Average Error: 0.2 → 0.2
Time: 23.5s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(4, \mathsf{fma}\left(1, \mathsf{fma}\left(a, a, {b}^{2}\right), {a}^{3}\right), {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1\]
\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
↓
\mathsf{fma}\left(4, \mathsf{fma}\left(1, \mathsf{fma}\left(a, a, {b}^{2}\right), {a}^{3}\right), {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1
double f(double a, double b) {
        double r183415 = a;
        double r183416 = r183415 * r183415;
        double r183417 = b;
        double r183418 = r183417 * r183417;
        double r183419 = r183416 + r183418;
        double r183420 = 2.0;
        double r183421 = pow(r183419, r183420);
        double r183422 = 4.0;
        double r183423 = 1.0;
        double r183424 = r183423 + r183415;
        double r183425 = r183416 * r183424;
        double r183426 = 3.0;
        double r183427 = r183426 * r183415;
        double r183428 = r183423 - r183427;
        double r183429 = r183418 * r183428;
        double r183430 = r183425 + r183429;
        double r183431 = r183422 * r183430;
        double r183432 = r183421 + r183431;
        double r183433 = r183432 - r183423;
        return r183433;
}
↓
double f(double a, double b) {
        double r183434 = 4.0;
        double r183435 = 1.0;
        double r183436 = a;
        double r183437 = b;
        double r183438 = 2.0;
        double r183439 = pow(r183437, r183438);
        double r183440 = fma(r183436, r183436, r183439);
        double r183441 = 3.0;
        double r183442 = pow(r183436, r183441);
        double r183443 = fma(r183435, r183440, r183442);
        double r183444 = r183437 * r183437;
        double r183445 = fma(r183436, r183436, r183444);
        double r183446 = 2.0;
        double r183447 = pow(r183445, r183446);
        double r183448 = fma(r183434, r183443, r183447);
        double r183449 = r183448 - r183435;
        return r183449;
}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 0.2

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

  2. Simplified0.2

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

  3. Taylor expanded around 0 0.2

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

  4. Simplified0.2

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

  5. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(4, \mathsf{fma}\left(1, \mathsf{fma}\left(a, a, {b}^{2}\right), {a}^{3}\right), {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))