Average Error: 0.2 → 0.2
Time: 23.9s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\sqrt{\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right)} \cdot \sqrt{\mathsf{fma}\left(4, \mathsf{fma}\left(a \cdot a, 1 + a, \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right), {\left(\mathsf{fma}\left(a, a, b \cdot b\right)\right)}^{2}\right)} - 1\]
\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
↓
\sqrt{\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)} \cdot \sqrt{\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 r126030 = a;
        double r126031 = r126030 * r126030;
        double r126032 = b;
        double r126033 = r126032 * r126032;
        double r126034 = r126031 + r126033;
        double r126035 = 2.0;
        double r126036 = pow(r126034, r126035);
        double r126037 = 4.0;
        double r126038 = 1.0;
        double r126039 = r126038 + r126030;
        double r126040 = r126031 * r126039;
        double r126041 = 3.0;
        double r126042 = r126041 * r126030;
        double r126043 = r126038 - r126042;
        double r126044 = r126033 * r126043;
        double r126045 = r126040 + r126044;
        double r126046 = r126037 * r126045;
        double r126047 = r126036 + r126046;
        double r126048 = r126047 - r126038;
        return r126048;
}
↓
double f(double a, double b) {
        double r126049 = 4.0;
        double r126050 = a;
        double r126051 = r126050 * r126050;
        double r126052 = 1.0;
        double r126053 = r126052 + r126050;
        double r126054 = b;
        double r126055 = r126054 * r126054;
        double r126056 = 3.0;
        double r126057 = r126056 * r126050;
        double r126058 = r126052 - r126057;
        double r126059 = r126055 * r126058;
        double r126060 = fma(r126051, r126053, r126059);
        double r126061 = fma(r126050, r126050, r126055);
        double r126062 = 2.0;
        double r126063 = pow(r126061, r126062);
        double r126064 = fma(r126049, r126060, r126063);
        double r126065 = sqrt(r126064);
        double r126066 = r126065 * r126065;
        double r126067 = r126066 - r126052;
        return r126067;
}

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. Using strategy rm

  4. Applied add-sqr-sqrt0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019305 +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))