Average Error: 0.2 → 1.5
Time: 5.3s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot a \le 5.308167373422853179984591298240806742507 \cdot 10^{-16}:\\ \;\;\;\;\left(\left({b}^{4} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left({a}^{4} + 4 \cdot {b}^{2}\right) - 1\\ \end{array}\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(b \cdot b\right)\right) - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot a \le 5.308167373422853179984591298240806742507 \cdot 10^{-16}:\\
\;\;\;\;\left(\left({b}^{4} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1\\

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

\end{array}
double f(double a, double b) {
        double r339543 = a;
        double r339544 = r339543 * r339543;
        double r339545 = b;
        double r339546 = r339545 * r339545;
        double r339547 = r339544 + r339546;
        double r339548 = 2.0;
        double r339549 = pow(r339547, r339548);
        double r339550 = 4.0;
        double r339551 = r339550 * r339546;
        double r339552 = r339549 + r339551;
        double r339553 = 1.0;
        double r339554 = r339552 - r339553;
        return r339554;
}
↓
double f(double a, double b) {
        double r339555 = a;
        double r339556 = r339555 * r339555;
        double r339557 = 5.308167373422853e-16;
        bool r339558 = r339556 <= r339557;
        double r339559 = b;
        double r339560 = 4.0;
        double r339561 = pow(r339559, r339560);
        double r339562 = 2.0;
        double r339563 = 2.0;
        double r339564 = pow(r339555, r339563);
        double r339565 = pow(r339559, r339563);
        double r339566 = r339564 * r339565;
        double r339567 = r339562 * r339566;
        double r339568 = r339561 + r339567;
        double r339569 = 4.0;
        double r339570 = r339559 * r339559;
        double r339571 = r339569 * r339570;
        double r339572 = r339568 + r339571;
        double r339573 = 1.0;
        double r339574 = r339572 - r339573;
        double r339575 = pow(r339555, r339560);
        double r339576 = r339569 * r339565;
        double r339577 = r339575 + r339576;
        double r339578 = r339577 - r339573;
        double r339579 = r339558 ? r339574 : r339578;
        return r339579;
}

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 (* a a) < 5.308167373422853e-16

    1. Initial program 0.1

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

    2. Taylor expanded around 0 0.0

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

    if 5.308167373422853e-16 < (* a a)

    1. Initial program 0.5

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

    2. Taylor expanded around 0 6.8

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \le 5.308167373422853179984591298240806742507 \cdot 10^{-16}:\\ \;\;\;\;\left(\left({b}^{4} + 2 \cdot \left({a}^{2} \cdot {b}^{2}\right)\right) + 4 \cdot \left(b \cdot b\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left({a}^{4} + 4 \cdot {b}^{2}\right) - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019346 
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (26)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (* b b))) 1))