Average Error: 0.2 → 0.2
Time: 5.9s
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 r333335 = a;
        double r333336 = r333335 * r333335;
        double r333337 = b;
        double r333338 = r333337 * r333337;
        double r333339 = r333336 + r333338;
        double r333340 = 2.0;
        double r333341 = pow(r333339, r333340);
        double r333342 = 4.0;
        double r333343 = 1.0;
        double r333344 = r333343 - r333335;
        double r333345 = r333336 * r333344;
        double r333346 = 3.0;
        double r333347 = r333346 + r333335;
        double r333348 = r333338 * r333347;
        double r333349 = r333345 + r333348;
        double r333350 = r333342 * r333349;
        double r333351 = r333341 + r333350;
        double r333352 = r333351 - r333343;
        return r333352;
}
↓
double f(double a, double b) {
        double r333353 = 4.0;
        double r333354 = a;
        double r333355 = r333354 * r333354;
        double r333356 = 1.0;
        double r333357 = r333356 - r333354;
        double r333358 = b;
        double r333359 = r333358 * r333358;
        double r333360 = 3.0;
        double r333361 = r333360 + r333354;
        double r333362 = r333359 * r333361;
        double r333363 = fma(r333355, r333357, r333362);
        double r333364 = r333355 + r333359;
        double r333365 = 2.0;
        double r333366 = pow(r333364, r333365);
        double r333367 = r333366 - r333356;
        double r333368 = fma(r333353, r333363, r333367);
        return r333368;
}

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 2020002 +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))