Average Error: 0.2 → 0.2
Time: 25.2s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(3 + a\right)\right), 4, {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1\]
\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(3 + a\right)\right)\right) - 1
↓
\mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, 1 - a, \left(b \cdot b\right) \cdot \left(3 + a\right)\right), 4, {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1
double f(double a, double b) {
        double r292472 = a;
        double r292473 = r292472 * r292472;
        double r292474 = b;
        double r292475 = r292474 * r292474;
        double r292476 = r292473 + r292475;
        double r292477 = 2.0;
        double r292478 = pow(r292476, r292477);
        double r292479 = 4.0;
        double r292480 = 1.0;
        double r292481 = r292480 - r292472;
        double r292482 = r292473 * r292481;
        double r292483 = 3.0;
        double r292484 = r292483 + r292472;
        double r292485 = r292475 * r292484;
        double r292486 = r292482 + r292485;
        double r292487 = r292479 * r292486;
        double r292488 = r292478 + r292487;
        double r292489 = r292488 - r292480;
        return r292489;
}
↓
double f(double a, double b) {
        double r292490 = a;
        double r292491 = r292490 * r292490;
        double r292492 = 1.0;
        double r292493 = r292492 - r292490;
        double r292494 = b;
        double r292495 = r292494 * r292494;
        double r292496 = 3.0;
        double r292497 = r292496 + r292490;
        double r292498 = r292495 * r292497;
        double r292499 = fma(r292491, r292493, r292498);
        double r292500 = 4.0;
        double r292501 = fma(r292490, r292490, r292495);
        double r292502 = 2.0;
        double r292503 = pow(r292501, r292502);
        double r292504 = fma(r292499, r292500, r292503);
        double r292505 = r292504 - r292492;
        return r292505;
}

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

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2019303 +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))