Average Error: 0.0 → 0.0
Time: 6.3s
Precision: 64
\[5 \le a \le 10 \land 0.0 \le b \le 0.001000000000000000020816681711721685132943\]
\[\left(a + b\right) \cdot \left(a + b\right)\]
↓
\[\mathsf{fma}\left(b, \mathsf{fma}\left(a, 2, b\right), a \cdot a\right)\]
\left(a + b\right) \cdot \left(a + b\right)
↓
\mathsf{fma}\left(b, \mathsf{fma}\left(a, 2, b\right), a \cdot a\right)
double f(double a, double b) {
        double r3752849 = a;
        double r3752850 = b;
        double r3752851 = r3752849 + r3752850;
        double r3752852 = r3752851 * r3752851;
        return r3752852;
}
↓
double f(double a, double b) {
        double r3752853 = b;
        double r3752854 = a;
        double r3752855 = 2.0;
        double r3752856 = fma(r3752854, r3752855, r3752853);
        double r3752857 = r3752854 * r3752854;
        double r3752858 = fma(r3752853, r3752856, r3752857);
        return r3752858;
}

Error

Bits error versus a

Bits error versus b

Target

Original0.0
Target0.0
Herbie0.0
\[\left(\left(b \cdot a + b \cdot b\right) + b \cdot a\right) + a \cdot a\]

Derivation

  1. Initial program 0.0

    \[\left(a + b\right) \cdot \left(a + b\right)\]

  2. Taylor expanded around 0 0.0

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

  3. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(a, 2, b\right), a \cdot a\right)}\]

  4. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(b, \mathsf{fma}\left(a, 2, b\right), a \cdot a\right)\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (a b)
  :name "Expression 4, p15"
  :pre (and (<= 5.0 a 10.0) (<= 0.0 b 0.001))

  :herbie-target
  (+ (+ (+ (* b a) (* b b)) (* b a)) (* a a))

  (* (+ a b) (+ a b)))