Jmat.Real.dawson

Average Error: 29.0 → 0.0

Time: 11.2s

Precision: 64

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

Math

\[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.042406060400000001 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.00726441819999999999 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 5.0640340000000002 \cdot 10^{-4} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 1.789971 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.77154710189999998 \cdot \left(x \cdot x\right)\right) + 0.29097386390000002 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.069455576099999999 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.014000544199999999 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 8.32794500000000044 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 1.789971 \cdot 10^{-4}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -16518388.9625579324 \lor \neg \left(x \le 613.008358988488794\right):\\ \;\;\;\;\frac{0.25141790006653753}{{x}^{3}} + \left(\frac{0.1529819634592933}{{x}^{5}} + \frac{0.5}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left({\left(x \cdot x\right)}^{4} \cdot \left(x \cdot \left(x \cdot 1.789971 \cdot 10^{-4}\right) + 5.0640340000000002 \cdot 10^{-4}\right) + \left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + {x}^{6} \cdot 0.00726441819999999999\right)\right) + {x}^{4} \cdot 0.042406060400000001\right) \cdot x}{\left(\left(2 \cdot 1.789971 \cdot 10^{-4}\right) \cdot {\left(x \cdot x\right)}^{6} + \left(\left(x \cdot x\right) \cdot \left(0.77154710189999998 + 0.29097386390000002 \cdot \left(x \cdot x\right)\right) + 1\right)\right) + \left({x}^{6} \cdot 0.069455576099999999 + {\left(x \cdot x\right)}^{4} \cdot \left(0.014000544199999999 + \left(x \cdot x\right) \cdot 8.32794500000000044 \cdot 10^{-4}\right)\right)}\\ \end{array}\]

C

double f(double x) {
        double r354189 = 1.0;
        double r354190 = 0.1049934947;
        double r354191 = x;
        double r354192 = r354191 * r354191;
        double r354193 = r354190 * r354192;
        double r354194 = r354189 + r354193;
        double r354195 = 0.0424060604;
        double r354196 = r354192 * r354192;
        double r354197 = r354195 * r354196;
        double r354198 = r354194 + r354197;
        double r354199 = 0.0072644182;
        double r354200 = r354196 * r354192;
        double r354201 = r354199 * r354200;
        double r354202 = r354198 + r354201;
        double r354203 = 0.0005064034;
        double r354204 = r354200 * r354192;
        double r354205 = r354203 * r354204;
        double r354206 = r354202 + r354205;
        double r354207 = 0.0001789971;
        double r354208 = r354204 * r354192;
        double r354209 = r354207 * r354208;
        double r354210 = r354206 + r354209;
        double r354211 = 0.7715471019;
        double r354212 = r354211 * r354192;
        double r354213 = r354189 + r354212;
        double r354214 = 0.2909738639;
        double r354215 = r354214 * r354196;
        double r354216 = r354213 + r354215;
        double r354217 = 0.0694555761;
        double r354218 = r354217 * r354200;
        double r354219 = r354216 + r354218;
        double r354220 = 0.0140005442;
        double r354221 = r354220 * r354204;
        double r354222 = r354219 + r354221;
        double r354223 = 0.0008327945;
        double r354224 = r354223 * r354208;
        double r354225 = r354222 + r354224;
        double r354226 = 2.0;
        double r354227 = r354226 * r354207;
        double r354228 = r354208 * r354192;
        double r354229 = r354227 * r354228;
        double r354230 = r354225 + r354229;
        double r354231 = r354210 / r354230;
        double r354232 = r354231 * r354191;
        return r354232;
}

