Bouland and Aaronson, Equation (25)

Average Error: 0.2 → 0.1

Time: 48.6s

Precision: 64

Math

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

↓

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

C

double f(double a, double b) {
        double r55404940 = a;
        double r55404941 = r55404940 * r55404940;
        double r55404942 = b;
        double r55404943 = r55404942 * r55404942;
        double r55404944 = r55404941 + r55404943;
        double r55404945 = 2.0;
        double r55404946 = pow(r55404944, r55404945);
        double r55404947 = 4.0;
        double r55404948 = 1.0;
        double r55404949 = r55404948 + r55404940;
        double r55404950 = r55404941 * r55404949;
        double r55404951 = 3.0;
        double r55404952 = r55404951 * r55404940;
        double r55404953 = r55404948 - r55404952;
        double r55404954 = r55404943 * r55404953;
        double r55404955 = r55404950 + r55404954;
        double r55404956 = r55404947 * r55404955;
        double r55404957 = r55404946 + r55404956;
        double r55404958 = r55404957 - r55404948;
        return r55404958;
}

↓

double f(double a, double b) {
        double r55404959 = 4.0;
        double r55404960 = a;
        double r55404961 = r55404960 * r55404960;
        double r55404962 = 1.0;
        double r55404963 = r55404962 + r55404960;
        double r55404964 = r55404961 * r55404963;
        double r55404965 = b;
        double r55404966 = r55404965 * r55404965;
        double r55404967 = 3.0;
        double r55404968 = r55404960 * r55404967;
        double r55404969 = r55404962 - r55404968;
        double r55404970 = r55404966 * r55404969;
        double r55404971 = r55404964 + r55404970;
        double r55404972 = r55404959 * r55404971;
        double r55404973 = 2.0;
        double r55404974 = r55404961 * r55404973;
        double r55404975 = fma(r55404965, r55404965, r55404974);
        double r55404976 = r55404975 * r55404965;
        double r55404977 = pow(r55404960, r55404959);
        double r55404978 = fma(r55404965, r55404976, r55404977);
        double r55404979 = r55404972 + r55404978;
        double r55404980 = r55404979 - r55404962;
        return r55404980;
}

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. Taylor expanded around 0 0.0

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

3. Simplified 0.2

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

5. Applied pow1 0.2

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

6. Applied pow1 0.2

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

7. Applied pow-prod-up 0.2

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

8. Applied pow1 0.2

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

9. Applied pow-plus 0.2

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

10. Applied pow-sqr 0.1

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

11. Simplified 0.1

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

12. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019121 +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))