Average Error: 0.2 → 0.2
Time: 38.1s
Precision: 64
\[\left({\left(a \cdot a + b \cdot b\right)}^{2} + 4 \cdot \left(\left(a \cdot a\right) \cdot \left(1 + a\right) + \left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right)\right)\right) - 1\]
↓
\[\mathsf{fma}\left(a \cdot a + b \cdot b, a \cdot a + b \cdot b, 4 \cdot \left(\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right) + \left(a \cdot a\right) \cdot \left(a + 1\right)\right)\right) - 1\]
\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(a \cdot a + b \cdot b, a \cdot a + b \cdot b, 4 \cdot \left(\left(b \cdot b\right) \cdot \left(1 - 3 \cdot a\right) + \left(a \cdot a\right) \cdot \left(a + 1\right)\right)\right) - 1
double f(double a, double b) {
        double r6000876 = a;
        double r6000877 = r6000876 * r6000876;
        double r6000878 = b;
        double r6000879 = r6000878 * r6000878;
        double r6000880 = r6000877 + r6000879;
        double r6000881 = 2.0;
        double r6000882 = pow(r6000880, r6000881);
        double r6000883 = 4.0;
        double r6000884 = 1.0;
        double r6000885 = r6000884 + r6000876;
        double r6000886 = r6000877 * r6000885;
        double r6000887 = 3.0;
        double r6000888 = r6000887 * r6000876;
        double r6000889 = r6000884 - r6000888;
        double r6000890 = r6000879 * r6000889;
        double r6000891 = r6000886 + r6000890;
        double r6000892 = r6000883 * r6000891;
        double r6000893 = r6000882 + r6000892;
        double r6000894 = r6000893 - r6000884;
        return r6000894;
}
↓
double f(double a, double b) {
        double r6000895 = a;
        double r6000896 = r6000895 * r6000895;
        double r6000897 = b;
        double r6000898 = r6000897 * r6000897;
        double r6000899 = r6000896 + r6000898;
        double r6000900 = 4.0;
        double r6000901 = 1.0;
        double r6000902 = 3.0;
        double r6000903 = r6000902 * r6000895;
        double r6000904 = r6000901 - r6000903;
        double r6000905 = r6000898 * r6000904;
        double r6000906 = r6000895 + r6000901;
        double r6000907 = r6000896 * r6000906;
        double r6000908 = r6000905 + r6000907;
        double r6000909 = r6000900 * r6000908;
        double r6000910 = fma(r6000899, r6000899, r6000909);
        double r6000911 = r6000910 - r6000901;
        return r6000911;
}

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 unpow20.2

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

  4. Applied fma-def0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (a b)
  :name "Bouland and Aaronson, Equation (25)"
  (- (+ (pow (+ (* a a) (* b b)) 2) (* 4 (+ (* (* a a) (+ 1 a)) (* (* b b) (- 1 (* 3 a)))))) 1))