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

Average Error: 5.5 → 0.8

Time: 38.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k = -\infty \lor \neg \left(\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \le 2.291150336217132422785785995759037448958 \cdot 10^{305}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(\left(t \cdot y\right) \cdot z\right) \cdot x, 18, \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(27 \cdot k\right) \cdot j\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r464163 = x;
        double r464164 = 18.0;
        double r464165 = r464163 * r464164;
        double r464166 = y;
        double r464167 = r464165 * r464166;
        double r464168 = z;
        double r464169 = r464167 * r464168;
        double r464170 = t;
        double r464171 = r464169 * r464170;
        double r464172 = a;
        double r464173 = 4.0;
        double r464174 = r464172 * r464173;
        double r464175 = r464174 * r464170;
        double r464176 = r464171 - r464175;
        double r464177 = b;
        double r464178 = c;
        double r464179 = r464177 * r464178;
        double r464180 = r464176 + r464179;
        double r464181 = r464163 * r464173;
        double r464182 = i;
        double r464183 = r464181 * r464182;
        double r464184 = r464180 - r464183;
        double r464185 = j;
        double r464186 = 27.0;
        double r464187 = r464185 * r464186;
        double r464188 = k;
        double r464189 = r464187 * r464188;
        double r464190 = r464184 - r464189;
        return r464190;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
        double r464191 = x;
        double r464192 = 18.0;
        double r464193 = r464191 * r464192;
        double r464194 = y;
        double r464195 = r464193 * r464194;
        double r464196 = z;
        double r464197 = r464195 * r464196;
        double r464198 = t;
        double r464199 = r464197 * r464198;
        double r464200 = a;
        double r464201 = 4.0;
        double r464202 = r464200 * r464201;
        double r464203 = r464202 * r464198;
        double r464204 = r464199 - r464203;
        double r464205 = b;
        double r464206 = c;
        double r464207 = r464205 * r464206;
        double r464208 = r464204 + r464207;
        double r464209 = r464191 * r464201;
        double r464210 = i;
        double r464211 = r464209 * r464210;
        double r464212 = r464208 - r464211;
        double r464213 = j;
        double r464214 = 27.0;
        double r464215 = r464213 * r464214;
        double r464216 = k;
        double r464217 = r464215 * r464216;
        double r464218 = r464212 - r464217;
        double r464219 = -inf.0;
        bool r464220 = r464218 <= r464219;
        double r464221 = 2.2911503362171324e+305;
        bool r464222 = r464218 <= r464221;
        double r464223 = !r464222;
        bool r464224 = r464220 || r464223;
        double r464225 = r464198 * r464194;
        double r464226 = r464225 * r464196;
        double r464227 = r464226 * r464191;
        double r464228 = r464191 * r464210;
        double r464229 = fma(r464198, r464200, r464228);
        double r464230 = r464214 * r464216;
        double r464231 = r464230 * r464213;
        double r464232 = fma(r464201, r464229, r464231);
        double r464233 = -r464232;
        double r464234 = fma(r464206, r464205, r464233);
        double r464235 = fma(r464227, r464192, r464234);
        double r464236 = r464224 ? r464235 : r464218;
        return r464236;
}

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	5.5
Target	1.6
Herbie	0.8

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

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)) (* (* j 27.0) k)) < -inf.0 or 2.2911503362171324e+305 < (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k))

   1. Initial program 59.9

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

   2. Simplified 14.9

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

   4. Applied associate-*r* 7.5

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

   5. Taylor expanded around 0 6.5

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

   7. Applied associate-*r* 6.6

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

   if -inf.0 < (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)) < 2.2911503362171324e+305

   1. Initial program 0.3

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k = -\infty \lor \neg \left(\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k \le 2.291150336217132422785785995759037448958 \cdot 10^{305}\right):\\ \;\;\;\;\mathsf{fma}\left(\left(\left(t \cdot y\right) \cdot z\right) \cdot x, 18, \mathsf{fma}\left(c, b, -\mathsf{fma}\left(4, \mathsf{fma}\left(t, a, x \cdot i\right), \left(27 \cdot k\right) \cdot j\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 +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)))