↓

double f(double x) {
        double r354233 = x;
        double r354234 = -16518388.962557932;
        bool r354235 = r354233 <= r354234;
        double r354236 = 613.0083589884888;
        bool r354237 = r354233 <= r354236;
        double r354238 = !r354237;
        bool r354239 = r354235 || r354238;
        double r354240 = 0.2514179000665375;
        double r354241 = 3.0;
        double r354242 = pow(r354233, r354241);
        double r354243 = r354240 / r354242;
        double r354244 = 0.15298196345929327;
        double r354245 = 5.0;
        double r354246 = pow(r354233, r354245);
        double r354247 = r354244 / r354246;
        double r354248 = 0.5;
        double r354249 = r354248 / r354233;
        double r354250 = r354247 + r354249;
        double r354251 = r354243 + r354250;
        double r354252 = r354233 * r354233;
        double r354253 = 4.0;
        double r354254 = pow(r354252, r354253);
        double r354255 = 0.0001789971;
        double r354256 = r354233 * r354255;
        double r354257 = r354233 * r354256;
        double r354258 = 0.0005064034;
        double r354259 = r354257 + r354258;
        double r354260 = r354254 * r354259;
        double r354261 = 1.0;
        double r354262 = 0.1049934947;
        double r354263 = r354262 * r354252;
        double r354264 = r354261 + r354263;
        double r354265 = 6.0;
        double r354266 = pow(r354233, r354265);
        double r354267 = 0.0072644182;
        double r354268 = r354266 * r354267;
        double r354269 = r354264 + r354268;
        double r354270 = r354260 + r354269;
        double r354271 = pow(r354233, r354253);
        double r354272 = 0.0424060604;
        double r354273 = r354271 * r354272;
        double r354274 = r354270 + r354273;
        double r354275 = r354274 * r354233;
        double r354276 = 2.0;
        double r354277 = r354276 * r354255;
        double r354278 = pow(r354252, r354265);
        double r354279 = r354277 * r354278;
        double r354280 = 0.7715471019;
        double r354281 = 0.2909738639;
        double r354282 = r354281 * r354252;
        double r354283 = r354280 + r354282;
        double r354284 = r354252 * r354283;
        double r354285 = r354284 + r354261;
        double r354286 = r354279 + r354285;
        double r354287 = 0.0694555761;
        double r354288 = r354266 * r354287;
        double r354289 = 0.0140005442;
        double r354290 = 0.0008327945;
        double r354291 = r354252 * r354290;
        double r354292 = r354289 + r354291;
        double r354293 = r354254 * r354292;
        double r354294 = r354288 + r354293;
        double r354295 = r354286 + r354294;
        double r354296 = r354275 / r354295;
        double r354297 = r354239 ? r354251 : r354296;
        return r354297;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -16518388.962557932 or 613.0083589884888 < x

   1. Initial program 58.9

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.042406060400000001 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.00726441819999999999 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 5.0640340000000002 \cdot 10^{-4} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 1.789971 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.77154710189999998 \cdot \left(x \cdot x\right)\right) + 0.29097386390000002 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.069455576099999999 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.014000544199999999 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 8.32794500000000044 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 1.789971 \cdot 10^{-4}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]

   2. Simplified 59.0

      \[\leadsto \color{blue}{\frac{\left(\left({\left(x \cdot x\right)}^{4} \cdot \left(\left(x \cdot x\right) \cdot 1.789971 \cdot 10^{-4} + 5.0640340000000002 \cdot 10^{-4}\right) + \left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + {x}^{6} \cdot 0.00726441819999999999\right)\right) + {x}^{4} \cdot 0.042406060400000001\right) \cdot x}{\left(\left(2 \cdot 1.789971 \cdot 10^{-4}\right) \cdot {\left(x \cdot x\right)}^{6} + \left(\left(x \cdot x\right) \cdot \left(0.77154710189999998 + 0.29097386390000002 \cdot \left(x \cdot x\right)\right) + 1\right)\right) + \left({x}^{6} \cdot 0.069455576099999999 + {\left(x \cdot x\right)}^{4} \cdot \left(0.014000544199999999 + \left(x \cdot x\right) \cdot 8.32794500000000044 \cdot 10^{-4}\right)\right)}}\]

   3. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{0.25141790006653753 \cdot \frac{1}{{x}^{3}} + \left(0.1529819634592933 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right)}\]

   4. Simplified 0.0

      \[\leadsto \color{blue}{\frac{0.25141790006653753}{{x}^{3}} + \left(\frac{0.1529819634592933}{{x}^{5}} + \frac{0.5}{x}\right)}\]

   if -16518388.962557932 < x < 613.0083589884888

   1. Initial program 0.0

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.042406060400000001 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.00726441819999999999 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 5.0640340000000002 \cdot 10^{-4} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 1.789971 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.77154710189999998 \cdot \left(x \cdot x\right)\right) + 0.29097386390000002 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.069455576099999999 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.014000544199999999 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 8.32794500000000044 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 1.789971 \cdot 10^{-4}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]

