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

Average Error: 5.2 → 5.7

Time: 18.3s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \cdot c \le -7.78697423890387318 \cdot 10^{161}:\\ \;\;\;\;\left(\left(\left(0 \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{elif}\;b \cdot c \le -4.72896944561745812 \cdot 10^{44}:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right)\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{elif}\;b \cdot c \le -1.6071174332412006 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(x \cdot 18\right), t \cdot y, b \cdot c - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{elif}\;b \cdot c \le 1.236802286 \cdot 10^{-314}:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(z \cdot t\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{elif}\;b \cdot c \le 7.832862067016947 \cdot 10^{-40}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \sqrt{27} \cdot \left(\sqrt{27} \cdot \left(k \cdot j\right)\right)\\ \mathbf{elif}\;b \cdot c \le 1.7946900526051252 \cdot 10^{23}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(x \cdot 18\right), t \cdot y, b \cdot c - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right)\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r139136 = x;
        double r139137 = 18.0;
        double r139138 = r139136 * r139137;
        double r139139 = y;
        double r139140 = r139138 * r139139;
        double r139141 = z;
        double r139142 = r139140 * r139141;
        double r139143 = t;
        double r139144 = r139142 * r139143;
        double r139145 = a;
        double r139146 = 4.0;
        double r139147 = r139145 * r139146;
        double r139148 = r139147 * r139143;
        double r139149 = r139144 - r139148;
        double r139150 = b;
        double r139151 = c;
        double r139152 = r139150 * r139151;
        double r139153 = r139149 + r139152;
        double r139154 = r139136 * r139146;
        double r139155 = i;
        double r139156 = r139154 * r139155;
        double r139157 = r139153 - r139156;
        double r139158 = j;
        double r139159 = 27.0;
        double r139160 = r139158 * r139159;
        double r139161 = k;
        double r139162 = r139160 * r139161;
        double r139163 = r139157 - r139162;
        return r139163;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r139164 = b;
        double r139165 = c;
        double r139166 = r139164 * r139165;
        double r139167 = -7.786974238903873e+161;
        bool r139168 = r139166 <= r139167;
        double r139169 = 0.0;
        double r139170 = t;
        double r139171 = r139169 * r139170;
        double r139172 = a;
        double r139173 = 4.0;
        double r139174 = r139172 * r139173;
        double r139175 = r139174 * r139170;
        double r139176 = r139171 - r139175;
        double r139177 = r139176 + r139166;
        double r139178 = x;
        double r139179 = r139178 * r139173;
        double r139180 = i;
        double r139181 = r139179 * r139180;
        double r139182 = r139177 - r139181;
        double r139183 = j;
        double r139184 = 27.0;
        double r139185 = k;
        double r139186 = r139184 * r139185;
        double r139187 = r139183 * r139186;
        double r139188 = r139182 - r139187;
        double r139189 = -4.728969445617458e+44;
        bool r139190 = r139166 <= r139189;
        double r139191 = 18.0;
        double r139192 = r139178 * r139191;
        double r139193 = y;
        double r139194 = z;
        double r139195 = r139193 * r139194;
        double r139196 = r139192 * r139195;
        double r139197 = r139196 * r139170;
        double r139198 = r139197 - r139175;
        double r139199 = r139198 + r139166;
        double r139200 = r139199 - r139181;
        double r139201 = r139200 - r139187;
        double r139202 = -1.6071174332412006e-10;
        bool r139203 = r139166 <= r139202;
        double r139204 = r139194 * r139192;
        double r139205 = r139170 * r139193;
        double r139206 = r139178 * r139180;
        double r139207 = fma(r139170, r139172, r139206);
        double r139208 = r139183 * r139184;
        double r139209 = r139208 * r139185;
        double r139210 = fma(r139173, r139207, r139209);
        double r139211 = r139166 - r139210;
        double r139212 = fma(r139204, r139205, r139211);
        double r139213 = 1.2368022860888e-314;
        bool r139214 = r139166 <= r139213;
        double r139215 = r139192 * r139193;
        double r139216 = r139194 * r139170;
        double r139217 = r139215 * r139216;
        double r139218 = r139217 - r139175;
        double r139219 = r139218 + r139166;
        double r139220 = r139219 - r139181;
        double r139221 = r139220 - r139187;
        double r139222 = 7.832862067016947e-40;
        bool r139223 = r139166 <= r139222;
        double r139224 = r139215 * r139194;
        double r139225 = r139224 * r139170;
        double r139226 = r139225 - r139175;
        double r139227 = r139226 + r139166;
        double r139228 = r139227 - r139181;
        double r139229 = sqrt(r139184);
        double r139230 = r139185 * r139183;
        double r139231 = r139229 * r139230;
        double r139232 = r139229 * r139231;
        double r139233 = r139228 - r139232;
        double r139234 = 1.7946900526051252e+23;
        bool r139235 = r139166 <= r139234;
        double r139236 = r139235 ? r139212 : r139201;
        double r139237 = r139223 ? r139233 : r139236;
        double r139238 = r139214 ? r139221 : r139237;
        double r139239 = r139203 ? r139212 : r139238;
        double r139240 = r139190 ? r139201 : r139239;
        double r139241 = r139168 ? r139188 : r139240;
        return r139241;
}

