Average Error: 0.2 → 0.2
Time: 28.2s
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 r197315 = a;
        double r197316 = r197315 * r197315;
        double r197317 = b;
        double r197318 = r197317 * r197317;
        double r197319 = r197316 + r197318;
        double r197320 = 2.0;
        double r197321 = pow(r197319, r197320);
        double r197322 = 4.0;
        double r197323 = 1.0;
        double r197324 = r197323 - r197315;
        double r197325 = r197316 * r197324;
        double r197326 = 3.0;
        double r197327 = r197326 + r197315;
        double r197328 = r197318 * r197327;
        double r197329 = r197325 + r197328;
        double r197330 = r197322 * r197329;
        double r197331 = r197321 + r197330;
        double r197332 = r197331 - r197323;
        return r197332;
}
↓
double f(double a, double b) {
        double r197333 = a;
        double r197334 = r197333 * r197333;
        double r197335 = 1.0;
        double r197336 = r197335 - r197333;
        double r197337 = b;
        double r197338 = r197337 * r197337;
        double r197339 = 3.0;
        double r197340 = r197339 + r197333;
        double r197341 = r197338 * r197340;
        double r197342 = fma(r197334, r197336, r197341);
        double r197343 = 4.0;
        double r197344 = fma(r197333, r197333, r197338);
        double r197345 = 2.0;
        double r197346 = pow(r197344, r197345);
        double r197347 = fma(r197342, r197343, r197346);
        double r197348 = r197347 - r197335;
        return r197348;
}

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 2019325 +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))