Jmat.Real.dawson

Average Error: 29.3 → 0.0

Time: 34.3s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -266320.39092243672 \lor \neg \left(x \le 4432.6458967592771\right):\\ \;\;\;\;\left(\frac{0.25141790006653753}{{x}^{3}} + \frac{0.5}{x}\right) + \frac{0.1529819634592933}{{x}^{5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(2 \cdot 1.789971 \cdot 10^{-4}\right) \cdot {x}^{12} + \left({x}^{8} \cdot 0.014000544199999999 + \left(\left(0.069455576099999999 \cdot {x}^{6} + \left(\left(1 + 0.77154710189999998 \cdot \left(x \cdot x\right)\right) + {x}^{4} \cdot 0.29097386390000002\right)\right) + 8.32794500000000044 \cdot 10^{-4} \cdot \left({x}^{6} \cdot {x}^{4}\right)\right)\right)}{\left(1.789971 \cdot 10^{-4} \cdot \left({x}^{6} \cdot {x}^{4}\right) + \left({x}^{6} \cdot 0.00726441819999999999 + \left({x}^{4} \cdot 0.042406060400000001 + \left(1 + \left(\sqrt[3]{0.1049934947 \cdot \left(x \cdot x\right)} \cdot \sqrt[3]{0.1049934947 \cdot \left(x \cdot x\right)}\right) \cdot \sqrt[3]{0.1049934947 \cdot \left(x \cdot x\right)}\right)\right)\right)\right) + 5.0640340000000002 \cdot 10^{-4} \cdot {x}^{8}}} \cdot x\\ \end{array}\]

C

double f(double x) {
        double r192499 = 1.0;
        double r192500 = 0.1049934947;
        double r192501 = x;
        double r192502 = r192501 * r192501;
        double r192503 = r192500 * r192502;
        double r192504 = r192499 + r192503;
        double r192505 = 0.0424060604;
        double r192506 = r192502 * r192502;
        double r192507 = r192505 * r192506;
        double r192508 = r192504 + r192507;
        double r192509 = 0.0072644182;
        double r192510 = r192506 * r192502;
        double r192511 = r192509 * r192510;
        double r192512 = r192508 + r192511;
        double r192513 = 0.0005064034;
        double r192514 = r192510 * r192502;
        double r192515 = r192513 * r192514;
        double r192516 = r192512 + r192515;
        double r192517 = 0.0001789971;
        double r192518 = r192514 * r192502;
        double r192519 = r192517 * r192518;
        double r192520 = r192516 + r192519;
        double r192521 = 0.7715471019;
        double r192522 = r192521 * r192502;
        double r192523 = r192499 + r192522;
        double r192524 = 0.2909738639;
        double r192525 = r192524 * r192506;
        double r192526 = r192523 + r192525;
        double r192527 = 0.0694555761;
        double r192528 = r192527 * r192510;
        double r192529 = r192526 + r192528;
        double r192530 = 0.0140005442;
        double r192531 = r192530 * r192514;
        double r192532 = r192529 + r192531;
        double r192533 = 0.0008327945;
        double r192534 = r192533 * r192518;
        double r192535 = r192532 + r192534;
        double r192536 = 2.0;
        double r192537 = r192536 * r192517;
        double r192538 = r192518 * r192502;
        double r192539 = r192537 * r192538;
        double r192540 = r192535 + r192539;
        double r192541 = r192520 / r192540;
        double r192542 = r192541 * r192501;
        return r192542;
}

↓

double f(double x) {
        double r192543 = x;
        double r192544 = -266320.3909224367;
        bool r192545 = r192543 <= r192544;
        double r192546 = 4432.645896759277;
        bool r192547 = r192543 <= r192546;
        double r192548 = !r192547;
        bool r192549 = r192545 || r192548;
        double r192550 = 0.2514179000665375;
        double r192551 = 3.0;
        double r192552 = pow(r192543, r192551);
        double r192553 = r192550 / r192552;
        double r192554 = 0.5;
        double r192555 = r192554 / r192543;
        double r192556 = r192553 + r192555;
        double r192557 = 0.15298196345929327;
        double r192558 = 5.0;
        double r192559 = pow(r192543, r192558);
        double r192560 = r192557 / r192559;
        double r192561 = r192556 + r192560;
        double r192562 = 1.0;
        double r192563 = 2.0;
        double r192564 = 0.0001789971;
        double r192565 = r192563 * r192564;
        double r192566 = 12.0;
        double r192567 = pow(r192543, r192566);
        double r192568 = r192565 * r192567;
        double r192569 = 8.0;
        double r192570 = pow(r192543, r192569);
        double r192571 = 0.0140005442;
        double r192572 = r192570 * r192571;
        double r192573 = 0.0694555761;
        double r192574 = 6.0;
        double r192575 = pow(r192543, r192574);
        double r192576 = r192573 * r192575;
        double r192577 = 1.0;
        double r192578 = 0.7715471019;
        double r192579 = r192543 * r192543;
        double r192580 = r192578 * r192579;
        double r192581 = r192577 + r192580;
        double r192582 = 4.0;
        double r192583 = pow(r192543, r192582);
        double r192584 = 0.2909738639;
        double r192585 = r192583 * r192584;
        double r192586 = r192581 + r192585;
        double r192587 = r192576 + r192586;
        double r192588 = 0.0008327945;
        double r192589 = r192575 * r192583;
        double r192590 = r192588 * r192589;
        double r192591 = r192587 + r192590;
        double r192592 = r192572 + r192591;
        double r192593 = r192568 + r192592;
        double r192594 = r192564 * r192589;
        double r192595 = 0.0072644182;
        double r192596 = r192575 * r192595;
        double r192597 = 0.0424060604;
        double r192598 = r192583 * r192597;
        double r192599 = 0.1049934947;
        double r192600 = r192599 * r192579;
        double r192601 = cbrt(r192600);
        double r192602 = r192601 * r192601;
        double r192603 = r192602 * r192601;
        double r192604 = r192577 + r192603;
        double r192605 = r192598 + r192604;
        double r192606 = r192596 + r192605;
        double r192607 = r192594 + r192606;
        double r192608 = 0.0005064034;
        double r192609 = r192608 * r192570;
        double r192610 = r192607 + r192609;
        double r192611 = r192593 / r192610;
        double r192612 = r192562 / r192611;
        double r192613 = r192612 * r192543;
        double r192614 = r192549 ? r192561 : r192613;
        return r192614;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -266320.3909224367 or 4432.645896759277 < x