   2. Simplified 0.0

      \[\leadsto \color{blue}{\frac{\left(\left({\left(x \cdot x\right)}^{4} \cdot \left(\left(x \cdot x\right) \cdot 1.789971 \cdot 10^{-4} + 5.0640340000000002 \cdot 10^{-4}\right) + \left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + {x}^{6} \cdot 0.00726441819999999999\right)\right) + {x}^{4} \cdot 0.042406060400000001\right) \cdot x}{\left(\left(2 \cdot 1.789971 \cdot 10^{-4}\right) \cdot {\left(x \cdot x\right)}^{6} + \left(\left(x \cdot x\right) \cdot \left(0.77154710189999998 + 0.29097386390000002 \cdot \left(x \cdot x\right)\right) + 1\right)\right) + \left({x}^{6} \cdot 0.069455576099999999 + {\left(x \cdot x\right)}^{4} \cdot \left(0.014000544199999999 + \left(x \cdot x\right) \cdot 8.32794500000000044 \cdot 10^{-4}\right)\right)}}\]
   3. Using strategy rm

   4. Applied associate-*l* 0.0

      \[\leadsto \frac{\left(\left({\left(x \cdot x\right)}^{4} \cdot \left(\color{blue}{x \cdot \left(x \cdot 1.789971 \cdot 10^{-4}\right)} + 5.0640340000000002 \cdot 10^{-4}\right) + \left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + {x}^{6} \cdot 0.00726441819999999999\right)\right) + {x}^{4} \cdot 0.042406060400000001\right) \cdot x}{\left(\left(2 \cdot 1.789971 \cdot 10^{-4}\right) \cdot {\left(x \cdot x\right)}^{6} + \left(\left(x \cdot x\right) \cdot \left(0.77154710189999998 + 0.29097386390000002 \cdot \left(x \cdot x\right)\right) + 1\right)\right) + \left({x}^{6} \cdot 0.069455576099999999 + {\left(x \cdot x\right)}^{4} \cdot \left(0.014000544199999999 + \left(x \cdot x\right) \cdot 8.32794500000000044 \cdot 10^{-4}\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -16518388.9625579324 \lor \neg \left(x \le 613.008358988488794\right):\\ \;\;\;\;\frac{0.25141790006653753}{{x}^{3}} + \left(\frac{0.1529819634592933}{{x}^{5}} + \frac{0.5}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left({\left(x \cdot x\right)}^{4} \cdot \left(x \cdot \left(x \cdot 1.789971 \cdot 10^{-4}\right) + 5.0640340000000002 \cdot 10^{-4}\right) + \left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + {x}^{6} \cdot 0.00726441819999999999\right)\right) + {x}^{4} \cdot 0.042406060400000001\right) \cdot x}{\left(\left(2 \cdot 1.789971 \cdot 10^{-4}\right) \cdot {\left(x \cdot x\right)}^{6} + \left(\left(x \cdot x\right) \cdot \left(0.77154710189999998 + 0.29097386390000002 \cdot \left(x \cdot x\right)\right) + 1\right)\right) + \left({x}^{6} \cdot 0.069455576099999999 + {\left(x \cdot x\right)}^{4} \cdot \left(0.014000544199999999 + \left(x \cdot x\right) \cdot 8.32794500000000044 \cdot 10^{-4}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (x)
  :name "Jmat.Real.dawson"
  :precision binary64
  (* (/ (+ (+ (+ (+ (+ 1 (* 0.1049934947 (* x x))) (* 0.0424060604 (* (* x x) (* x x)))) (* 0.0072644182 (* (* (* x x) (* x x)) (* x x)))) (* 0.0005064034 (* (* (* (* x x) (* x x)) (* x x)) (* x x)))) (* 0.0001789971 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)))) (+ (+ (+ (+ (+ (+ 1 (* 0.7715471019 (* x x))) (* 0.2909738639 (* (* x x) (* x x)))) (* 0.0694555761 (* (* (* x x) (* x x)) (* x x)))) (* 0.0140005442 (* (* (* (* x x) (* x x)) (* x x)) (* x x)))) (* 0.0008327945 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)))) (* (* 2 0.0001789971) (* (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)) (* x x))))) x))