Jmat.Real.dawson

Average Error: 29.6 → 0.0

Time: 1.2m

Precision: 64

Math

\[\frac{\left(\left(\left(\left(1 + 0.1049934946999999951788851149103720672429 \cdot \left(x \cdot x\right)\right) + 0.04240606040000000076517494562722276896238 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.007264418199999999985194687468492702464573 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 5.064034000000000243502107366566633572802 \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) + 1.789971000000000009994005623070734145585 \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 + 0.7715471018999999763821051601553335785866 \cdot \left(x \cdot x\right)\right) + 0.2909738639000000182122107617033179849386 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.06945557609999999937322456844412954524159 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.01400054419999999938406531896362139377743 \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) + 8.327945000000000442749725770852364803432 \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 1.789971000000000009994005623070734145585 \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 -17126785.86309130489826202392578125 \lor \neg \left(x \le 769.1144380026898943469859659671783447266\right):\\ \;\;\;\;\left(\frac{0.2514179000665375252054900556686334311962}{{x}^{3}} + \frac{0.5}{x}\right) + \frac{0.1529819634592932686700805788859724998474}{{x}^{5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(\left(1 + 0.1049934946999999951788851149103720672429 \cdot \left(x \cdot x\right)\right) + 0.04240606040000000076517494562722276896238 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.007264418199999999985194687468492702464573 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 5.064034000000000243502107366566633572802 \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) + 1.789971000000000009994005623070734145585 \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 + 0.7715471018999999763821051601553335785866 \cdot \left(x \cdot x\right)\right) + 0.2909738639000000182122107617033179849386 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.06945557609999999937322456844412954524159 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.01400054419999999938406531896362139377743 \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) + 8.327945000000000442749725770852364803432 \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 1.789971000000000009994005623070734145585 \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\\ \end{array}\]

C

double f(double x) {
        double r139162 = 1.0;
        double r139163 = 0.1049934947;
        double r139164 = x;
        double r139165 = r139164 * r139164;
        double r139166 = r139163 * r139165;
        double r139167 = r139162 + r139166;
        double r139168 = 0.0424060604;
        double r139169 = r139165 * r139165;
        double r139170 = r139168 * r139169;
        double r139171 = r139167 + r139170;
        double r139172 = 0.0072644182;
        double r139173 = r139169 * r139165;
        double r139174 = r139172 * r139173;
        double r139175 = r139171 + r139174;
        double r139176 = 0.0005064034;
        double r139177 = r139173 * r139165;
        double r139178 = r139176 * r139177;
        double r139179 = r139175 + r139178;
        double r139180 = 0.0001789971;
        double r139181 = r139177 * r139165;
        double r139182 = r139180 * r139181;
        double r139183 = r139179 + r139182;
        double r139184 = 0.7715471019;
        double r139185 = r139184 * r139165;
        double r139186 = r139162 + r139185;
        double r139187 = 0.2909738639;
        double r139188 = r139187 * r139169;
        double r139189 = r139186 + r139188;
        double r139190 = 0.0694555761;
        double r139191 = r139190 * r139173;
        double r139192 = r139189 + r139191;
        double r139193 = 0.0140005442;
        double r139194 = r139193 * r139177;
        double r139195 = r139192 + r139194;
        double r139196 = 0.0008327945;
        double r139197 = r139196 * r139181;
        double r139198 = r139195 + r139197;
        double r139199 = 2.0;
        double r139200 = r139199 * r139180;
        double r139201 = r139181 * r139165;
        double r139202 = r139200 * r139201;
        double r139203 = r139198 + r139202;
        double r139204 = r139183 / r139203;
        double r139205 = r139204 * r139164;
        return r139205;
}

↓

double f(double x) {
        double r139206 = x;
        double r139207 = -17126785.863091305;
        bool r139208 = r139206 <= r139207;
        double r139209 = 769.1144380026899;
        bool r139210 = r139206 <= r139209;
        double r139211 = !r139210;
        bool r139212 = r139208 || r139211;
        double r139213 = 0.2514179000665375;
        double r139214 = 3.0;
        double r139215 = pow(r139206, r139214);
        double r139216 = r139213 / r139215;
        double r139217 = 0.5;
        double r139218 = r139217 / r139206;
        double r139219 = r139216 + r139218;
        double r139220 = 0.15298196345929327;
        double r139221 = 5.0;
        double r139222 = pow(r139206, r139221);
        double r139223 = r139220 / r139222;
        double r139224 = r139219 + r139223;
        double r139225 = 1.0;
        double r139226 = 0.1049934947;
        double r139227 = r139206 * r139206;
        double r139228 = r139226 * r139227;
        double r139229 = r139225 + r139228;
        double r139230 = 0.0424060604;
        double r139231 = r139227 * r139227;
        double r139232 = r139230 * r139231;
        double r139233 = r139229 + r139232;
        double r139234 = 0.0072644182;
        double r139235 = r139231 * r139227;
        double r139236 = r139234 * r139235;
        double r139237 = r139233 + r139236;
        double r139238 = 0.0005064034;
        double r139239 = r139235 * r139227;
        double r139240 = r139238 * r139239;
        double r139241 = r139237 + r139240;
        double r139242 = 0.0001789971;
        double r139243 = r139239 * r139227;
        double r139244 = r139242 * r139243;
        double r139245 = r139241 + r139244;
        double r139246 = 0.7715471019;
        double r139247 = r139246 * r139227;
        double r139248 = r139225 + r139247;
        double r139249 = 0.2909738639;
        double r139250 = r139249 * r139231;
        double r139251 = r139248 + r139250;
        double r139252 = 0.0694555761;
        double r139253 = r139252 * r139235;
        double r139254 = r139251 + r139253;
        double r139255 = 0.0140005442;
        double r139256 = r139255 * r139239;
        double r139257 = r139254 + r139256;
        double r139258 = 0.0008327945;
        double r139259 = r139258 * r139243;
        double r139260 = r139257 + r139259;
        double r139261 = 2.0;
        double r139262 = r139261 * r139242;
        double r139263 = r139243 * r139227;
        double r139264 = r139262 * r139263;
        double r139265 = r139260 + r139264;
        double r139266 = r139245 / r139265;
        double r139267 = r139266 * r139206;
        double r139268 = r139212 ? r139224 : r139267;
        return r139268;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -17126785.863091305 or 769.1144380026899 < x

