Jmat.Real.dawson

Average Error: 29.8 → 0.0

Time: 1.0m

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\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1019.119841081965660123387351632118225098 \lor \neg \left(x \le 57958238.795458965003490447998046875\right):\\ \;\;\;\;\frac{0.5}{x} + \left(\frac{0.2514179000665375252054900556686334311962}{{x}^{3}} + \frac{0.1529819634592932686700805788859724998474}{{x}^{5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(1.789971000000000009994005623070734145585 \cdot 10^{-4}, {x}^{6} \cdot {x}^{4}, \mathsf{fma}\left({x}^{8}, 5.064034000000000243502107366566633572802 \cdot 10^{-4}, \mathsf{fma}\left(0.007264418199999999985194687468492702464573, {x}^{6}, \mathsf{fma}\left({x}^{4}, 0.04240606040000000076517494562722276896238, \mathsf{fma}\left(0.1049934946999999951788851149103720672429, x \cdot x, 1\right)\right)\right)\right)\right)}{\mathsf{fma}\left(1.789971000000000009994005623070734145585 \cdot 10^{-4}, 2 \cdot {\left({x}^{4}\right)}^{3}, \mathsf{fma}\left({x}^{6} \cdot {x}^{4}, 8.327945000000000442749725770852364803432 \cdot 10^{-4}, \mathsf{fma}\left(0.01400054419999999938406531896362139377743, {x}^{8}, \mathsf{fma}\left(0.06945557609999999937322456844412954524159, {x}^{6}, \mathsf{fma}\left(0.2909738639000000182122107617033179849386, {x}^{4}, \mathsf{fma}\left(0.7715471018999999763821051601553335785866 \cdot x, x, 1\right)\right)\right)\right)\right)\right)}\right)\right) \cdot x\\ \end{array}\]

C

double f(double x) {
        double r116192 = 1.0;
        double r116193 = 0.1049934947;
        double r116194 = x;
        double r116195 = r116194 * r116194;
        double r116196 = r116193 * r116195;
        double r116197 = r116192 + r116196;
        double r116198 = 0.0424060604;
        double r116199 = r116195 * r116195;
        double r116200 = r116198 * r116199;
        double r116201 = r116197 + r116200;
        double r116202 = 0.0072644182;
        double r116203 = r116199 * r116195;
        double r116204 = r116202 * r116203;
        double r116205 = r116201 + r116204;
        double r116206 = 0.0005064034;
        double r116207 = r116203 * r116195;
        double r116208 = r116206 * r116207;
        double r116209 = r116205 + r116208;
        double r116210 = 0.0001789971;
        double r116211 = r116207 * r116195;
        double r116212 = r116210 * r116211;
        double r116213 = r116209 + r116212;
        double r116214 = 0.7715471019;
        double r116215 = r116214 * r116195;
        double r116216 = r116192 + r116215;
        double r116217 = 0.2909738639;
        double r116218 = r116217 * r116199;
        double r116219 = r116216 + r116218;
        double r116220 = 0.0694555761;
        double r116221 = r116220 * r116203;
        double r116222 = r116219 + r116221;
        double r116223 = 0.0140005442;
        double r116224 = r116223 * r116207;
        double r116225 = r116222 + r116224;
        double r116226 = 0.0008327945;
        double r116227 = r116226 * r116211;
        double r116228 = r116225 + r116227;
        double r116229 = 2.0;
        double r116230 = r116229 * r116210;
        double r116231 = r116211 * r116195;
        double r116232 = r116230 * r116231;
        double r116233 = r116228 + r116232;
        double r116234 = r116213 / r116233;
        double r116235 = r116234 * r116194;
        return r116235;
}

