Average Error: 0.2 → 0.7
Time: 20.2s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;b \cdot b \le 1.02236235983317489029397541438357559891 \cdot 10^{-37} \lor \neg \left(b \cdot b \le 15928146782686.330078125\right):\\ \;\;\;\;\left({a}^{4} + \left({b}^{4} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left({b}^{4} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right) + \left(\sqrt[3]{4 \cdot \left(b \cdot b\right)} \cdot \sqrt[3]{4 \cdot \left(b \cdot b\right)}\right) \cdot \sqrt[3]{4 \cdot \left(b \cdot b\right)}\right) - 1\\ \end{array}\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\begin{array}{l}
\mathbf{if}\;b \cdot b \le 1.02236235983317489029397541438357559891 \cdot 10^{-37} \lor \neg \left(b \cdot b \le 15928146782686.330078125\right):\\
\;\;\;\;\left({a}^{4} + \left({b}^{4} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)\right) - 1\\

\mathbf{else}:\\
\;\;\;\;\left(\left({b}^{4} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right) + \left(\sqrt[3]{4 \cdot \left(b \cdot b\right)} \cdot \sqrt[3]{4 \cdot \left(b \cdot b\right)}\right) \cdot \sqrt[3]{4 \cdot \left(b \cdot b\right)}\right) - 1\\

\end{array}
double f(double a, double b) {
        double r136471 = a;
        double r136472 = r136471 * r136471;
        double r136473 = b;
        double r136474 = r136473 * r136473;
        double r136475 = r136472 + r136474;
        double r136476 = 2.0;
        double r136477 = pow(r136475, r136476);
        double r136478 = 4.0;
        double r136479 = r136478 * r136474;
        double r136480 = r136477 + r136479;
        double r136481 = 1.0;
        double r136482 = r136480 - r136481;
        return r136482;
}
↓
double f(double a, double b) {
        double r136483 = b;
        double r136484 = r136483 * r136483;
        double r136485 = 1.0223623598331749e-37;
        bool r136486 = r136484 <= r136485;
        double r136487 = 15928146782686.33;
        bool r136488 = r136484 <= r136487;
        double r136489 = !r136488;
        bool r136490 = r136486 || r136489;
        double r136491 = a;
        double r136492 = 4.0;
        double r136493 = pow(r136491, r136492);
        double r136494 = pow(r136483, r136492);
        double r136495 = 2.0;
        double r136496 = pow(r136491, r136495);
        double r136497 = pow(r136483, r136495);
        double r136498 = r136496 * r136497;
        double r136499 = r136495 * r136498;
        double r136500 = r136494 + r136499;
        double r136501 = r136493 + r136500;
        double r136502 = 1.0;
        double r136503 = r136501 - r136502;
        double r136504 = 2.0;
        double r136505 = r136504 * r136498;
        double r136506 = r136494 + r136505;
        double r136507 = 4.0;
        double r136508 = r136507 * r136484;
        double r136509 = cbrt(r136508);
        double r136510 = r136509 * r136509;
        double r136511 = r136510 * r136509;
        double r136512 = r136506 + r136511;
        double r136513 = r136512 - r136502;
        double r136514 = r136490 ? r136503 : r136513;
        return r136514;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (* b b) < 1.0223623598331749e-37 or 15928146782686.33 < (* b b)

    1. Initial program 0.2

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left({a}^{4} + \left({b}^{4} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)\right)} - 1\]

    if 1.0223623598331749e-37 < (* b b) < 15928146782686.33

    1. Initial program 0.2

      \[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]

    2. Taylor expanded around 0 10.6

      \[\leadsto \left(\color{blue}{\left({b}^{4} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
    3. Using strategy rm

    4. Applied add-cube-cbrt10.6

      \[\leadsto \left(\left({b}^{4} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right) + \color{blue}{\left(\sqrt[3]{4 \cdot \left(b \cdot b\right)} \cdot \sqrt[3]{4 \cdot \left(b \cdot b\right)}\right) \cdot \sqrt[3]{4 \cdot \left(b \cdot b\right)}}\right) - 1\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \le 1.02236235983317489029397541438357559891 \cdot 10^{-37} \lor \neg \left(b \cdot b \le 15928146782686.330078125\right):\\ \;\;\;\;\left({a}^{4} + \left({b}^{4} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left({b}^{4} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right) + \left(\sqrt[3]{4 \cdot \left(b \cdot b\right)} \cdot \sqrt[3]{4 \cdot \left(b \cdot b\right)}\right) \cdot \sqrt[3]{4 \cdot \left(b \cdot b\right)}\right) - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))