   1. Initial program 59.5

      \[\frac{\left(\left(\left(\left(1 + 0.1049934946999999951788851149103720672429 \cdot \left(x \cdot x\right)\right) + 0.04240606040000000076517494562722276896238 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.007264418199999999985194687468492702464573 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 5.064034000000000243502107366566633572802 \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) + 1.789971000000000009994005623070734145585 \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 + 0.7715471018999999763821051601553335785866 \cdot \left(x \cdot x\right)\right) + 0.2909738639000000182122107617033179849386 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.06945557609999999937322456844412954524159 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.01400054419999999938406531896362139377743 \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) + 8.327945000000000442749725770852364803432 \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 1.789971000000000009994005623070734145585 \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. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{0.2514179000665375252054900556686334311962 \cdot \frac{1}{{x}^{3}} + \left(0.1529819634592932686700805788859724998474 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right)}\]

   3. Simplified 0.0

      \[\leadsto \color{blue}{\left(\frac{0.2514179000665375252054900556686334311962}{{x}^{3}} + \frac{0.5}{x}\right) + \frac{0.1529819634592932686700805788859724998474}{{x}^{5}}}\]

   if -17126785.863091305 < x < 769.1144380026899

   1. Initial program 0.0

      \[\frac{\left(\left(\left(\left(1 + 0.1049934946999999951788851149103720672429 \cdot \left(x \cdot x\right)\right) + 0.04240606040000000076517494562722276896238 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.007264418199999999985194687468492702464573 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 5.064034000000000243502107366566633572802 \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) + 1.789971000000000009994005623070734145585 \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 + 0.7715471018999999763821051601553335785866 \cdot \left(x \cdot x\right)\right) + 0.2909738639000000182122107617033179849386 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.06945557609999999937322456844412954524159 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.01400054419999999938406531896362139377743 \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) + 8.327945000000000442749725770852364803432 \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 1.789971000000000009994005623070734145585 \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\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -17126785.86309130489826202392578125 \lor \neg \left(x \le 769.1144380026898943469859659671783447266\right):\\ \;\;\;\;\left(\frac{0.2514179000665375252054900556686334311962}{{x}^{3}} + \frac{0.5}{x}\right) + \frac{0.1529819634592932686700805788859724998474}{{x}^{5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(\left(1 + 0.1049934946999999951788851149103720672429 \cdot \left(x \cdot x\right)\right) + 0.04240606040000000076517494562722276896238 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.007264418199999999985194687468492702464573 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 5.064034000000000243502107366566633572802 \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) + 1.789971000000000009994005623070734145585 \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 + 0.7715471018999999763821051601553335785866 \cdot \left(x \cdot x\right)\right) + 0.2909738639000000182122107617033179849386 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.06945557609999999937322456844412954524159 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.01400054419999999938406531896362139377743 \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) + 8.327945000000000442749725770852364803432 \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 1.789971000000000009994005623070734145585 \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\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.dawson"
  :precision binary64
  (* (/ (+ (+ (+ (+ (+ 1 (* 0.1049934947 (* x x))) (* 0.042406060400000001 (* (* x x) (* x x)))) (* 0.00726441819999999999 (* (* (* x x) (* x x)) (* x x)))) (* 5.0640340000000002e-4 (* (* (* (* x x) (* x x)) (* x x)) (* x x)))) (* 1.789971e-4 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)))) (+ (+ (+ (+ (+ (+ 1 (* 0.77154710189999998 (* x x))) (* 0.29097386390000002 (* (* x x) (* x x)))) (* 0.069455576099999999 (* (* (* x x) (* x x)) (* x x)))) (* 0.014000544199999999 (* (* (* (* x x) (* x x)) (* x x)) (* x x)))) (* 8.32794500000000044e-4 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)))) (* (* 2 1.789971e-4) (* (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)) (* x x))))) x))