Average Error: 0.2 → 0.2
Time: 6.1s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(3 + a\right)\right), {\left(a \cdot a + b \cdot b\right)}^{2} - 1\right)\]
\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(3 + a\right)\right)\right) - 1
↓
\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(3 + a\right)\right), {\left(a \cdot a + b \cdot b\right)}^{2} - 1\right)
double f(double a, double b) {
        double r338508 = a;
        double r338509 = r338508 * r338508;
        double r338510 = b;
        double r338511 = r338510 * r338510;
        double r338512 = r338509 + r338511;
        double r338513 = 2.0;
        double r338514 = pow(r338512, r338513);
        double r338515 = 4.0;
        double r338516 = 1.0;
        double r338517 = r338516 - r338508;
        double r338518 = r338509 * r338517;
        double r338519 = 3.0;
        double r338520 = r338519 + r338508;
        double r338521 = r338511 * r338520;
        double r338522 = r338518 + r338521;
        double r338523 = r338515 * r338522;
        double r338524 = r338514 + r338523;
        double r338525 = r338524 - r338516;
        return r338525;
}
↓
double f(double a, double b) {
        double r338526 = 4.0;
        double r338527 = a;
        double r338528 = r338527 * r338527;
        double r338529 = 1.0;
        double r338530 = r338529 - r338527;
        double r338531 = b;
        double r338532 = r338531 * r338531;
        double r338533 = 3.0;
        double r338534 = r338533 + r338527;
        double r338535 = r338532 * r338534;
        double r338536 = fma(r338528, r338530, r338535);
        double r338537 = r338528 + r338532;
        double r338538 = 2.0;
        double r338539 = pow(r338537, r338538);
        double r338540 = r338539 - r338529;
        double r338541 = fma(r338526, r338536, r338540);
        return r338541;
}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 0.2

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

  2. Simplified0.2

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2020021 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  :precision binary64
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1))