Average Error: 0.2 → 0.2
Time: 6.6s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\left(a \cdot a + b \cdot b\right)}^{2} - 1\right)\]
\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(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(a \cdot a + b \cdot b\right)}^{2} - 1\right)
double f(double a, double b) {
        double r128246 = a;
        double r128247 = r128246 * r128246;
        double r128248 = b;
        double r128249 = r128248 * r128248;
        double r128250 = r128247 + r128249;
        double r128251 = 2.0;
        double r128252 = pow(r128250, r128251);
        double r128253 = 4.0;
        double r128254 = 1.0;
        double r128255 = r128254 + r128246;
        double r128256 = r128247 * r128255;
        double r128257 = 3.0;
        double r128258 = r128257 * r128246;
        double r128259 = r128254 - r128258;
        double r128260 = r128249 * r128259;
        double r128261 = r128256 + r128260;
        double r128262 = r128253 * r128261;
        double r128263 = r128252 + r128262;
        double r128264 = r128263 - r128254;
        return r128264;
}
↓
double f(double a, double b) {
        double r128265 = 4.0;
        double r128266 = a;
        double r128267 = r128266 * r128266;
        double r128268 = 1.0;
        double r128269 = r128268 + r128266;
        double r128270 = b;
        double r128271 = r128270 * r128270;
        double r128272 = 3.0;
        double r128273 = r128272 * r128266;
        double r128274 = r128268 - r128273;
        double r128275 = r128271 * r128274;
        double r128276 = fma(r128267, r128269, r128275);
        double r128277 = r128267 + r128271;
        double r128278 = 2.0;
        double r128279 = pow(r128277, r128278);
        double r128280 = r128279 - r128268;
        double r128281 = fma(r128265, r128276, r128280);
        return r128281;
}

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

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2020001 +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))