   1. Initial program 59.2

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

   2. Taylor expanded around inf 0.0

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

   3. Simplified 0.0

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

   if -266320.3909224367 < x < 4432.645896759277

   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. Using strategy rm

   3. Applied clear-num 0.0

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

   4. Simplified 0.0

      \[\leadsto \frac{1}{\color{blue}{\frac{\left(2 \cdot 1.789971 \cdot 10^{-4}\right) \cdot {x}^{12} + \left({x}^{8} \cdot 0.014000544199999999 + \left(\left(0.069455576099999999 \cdot {x}^{6} + \left(\left(1 + 0.77154710189999998 \cdot \left(x \cdot x\right)\right) + {x}^{4} \cdot 0.29097386390000002\right)\right) + 8.32794500000000044 \cdot 10^{-4} \cdot \left({x}^{6} \cdot {x}^{4}\right)\right)\right)}{\left(1.789971 \cdot 10^{-4} \cdot \left({x}^{6} \cdot {x}^{4}\right) + \left({x}^{6} \cdot 0.00726441819999999999 + \left({x}^{4} \cdot 0.042406060400000001 + \left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right)\right)\right)\right) + 5.0640340000000002 \cdot 10^{-4} \cdot {x}^{8}}}} \cdot x\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 0.0

      \[\leadsto \frac{1}{\frac{\left(2 \cdot 1.789971 \cdot 10^{-4}\right) \cdot {x}^{12} + \left({x}^{8} \cdot 0.014000544199999999 + \left(\left(0.069455576099999999 \cdot {x}^{6} + \left(\left(1 + 0.77154710189999998 \cdot \left(x \cdot x\right)\right) + {x}^{4} \cdot 0.29097386390000002\right)\right) + 8.32794500000000044 \cdot 10^{-4} \cdot \left({x}^{6} \cdot {x}^{4}\right)\right)\right)}{\left(1.789971 \cdot 10^{-4} \cdot \left({x}^{6} \cdot {x}^{4}\right) + \left({x}^{6} \cdot 0.00726441819999999999 + \left({x}^{4} \cdot 0.042406060400000001 + \left(1 + \color{blue}{\left(\sqrt[3]{0.1049934947 \cdot \left(x \cdot x\right)} \cdot \sqrt[3]{0.1049934947 \cdot \left(x \cdot x\right)}\right) \cdot \sqrt[3]{0.1049934947 \cdot \left(x \cdot x\right)}}\right)\right)\right)\right) + 5.0640340000000002 \cdot 10^{-4} \cdot {x}^{8}}} \cdot x\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -266320.39092243672 \lor \neg \left(x \le 4432.6458967592771\right):\\ \;\;\;\;\left(\frac{0.25141790006653753}{{x}^{3}} + \frac{0.5}{x}\right) + \frac{0.1529819634592933}{{x}^{5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(2 \cdot 1.789971 \cdot 10^{-4}\right) \cdot {x}^{12} + \left({x}^{8} \cdot 0.014000544199999999 + \left(\left(0.069455576099999999 \cdot {x}^{6} + \left(\left(1 + 0.77154710189999998 \cdot \left(x \cdot x\right)\right) + {x}^{4} \cdot 0.29097386390000002\right)\right) + 8.32794500000000044 \cdot 10^{-4} \cdot \left({x}^{6} \cdot {x}^{4}\right)\right)\right)}{\left(1.789971 \cdot 10^{-4} \cdot \left({x}^{6} \cdot {x}^{4}\right) + \left({x}^{6} \cdot 0.00726441819999999999 + \left({x}^{4} \cdot 0.042406060400000001 + \left(1 + \left(\sqrt[3]{0.1049934947 \cdot \left(x \cdot x\right)} \cdot \sqrt[3]{0.1049934947 \cdot \left(x \cdot x\right)}\right) \cdot \sqrt[3]{0.1049934947 \cdot \left(x \cdot x\right)}\right)\right)\right)\right) + 5.0640340000000002 \cdot 10^{-4} \cdot {x}^{8}}} \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(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))