Average Error: 0.2 → 0.9
Time: 23.5s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.70669310275382700610447006539516223711 \cdot 10^{-4} \lor \neg \left(b \le 2.006560387622814189967357378918677568436\right):\\ \;\;\;\;\left({a}^{4} + \mathsf{fma}\left(2, {a}^{2} \cdot {b}^{2}, {b}^{4}\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{-1}{a}\right)}^{-4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\\ \end{array}\]
\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1
↓
\begin{array}{l}
\mathbf{if}\;b \le -4.70669310275382700610447006539516223711 \cdot 10^{-4} \lor \neg \left(b \le 2.006560387622814189967357378918677568436\right):\\
\;\;\;\;\left({a}^{4} + \mathsf{fma}\left(2, {a}^{2} \cdot {b}^{2}, {b}^{4}\right)\right) - 1\\

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

\end{array}
double f(double a, double b) {
        double r87408 = a;
        double r87409 = r87408 * r87408;
        double r87410 = b;
        double r87411 = r87410 * r87410;
        double r87412 = r87409 + r87411;
        double r87413 = 2.0;
        double r87414 = pow(r87412, r87413);
        double r87415 = 4.0;
        double r87416 = 1.0;
        double r87417 = r87416 + r87408;
        double r87418 = r87409 * r87417;
        double r87419 = 3.0;
        double r87420 = r87419 * r87408;
        double r87421 = r87416 - r87420;
        double r87422 = r87411 * r87421;
        double r87423 = r87418 + r87422;
        double r87424 = r87415 * r87423;
        double r87425 = r87414 + r87424;
        double r87426 = r87425 - r87416;
        return r87426;
}
↓
double f(double a, double b) {
        double r87427 = b;
        double r87428 = -0.0004706693102753827;
        bool r87429 = r87427 <= r87428;
        double r87430 = 2.006560387622814;
        bool r87431 = r87427 <= r87430;
        double r87432 = !r87431;
        bool r87433 = r87429 || r87432;
        double r87434 = a;
        double r87435 = 4.0;
        double r87436 = pow(r87434, r87435);
        double r87437 = 2.0;
        double r87438 = pow(r87434, r87437);
        double r87439 = pow(r87427, r87437);
        double r87440 = r87438 * r87439;
        double r87441 = pow(r87427, r87435);
        double r87442 = fma(r87437, r87440, r87441);
        double r87443 = r87436 + r87442;
        double r87444 = 1.0;
        double r87445 = r87443 - r87444;
        double r87446 = -1.0;
        double r87447 = r87446 / r87434;
        double r87448 = -4.0;
        double r87449 = pow(r87447, r87448);
        double r87450 = 4.0;
        double r87451 = r87434 * r87434;
        double r87452 = r87444 + r87434;
        double r87453 = r87451 * r87452;
        double r87454 = r87427 * r87427;
        double r87455 = 3.0;
        double r87456 = r87455 * r87434;
        double r87457 = r87444 - r87456;
        double r87458 = r87454 * r87457;
        double r87459 = r87453 + r87458;
        double r87460 = r87450 * r87459;
        double r87461 = r87449 + r87460;
        double r87462 = r87461 - r87444;
        double r87463 = r87433 ? r87445 : r87462;
        return r87463;
}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Split input into 2 regimes

  2. if b < -0.0004706693102753827 or 2.006560387622814 < b

    1. Initial program 0.5

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

    2. Taylor expanded around inf 3.1

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

    3. Simplified3.1

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

    if -0.0004706693102753827 < b < 2.006560387622814

    1. Initial program 0.1

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

    2. Taylor expanded around -inf 0.4

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.70669310275382700610447006539516223711 \cdot 10^{-4} \lor \neg \left(b \le 2.006560387622814189967357378918677568436\right):\\ \;\;\;\;\left({a}^{4} + \mathsf{fma}\left(2, {a}^{2} \cdot {b}^{2}, {b}^{4}\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{-1}{a}\right)}^{-4} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (+ 1 a)) (* (* b b) (- 1 (* 3 a)))))) 1))