Average Error: 0.2 → 0.2
Time: 56.9s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(4, \mathsf{fma}\left(1 - 3 \cdot a, b \cdot b, \left(a \cdot a\right) \cdot \left(a + 1\right)\right), {\left(\mathsf{fma}\left(b, b, a \cdot a\right)\right)}^{2}\right) - 1\]
\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(1 - 3 \cdot a, b \cdot b, \left(a \cdot a\right) \cdot \left(a + 1\right)\right), {\left(\mathsf{fma}\left(b, b, a \cdot a\right)\right)}^{2}\right) - 1
double f(double a, double b) {
        double r9749035 = a;
        double r9749036 = r9749035 * r9749035;
        double r9749037 = b;
        double r9749038 = r9749037 * r9749037;
        double r9749039 = r9749036 + r9749038;
        double r9749040 = 2.0;
        double r9749041 = pow(r9749039, r9749040);
        double r9749042 = 4.0;
        double r9749043 = 1.0;
        double r9749044 = r9749043 + r9749035;
        double r9749045 = r9749036 * r9749044;
        double r9749046 = 3.0;
        double r9749047 = r9749046 * r9749035;
        double r9749048 = r9749043 - r9749047;
        double r9749049 = r9749038 * r9749048;
        double r9749050 = r9749045 + r9749049;
        double r9749051 = r9749042 * r9749050;
        double r9749052 = r9749041 + r9749051;
        double r9749053 = r9749052 - r9749043;
        return r9749053;
}
↓
double f(double a, double b) {
        double r9749054 = 4.0;
        double r9749055 = 1.0;
        double r9749056 = 3.0;
        double r9749057 = a;
        double r9749058 = r9749056 * r9749057;
        double r9749059 = r9749055 - r9749058;
        double r9749060 = b;
        double r9749061 = r9749060 * r9749060;
        double r9749062 = r9749057 * r9749057;
        double r9749063 = r9749057 + r9749055;
        double r9749064 = r9749062 * r9749063;
        double r9749065 = fma(r9749059, r9749061, r9749064);
        double r9749066 = fma(r9749060, r9749060, r9749062);
        double r9749067 = 2.0;
        double r9749068 = pow(r9749066, r9749067);
        double r9749069 = fma(r9749054, r9749065, r9749068);
        double r9749070 = r9749069 - r9749055;
        return r9749070;
}

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

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  (- (+ (pow (+ (* a a) (* b b)) 2.0) (* 4.0 (+ (* (* a a) (+ 1.0 a)) (* (* b b) (- 1.0 (* 3.0 a)))))) 1.0))