Average Error: 0.2 → 1.5
Time: 25.0s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -0.003108479216049080465222864688712434144691 \lor \neg \left(a \le 8.829970458282619509299965487869599201076 \cdot 10^{-5}\right):\\ \;\;\;\;\sqrt{\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\left(\frac{-1}{a}\right)}^{-4}\right)} \cdot \sqrt{\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\left(\frac{-1}{a}\right)}^{-4}\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(2, {a}^{2} \cdot {b}^{2}, {b}^{4}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\\ \end{array}\]
\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}\;a \le -0.003108479216049080465222864688712434144691 \lor \neg \left(a \le 8.829970458282619509299965487869599201076 \cdot 10^{-5}\right):\\
\;\;\;\;\sqrt{\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\left(\frac{-1}{a}\right)}^{-4}\right)} \cdot \sqrt{\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\left(\frac{-1}{a}\right)}^{-4}\right)} - 1\\

\mathbf{else}:\\
\;\;\;\;\left(\mathsf{fma}\left(2, {a}^{2} \cdot {b}^{2}, {b}^{4}\right) + 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 r74547 = a;
        double r74548 = r74547 * r74547;
        double r74549 = b;
        double r74550 = r74549 * r74549;
        double r74551 = r74548 + r74550;
        double r74552 = 2.0;
        double r74553 = pow(r74551, r74552);
        double r74554 = 4.0;
        double r74555 = 1.0;
        double r74556 = r74555 + r74547;
        double r74557 = r74548 * r74556;
        double r74558 = 3.0;
        double r74559 = r74558 * r74547;
        double r74560 = r74555 - r74559;
        double r74561 = r74550 * r74560;
        double r74562 = r74557 + r74561;
        double r74563 = r74554 * r74562;
        double r74564 = r74553 + r74563;
        double r74565 = r74564 - r74555;
        return r74565;
}
↓
double f(double a, double b) {
        double r74566 = a;
        double r74567 = -0.0031084792160490805;
        bool r74568 = r74566 <= r74567;
        double r74569 = 8.82997045828262e-05;
        bool r74570 = r74566 <= r74569;
        double r74571 = !r74570;
        bool r74572 = r74568 || r74571;
        double r74573 = 4.0;
        double r74574 = r74566 * r74566;
        double r74575 = 1.0;
        double r74576 = r74575 + r74566;
        double r74577 = b;
        double r74578 = r74577 * r74577;
        double r74579 = 3.0;
        double r74580 = r74579 * r74566;
        double r74581 = r74575 - r74580;
        double r74582 = r74578 * r74581;
        double r74583 = fma(r74574, r74576, r74582);
        double r74584 = -1.0;
        double r74585 = r74584 / r74566;
        double r74586 = -4.0;
        double r74587 = pow(r74585, r74586);
        double r74588 = fma(r74573, r74583, r74587);
        double r74589 = sqrt(r74588);
        double r74590 = r74589 * r74589;
        double r74591 = r74590 - r74575;
        double r74592 = 2.0;
        double r74593 = 2.0;
        double r74594 = pow(r74566, r74593);
        double r74595 = pow(r74577, r74593);
        double r74596 = r74594 * r74595;
        double r74597 = 4.0;
        double r74598 = pow(r74577, r74597);
        double r74599 = fma(r74592, r74596, r74598);
        double r74600 = r74574 * r74576;
        double r74601 = r74600 + r74582;
        double r74602 = r74573 * r74601;
        double r74603 = r74599 + r74602;
        double r74604 = r74603 - r74575;
        double r74605 = r74572 ? r74591 : r74604;
        return r74605;
}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Split input into 2 regimes

  2. if a < -0.0031084792160490805 or 8.82997045828262e-05 < a

    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 7.5

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

    4. Applied add-sqr-sqrt7.6

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

    5. Simplified7.6

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

    6. Simplified7.6

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

    if -0.0031084792160490805 < a < 8.82997045828262e-05

    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 0 0.0

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

    3. Simplified0.0

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -0.003108479216049080465222864688712434144691 \lor \neg \left(a \le 8.829970458282619509299965487869599201076 \cdot 10^{-5}\right):\\ \;\;\;\;\sqrt{\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\left(\frac{-1}{a}\right)}^{-4}\right)} \cdot \sqrt{\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\left(\frac{-1}{a}\right)}^{-4}\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(2, {a}^{2} \cdot {b}^{2}, {b}^{4}\right) + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +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))