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r457101 = x;
        double r457102 = 18.0;
        double r457103 = r457101 * r457102;
        double r457104 = y;
        double r457105 = r457103 * r457104;
        double r457106 = z;
        double r457107 = r457105 * r457106;
        double r457108 = t;
        double r457109 = r457107 * r457108;
        double r457110 = a;
        double r457111 = 4.0;
        double r457112 = r457110 * r457111;
        double r457113 = r457112 * r457108;
        double r457114 = r457109 - r457113;
        double r457115 = b;
        double r457116 = c;
        double r457117 = r457115 * r457116;
        double r457118 = r457114 + r457117;
        double r457119 = r457101 * r457111;
        double r457120 = i;
        double r457121 = r457119 * r457120;
        double r457122 = r457118 - r457121;
        double r457123 = j;
        double r457124 = 27.0;
        double r457125 = r457123 * r457124;
        double r457126 = k;
        double r457127 = r457125 * r457126;
        double r457128 = r457122 - r457127;
        return r457128;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r457129 = x;
        double r457130 = -9.655457606217992e+111;
        bool r457131 = r457129 <= r457130;
        double r457132 = 18.0;
        double r457133 = r457129 * r457132;
        double r457134 = y;
        double r457135 = z;
        double r457136 = r457134 * r457135;
        double r457137 = t;
        double r457138 = r457136 * r457137;
        double r457139 = r457133 * r457138;
        double r457140 = a;
        double r457141 = 4.0;
        double r457142 = r457140 * r457141;
        double r457143 = r457142 * r457137;
        double r457144 = r457139 - r457143;
        double r457145 = b;
        double r457146 = c;
        double r457147 = r457145 * r457146;
        double r457148 = r457144 + r457147;
        double r457149 = r457129 * r457141;
        double r457150 = i;
        double r457151 = r457149 * r457150;
        double r457152 = r457148 - r457151;
        double r457153 = 27.0;
        double r457154 = k;
        double r457155 = j;
        double r457156 = r457154 * r457155;
        double r457157 = r457153 * r457156;
        double r457158 = r457152 - r457157;
        double r457159 = 3.1018628897132834e+79;
        bool r457160 = r457129 <= r457159;
        double r457161 = r457134 * r457132;
        double r457162 = r457129 * r457161;
        double r457163 = r457162 * r457135;
        double r457164 = r457163 * r457137;
        double r457165 = r457164 - r457143;
        double r457166 = r457165 + r457147;
        double r457167 = r457166 - r457151;
        double r457168 = r457153 * r457154;
        double r457169 = r457155 * r457168;
        double r457170 = r457167 - r457169;
        double r457171 = r457135 * r457137;
        double r457172 = r457134 * r457171;
        double r457173 = r457133 * r457172;
        double r457174 = r457173 - r457143;
        double r457175 = r457174 + r457147;
        double r457176 = r457175 - r457151;
        double r457177 = r457176 - r457169;
        double r457178 = r457160 ? r457170 : r457177;
        double r457179 = r457131 ? r457158 : r457178;
        return r457179;
}

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	1.4
Herbie	2.7

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

Derivation

1. Split input into 3 regimes

2. if x < -9.655457606217992e+111

   1. Initial program 17.0

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

   3. Applied associate-*l* 17.1

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

   5. Applied associate-*l* 10.0

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

   7. Applied associate-*l* 1.6

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

   9. Applied pow1 1.6

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

   10. Applied pow1 1.6

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

   11. Applied pow-prod-down 1.6

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

   12. Applied pow1 1.6

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

   13. Applied pow-prod-down 1.6

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

   14. Simplified 1.4

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

   if -9.655457606217992e+111 < x < 3.1018628897132834e+79

   1. Initial program 2.8

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

   3. Applied associate-*l* 2.8

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

   5. Applied associate-*l* 2.9

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

   6. Simplified 2.9

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

   if 3.1018628897132834e+79 < x

   1. Initial program 15.6

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

   3. Applied associate-*l* 15.8

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

   5. Applied associate-*l* 8.6

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

   7. Applied associate-*l* 1.5

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

   9. Applied associate-*l* 2.1

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

4. Final simplification 2.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9.65545760621799240184121524041608806323 \cdot 10^{111}:\\ \;\;\;\;\left(\left(\left(\left(x \cdot 18\right) \cdot \left(\left(y \cdot z\right) \cdot t\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - 27 \cdot \left(k \cdot j\right)\\ \mathbf{elif}\;x \le 3.101862889713283380041023072579951440529 \cdot 10^{79}:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot \left(y \cdot 18\right)\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(x \cdot 18\right) \cdot \left(y \cdot \left(z \cdot t\right)\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(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)))