Average Error: 0.2 → 0.2
Time: 6.3s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\left(\mathsf{fma}\left({a}^{2}, \left(1 + a\right) \cdot 4, {\left(a \cdot a + b \cdot b\right)}^{2}\right) + \left(\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) \cdot 4\right) - 1\]
\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
↓
\left(\mathsf{fma}\left({a}^{2}, \left(1 + a\right) \cdot 4, {\left(a \cdot a + b \cdot b\right)}^{2}\right) + \left(\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right) \cdot 4\right) - 1
double f(double a, double b) {
        double r150304 = a;
        double r150305 = r150304 * r150304;
        double r150306 = b;
        double r150307 = r150306 * r150306;
        double r150308 = r150305 + r150307;
        double r150309 = 2.0;
        double r150310 = pow(r150308, r150309);
        double r150311 = 4.0;
        double r150312 = 1.0;
        double r150313 = r150312 + r150304;
        double r150314 = r150305 * r150313;
        double r150315 = 3.0;
        double r150316 = r150315 * r150304;
        double r150317 = r150312 - r150316;
        double r150318 = r150307 * r150317;
        double r150319 = r150314 + r150318;
        double r150320 = r150311 * r150319;
        double r150321 = r150310 + r150320;
        double r150322 = r150321 - r150312;
        return r150322;
}
↓
double f(double a, double b) {
        double r150323 = a;
        double r150324 = 2.0;
        double r150325 = pow(r150323, r150324);
        double r150326 = 1.0;
        double r150327 = r150326 + r150323;
        double r150328 = 4.0;
        double r150329 = r150327 * r150328;
        double r150330 = r150323 * r150323;
        double r150331 = b;
        double r150332 = r150331 * r150331;
        double r150333 = r150330 + r150332;
        double r150334 = 2.0;
        double r150335 = pow(r150333, r150334);
        double r150336 = fma(r150325, r150329, r150335);
        double r150337 = 3.0;
        double r150338 = r150337 * r150323;
        double r150339 = r150326 - r150338;
        double r150340 = r150332 * r150339;
        double r150341 = r150340 * r150328;
        double r150342 = r150336 + r150341;
        double r150343 = r150342 - r150326;
        return r150343;
}

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. Using strategy rm

  3. Applied distribute-rgt-in0.2

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

  4. Applied associate-+r+0.2

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2020064 +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))