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

Derivation

1. Split input into 5 regimes

2. if (* b c) < -7.786974238903873e+161

   1. Initial program 5.8

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

   3. Applied associate-*l* 5.8

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

   4. Taylor expanded around 0 3.8

      \[\leadsto \left(\left(\left(\color{blue}{0} \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\]

   if -7.786974238903873e+161 < (* b c) < -4.728969445617458e+44 or 1.7946900526051252e+23 < (* b c)

   1. Initial program 4.8

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

   3. Applied associate-*l* 4.8

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

   5. Applied associate-*l* 6.2

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

   if -4.728969445617458e+44 < (* b c) < -1.6071174332412006e-10 or 7.832862067016947e-40 < (* b c) < 1.7946900526051252e+23

   1. Initial program 5.1

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

   2. Simplified 6.5

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

   if -1.6071174332412006e-10 < (* b c) < 1.2368022860888e-314

   1. Initial program 5.1

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

   3. Applied associate-*l* 5.1

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

   5. Applied associate-*l* 5.5

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

   if 1.2368022860888e-314 < (* b c) < 7.832862067016947e-40

   1. Initial program 5.6

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

   2. Taylor expanded around 0 5.6

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

   4. Applied add-sqr-sqrt 5.6

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

   5. Applied associate-*l* 5.7

      \[\leadsto \left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \color{blue}{\sqrt{27} \cdot \left(\sqrt{27} \cdot \left(k \cdot j\right)\right)}\]
3. Recombined 5 regimes into one program.

4. Final simplification 5.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot c \le -7.78697423890387318 \cdot 10^{161}:\\ \;\;\;\;\left(\left(\left(0 \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{elif}\;b \cdot c \le -4.72896944561745812 \cdot 10^{44}:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right)\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{elif}\;b \cdot c \le -1.6071174332412006 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(x \cdot 18\right), t \cdot y, b \cdot c - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{elif}\;b \cdot c \le 1.236802286 \cdot 10^{-314}:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot \left(z \cdot t\right) - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \mathbf{elif}\;b \cdot c \le 7.832862067016947 \cdot 10^{-40}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \sqrt{27} \cdot \left(\sqrt{27} \cdot \left(k \cdot j\right)\right)\\ \mathbf{elif}\;b \cdot c \le 1.7946900526051252 \cdot 10^{23}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \left(x \cdot 18\right), t \cdot y, b \cdot c - \mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(x \cdot 18\right) \cdot \left(y \cdot z\right)\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - j \cdot \left(27 \cdot k\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1"
  :precision binary64
  (- (- (+ (- (* (* (* (* x 18) y) z) t) (* (* a 4) t)) (* b c)) (* (* x 4) i)) (* (* j 27) k)))