Jmat.Real.dawson

Average Error: 29.5 → 0.0

Time: 2.0m

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 -161298189120609567375360 \lor \neg \left(x \le 8124.998191315608892182353883981704711914\right):\\ \;\;\;\;\frac{0.5}{x} + \left(\frac{0.2514179000665375252054900556686334311962}{{x}^{3}} + \frac{0.1529819634592932686700805788859724998474}{{x}^{5}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\left(5.064034000000000243502107366566633572802 \cdot 10^{-4} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot x\right)\right) + \left(\left(\left(1 + 0.1049934946999999951788851149103720672429 \cdot \left(x \cdot x\right)\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.04240606040000000076517494562722276896238\right) + \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 0.007264418199999999985194687468492702464573\right)\right) + \left(\left(x \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}}{\left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot x\right)\right) + \left(\left(\left(x \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 8.327945000000000442749725770852364803432 \cdot 10^{-4} + \left(\left(0.06945557609999999937322456844412954524159 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.2909738639000000182122107617033179849386 + \left(1 + 0.7715471018999999763821051601553335785866 \cdot \left(x \cdot x\right)\right)\right)\right) + 0.01400054419999999938406531896362139377743 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \end{array}\]

C

double f(double x) {
        double r250248 = 1.0;
        double r250249 = 0.1049934947;
        double r250250 = x;
        double r250251 = r250250 * r250250;
        double r250252 = r250249 * r250251;
        double r250253 = r250248 + r250252;
        double r250254 = 0.0424060604;
        double r250255 = r250251 * r250251;
        double r250256 = r250254 * r250255;
        double r250257 = r250253 + r250256;
        double r250258 = 0.0072644182;
        double r250259 = r250255 * r250251;
        double r250260 = r250258 * r250259;
        double r250261 = r250257 + r250260;
        double r250262 = 0.0005064034;
        double r250263 = r250259 * r250251;
        double r250264 = r250262 * r250263;
        double r250265 = r250261 + r250264;
        double r250266 = 0.0001789971;
        double r250267 = r250263 * r250251;
        double r250268 = r250266 * r250267;
        double r250269 = r250265 + r250268;
        double r250270 = 0.7715471019;
        double r250271 = r250270 * r250251;
        double r250272 = r250248 + r250271;
        double r250273 = 0.2909738639;
        double r250274 = r250273 * r250255;
        double r250275 = r250272 + r250274;
        double r250276 = 0.0694555761;
        double r250277 = r250276 * r250259;
        double r250278 = r250275 + r250277;
        double r250279 = 0.0140005442;
        double r250280 = r250279 * r250263;
        double r250281 = r250278 + r250280;
        double r250282 = 0.0008327945;
        double r250283 = r250282 * r250267;
        double r250284 = r250281 + r250283;
        double r250285 = 2.0;
        double r250286 = r250285 * r250266;
        double r250287 = r250267 * r250251;
        double r250288 = r250286 * r250287;
        double r250289 = r250284 + r250288;
        double r250290 = r250269 / r250289;
        double r250291 = r250290 * r250250;
        return r250291;
}

↓

double f(double x) {
        double r250292 = x;
        double r250293 = -1.6129818912060957e+23;
        bool r250294 = r250292 <= r250293;
        double r250295 = 8124.998191315609;
        bool r250296 = r250292 <= r250295;
        double r250297 = !r250296;
        bool r250298 = r250294 || r250297;
        double r250299 = 0.5;
        double r250300 = r250299 / r250292;
        double r250301 = 0.2514179000665375;
        double r250302 = 3.0;
        double r250303 = pow(r250292, r250302);
        double r250304 = r250301 / r250303;
        double r250305 = 0.15298196345929327;
        double r250306 = 5.0;
        double r250307 = pow(r250292, r250306);
        double r250308 = r250305 / r250307;
        double r250309 = r250304 + r250308;
        double r250310 = r250300 + r250309;
        double r250311 = 0.0005064034;
        double r250312 = r250292 * r250292;
        double r250313 = r250312 * r250312;
        double r250314 = r250312 * r250313;
        double r250315 = r250314 * r250312;
        double r250316 = r250311 * r250315;
        double r250317 = 1.0;
        double r250318 = 0.1049934947;
        double r250319 = r250318 * r250312;
        double r250320 = r250317 + r250319;
        double r250321 = 0.0424060604;
        double r250322 = r250313 * r250321;
        double r250323 = r250320 + r250322;
        double r250324 = 0.0072644182;
        double r250325 = r250314 * r250324;
        double r250326 = r250323 + r250325;
        double r250327 = r250316 + r250326;
        double r250328 = r250312 * r250315;
        double r250329 = 0.0001789971;
        double r250330 = r250328 * r250329;
        double r250331 = r250327 + r250330;
        double r250332 = 2.0;
        double r250333 = r250332 * r250329;
        double r250334 = r250328 * r250312;
        double r250335 = r250333 * r250334;
        double r250336 = 0.0008327945;
        double r250337 = r250328 * r250336;
        double r250338 = 0.0694555761;
        double r250339 = r250338 * r250314;
        double r250340 = 0.2909738639;
        double r250341 = r250313 * r250340;
        double r250342 = 0.7715471019;
        double r250343 = r250342 * r250312;
        double r250344 = r250317 + r250343;
        double r250345 = r250341 + r250344;
        double r250346 = r250339 + r250345;
        double r250347 = 0.0140005442;
        double r250348 = r250347 * r250315;
        double r250349 = r250346 + r250348;
        double r250350 = r250337 + r250349;
        double r250351 = r250335 + r250350;
        double r250352 = r250331 / r250351;
        double r250353 = r250292 * r250352;
        double r250354 = r250298 ? r250310 : r250353;
        return r250354;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -1.6129818912060957e+23 or 8124.998191315609 < x

   1. Initial program 61.1

      \[\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

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

   3. Simplified 0.0

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

   if -1.6129818912060957e+23 < x < 8124.998191315609

   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 -161298189120609567375360 \lor \neg \left(x \le 8124.998191315608892182353883981704711914\right):\\ \;\;\;\;\frac{0.5}{x} + \left(\frac{0.2514179000665375252054900556686334311962}{{x}^{3}} + \frac{0.1529819634592932686700805788859724998474}{{x}^{5}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\left(5.064034000000000243502107366566633572802 \cdot 10^{-4} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot x\right)\right) + \left(\left(\left(1 + 0.1049934946999999951788851149103720672429 \cdot \left(x \cdot x\right)\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.04240606040000000076517494562722276896238\right) + \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 0.007264418199999999985194687468492702464573\right)\right) + \left(\left(x \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}}{\left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot x\right)\right) + \left(\left(\left(x \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 8.327945000000000442749725770852364803432 \cdot 10^{-4} + \left(\left(0.06945557609999999937322456844412954524159 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.2909738639000000182122107617033179849386 + \left(1 + 0.7715471018999999763821051601553335785866 \cdot \left(x \cdot x\right)\right)\right)\right) + 0.01400054419999999938406531896362139377743 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.dawson"
  (* (/ (+ (+ (+ (+ (+ 1.0 (* 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 (* 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 0.0001789971) (* (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)) (* x x))))) x))