Average Error: 0.2 → 0.2
Time: 16.8s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 - a\right) + \left(b \cdot b\right) \cdot \left(3 + a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(4, \mathsf{fma}\left(a, a - a \cdot a, \left(b \cdot b\right) \cdot \left(a + 3\right)\right), \mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), -1\right)\right)\]
\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(3 + a\right)\right)\right) - 1
↓
\mathsf{fma}\left(4, \mathsf{fma}\left(a, a - a \cdot a, \left(b \cdot b\right) \cdot \left(a + 3\right)\right), \mathsf{fma}\left(\mathsf{fma}\left(a, a, b \cdot b\right), \mathsf{fma}\left(a, a, b \cdot b\right), -1\right)\right)
double f(double a, double b) {
        double r6538073 = a;
        double r6538074 = r6538073 * r6538073;
        double r6538075 = b;
        double r6538076 = r6538075 * r6538075;
        double r6538077 = r6538074 + r6538076;
        double r6538078 = 2.0;
        double r6538079 = pow(r6538077, r6538078);
        double r6538080 = 4.0;
        double r6538081 = 1.0;
        double r6538082 = r6538081 - r6538073;
        double r6538083 = r6538074 * r6538082;
        double r6538084 = 3.0;
        double r6538085 = r6538084 + r6538073;
        double r6538086 = r6538076 * r6538085;
        double r6538087 = r6538083 + r6538086;
        double r6538088 = r6538080 * r6538087;
        double r6538089 = r6538079 + r6538088;
        double r6538090 = r6538089 - r6538081;
        return r6538090;
}
↓
double f(double a, double b) {
        double r6538091 = 4.0;
        double r6538092 = a;
        double r6538093 = r6538092 * r6538092;
        double r6538094 = r6538092 - r6538093;
        double r6538095 = b;
        double r6538096 = r6538095 * r6538095;
        double r6538097 = 3.0;
        double r6538098 = r6538092 + r6538097;
        double r6538099 = r6538096 * r6538098;
        double r6538100 = fma(r6538092, r6538094, r6538099);
        double r6538101 = fma(r6538092, r6538092, r6538096);
        double r6538102 = -1.0;
        double r6538103 = fma(r6538101, r6538101, r6538102);
        double r6538104 = fma(r6538091, r6538100, r6538103);
        return r6538104;
}

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

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2019143 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (24)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (- 1 a)) (* (* b b) (+ 3 a))))) 1))