↓

double f(double x) {
        double r116236 = x;
        double r116237 = -1019.1198410819657;
        bool r116238 = r116236 <= r116237;
        double r116239 = 57958238.795458965;
        bool r116240 = r116236 <= r116239;
        double r116241 = !r116240;
        bool r116242 = r116238 || r116241;
        double r116243 = 0.5;
        double r116244 = r116243 / r116236;
        double r116245 = 0.2514179000665375;
        double r116246 = 3.0;
        double r116247 = pow(r116236, r116246);
        double r116248 = r116245 / r116247;
        double r116249 = 0.15298196345929327;
        double r116250 = 5.0;
        double r116251 = pow(r116236, r116250);
        double r116252 = r116249 / r116251;
        double r116253 = r116248 + r116252;
        double r116254 = r116244 + r116253;
        double r116255 = 0.0001789971;
        double r116256 = 6.0;
        double r116257 = pow(r116236, r116256);
        double r116258 = 4.0;
        double r116259 = pow(r116236, r116258);
        double r116260 = r116257 * r116259;
        double r116261 = 8.0;
        double r116262 = pow(r116236, r116261);
        double r116263 = 0.0005064034;
        double r116264 = 0.0072644182;
        double r116265 = 0.0424060604;
        double r116266 = 0.1049934947;
        double r116267 = r116236 * r116236;
        double r116268 = 1.0;
        double r116269 = fma(r116266, r116267, r116268);
        double r116270 = fma(r116259, r116265, r116269);
        double r116271 = fma(r116264, r116257, r116270);
        double r116272 = fma(r116262, r116263, r116271);
        double r116273 = fma(r116255, r116260, r116272);
        double r116274 = 2.0;
        double r116275 = pow(r116259, r116246);
        double r116276 = r116274 * r116275;
        double r116277 = 0.0008327945;
        double r116278 = 0.0140005442;
        double r116279 = 0.0694555761;
        double r116280 = 0.2909738639;
        double r116281 = 0.7715471019;
        double r116282 = r116281 * r116236;
        double r116283 = fma(r116282, r116236, r116268);
        double r116284 = fma(r116280, r116259, r116283);
        double r116285 = fma(r116279, r116257, r116284);
        double r116286 = fma(r116278, r116262, r116285);
        double r116287 = fma(r116260, r116277, r116286);
        double r116288 = fma(r116255, r116276, r116287);
        double r116289 = r116273 / r116288;
        double r116290 = log1p(r116289);
        double r116291 = expm1(r116290);
        double r116292 = r116291 * r116236;
        double r116293 = r116242 ? r116254 : r116292;
        return r116293;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -1019.1198410819657 or 57958238.795458965 < x

   1. Initial program 59.9

      \[\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

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

   if -1019.1198410819657 < x < 57958238.795458965

   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\]
   2. Using strategy rm

   3. Applied expm1-log1p-u 0.0

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\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)}\right)\right)} \cdot x\]

   4. Simplified 0.0

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(1.789971000000000009994005623070734145585 \cdot 10^{-4}, {x}^{6} \cdot {x}^{4}, \mathsf{fma}\left({x}^{8}, 5.064034000000000243502107366566633572802 \cdot 10^{-4}, \mathsf{fma}\left(0.007264418199999999985194687468492702464573, {x}^{6}, \mathsf{fma}\left({x}^{4}, 0.04240606040000000076517494562722276896238, \mathsf{fma}\left(0.1049934946999999951788851149103720672429, x \cdot x, 1\right)\right)\right)\right)\right)}{\mathsf{fma}\left(1.789971000000000009994005623070734145585 \cdot 10^{-4}, 2 \cdot {\left({x}^{4}\right)}^{3}, \mathsf{fma}\left({x}^{6} \cdot {x}^{4}, 8.327945000000000442749725770852364803432 \cdot 10^{-4}, \mathsf{fma}\left(0.01400054419999999938406531896362139377743, {x}^{8}, \mathsf{fma}\left(0.06945557609999999937322456844412954524159, {x}^{6}, \mathsf{fma}\left(0.2909738639000000182122107617033179849386, {x}^{4}, \mathsf{fma}\left(0.7715471018999999763821051601553335785866 \cdot x, x, 1\right)\right)\right)\right)\right)\right)}\right)}\right) \cdot x\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1019.119841081965660123387351632118225098 \lor \neg \left(x \le 57958238.795458965003490447998046875\right):\\ \;\;\;\;\frac{0.5}{x} + \left(\frac{0.2514179000665375252054900556686334311962}{{x}^{3}} + \frac{0.1529819634592932686700805788859724998474}{{x}^{5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(1.789971000000000009994005623070734145585 \cdot 10^{-4}, {x}^{6} \cdot {x}^{4}, \mathsf{fma}\left({x}^{8}, 5.064034000000000243502107366566633572802 \cdot 10^{-4}, \mathsf{fma}\left(0.007264418199999999985194687468492702464573, {x}^{6}, \mathsf{fma}\left({x}^{4}, 0.04240606040000000076517494562722276896238, \mathsf{fma}\left(0.1049934946999999951788851149103720672429, x \cdot x, 1\right)\right)\right)\right)\right)}{\mathsf{fma}\left(1.789971000000000009994005623070734145585 \cdot 10^{-4}, 2 \cdot {\left({x}^{4}\right)}^{3}, \mathsf{fma}\left({x}^{6} \cdot {x}^{4}, 8.327945000000000442749725770852364803432 \cdot 10^{-4}, \mathsf{fma}\left(0.01400054419999999938406531896362139377743, {x}^{8}, \mathsf{fma}\left(0.06945557609999999937322456844412954524159, {x}^{6}, \mathsf{fma}\left(0.2909738639000000182122107617033179849386, {x}^{4}, \mathsf{fma}\left(0.7715471018999999763821051601553335785866 \cdot x, x, 1\right)\right)\right)\right)\right)\right)}\right)\right) \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(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))