Average Error: 0.2 → 0.5
Time: 20.6s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;b \cdot b \le 1.061383019026471481028624498784720164347 \cdot 10^{-13} \lor \neg \left(b \cdot b \le 911.531239174287975401966832578182220459\right):\\ \;\;\;\;\left({a}^{4} + \left({b}^{4} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{{\left({\left(a \cdot a + b \cdot b\right)}^{2}\right)}^{3}} + 4 \cdot \left(b \cdot b\right)\right) - 1\\ \end{array}\]
\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 1.061383019026471481028624498784720164347 \cdot 10^{-13} \lor \neg \left(b \cdot b \le 911.531239174287975401966832578182220459\right):\\
\;\;\;\;\left({a}^{4} + \left({b}^{4} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)\right) - 1\\

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

\end{array}
double f(double a, double b) {
        double r126478 = a;
        double r126479 = r126478 * r126478;
        double r126480 = b;
        double r126481 = r126480 * r126480;
        double r126482 = r126479 + r126481;
        double r126483 = 2.0;
        double r126484 = pow(r126482, r126483);
        double r126485 = 4.0;
        double r126486 = r126485 * r126481;
        double r126487 = r126484 + r126486;
        double r126488 = 1.0;
        double r126489 = r126487 - r126488;
        return r126489;
}
↓
double f(double a, double b) {
        double r126490 = b;
        double r126491 = r126490 * r126490;
        double r126492 = 1.0613830190264715e-13;
        bool r126493 = r126491 <= r126492;
        double r126494 = 911.531239174288;
        bool r126495 = r126491 <= r126494;
        double r126496 = !r126495;
        bool r126497 = r126493 || r126496;
        double r126498 = a;
        double r126499 = 4.0;
        double r126500 = pow(r126498, r126499);
        double r126501 = pow(r126490, r126499);
        double r126502 = 2.0;
        double r126503 = pow(r126498, r126502);
        double r126504 = pow(r126490, r126502);
        double r126505 = r126503 * r126504;
        double r126506 = r126502 * r126505;
        double r126507 = r126501 + r126506;
        double r126508 = r126500 + r126507;
        double r126509 = 1.0;
        double r126510 = r126508 - r126509;
        double r126511 = r126498 * r126498;
        double r126512 = r126511 + r126491;
        double r126513 = 2.0;
        double r126514 = pow(r126512, r126513);
        double r126515 = 3.0;
        double r126516 = pow(r126514, r126515);
        double r126517 = cbrt(r126516);
        double r126518 = 4.0;
        double r126519 = r126518 * r126491;
        double r126520 = r126517 + r126519;
        double r126521 = r126520 - r126509;
        double r126522 = r126497 ? r126510 : r126521;
        return r126522;
}

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) < 1.0613830190264715e-13 or 911.531239174288 < (* 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 0.3

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

    if 1.0613830190264715e-13 < (* b b) < 911.531239174288

    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. Using strategy rm

    3. Applied add-cbrt-cube9.0

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

    4. Simplified9.0

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \cdot b \le 1.061383019026471481028624498784720164347 \cdot 10^{-13} \lor \neg \left(b \cdot b \le 911.531239174287975401966832578182220459\right):\\ \;\;\;\;\left({a}^{4} + \left({b}^{4} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{{\left({\left(a \cdot a + b \cdot b\right)}^{2}\right)}^{3}} + 4 \cdot \left(b \cdot b\right)\right) - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))