Average Error: 0.0 → 0
Time: 1.3s
Precision: 64
\[5 \le a \le 10 \land 0.0 \le b \le 10^{-3}\]
\[\left(a + b\right) \cdot \left(a + b\right)\]
↓
\[\mathsf{fma}\left(a, a, \mathsf{fma}\left(2, a \cdot b, {b}^{2}\right)\right)\]
\left(a + b\right) \cdot \left(a + b\right)
↓
\mathsf{fma}\left(a, a, \mathsf{fma}\left(2, a \cdot b, {b}^{2}\right)\right)
double f(double a, double b) {
        double r99189 = a;
        double r99190 = b;
        double r99191 = r99189 + r99190;
        double r99192 = r99191 * r99191;
        return r99192;
}
↓
double f(double a, double b) {
        double r99193 = a;
        double r99194 = 2.0;
        double r99195 = b;
        double r99196 = r99193 * r99195;
        double r99197 = pow(r99195, r99194);
        double r99198 = fma(r99194, r99196, r99197);
        double r99199 = fma(r99193, r99193, r99198);
        return r99199;
}

Error

Bits error versus a

Bits error versus b

Target

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

Derivation

  1. Initial program 0.0

    \[\left(a + b\right) \cdot \left(a + b\right)\]
  2. Using strategy rm

  3. Applied flip-+0.0

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

  4. Applied associate-*r/0.1

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

  5. Simplified0.1

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

  6. Taylor expanded around 0 0.0

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

  7. Simplified0

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

  8. Final simplification0

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

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (a b)
  :name "Expression 4, p15"
  :precision binary64
  :pre (and (<= 5 a 10) (<= 0.0 b 0.001))

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

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