Average Error: 0.2 → 0.2
Time: 28.6s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right) - 1\]
\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(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
double f(double a, double b) {
        double r107341 = a;
        double r107342 = r107341 * r107341;
        double r107343 = b;
        double r107344 = r107343 * r107343;
        double r107345 = r107342 + r107344;
        double r107346 = 2.0;
        double r107347 = pow(r107345, r107346);
        double r107348 = 4.0;
        double r107349 = 1.0;
        double r107350 = r107349 + r107341;
        double r107351 = r107342 * r107350;
        double r107352 = 3.0;
        double r107353 = r107352 * r107341;
        double r107354 = r107349 - r107353;
        double r107355 = r107344 * r107354;
        double r107356 = r107351 + r107355;
        double r107357 = r107348 * r107356;
        double r107358 = r107347 + r107357;
        double r107359 = r107358 - r107349;
        return r107359;
}
↓
double f(double a, double b) {
        double r107360 = 4.0;
        double r107361 = a;
        double r107362 = r107361 * r107361;
        double r107363 = 1.0;
        double r107364 = r107363 + r107361;
        double r107365 = b;
        double r107366 = r107365 * r107365;
        double r107367 = 3.0;
        double r107368 = r107367 * r107361;
        double r107369 = r107363 - r107368;
        double r107370 = r107366 * r107369;
        double r107371 = fma(r107362, r107364, r107370);
        double r107372 = fma(r107361, r107361, r107366);
        double r107373 = 2.0;
        double r107374 = pow(r107372, r107373);
        double r107375 = fma(r107360, r107371, r107374);
        double r107376 = r107375 - r107363;
        return r107376;
}

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. Final simplification0.2

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

Reproduce

herbie shell --seed 2019303 +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))