Average Error: 0.0 → 0.0
Time: 5.7s
Precision: 64
\[5 \le a \le 10 \land 0 \le b \le 0.001\]
\[\left(a + b\right) \cdot \left(a + b\right)\]
↓
\[(\left((2 \cdot a + b)_*\right) \cdot b + \left(a \cdot a\right))_*\]
\left(a + b\right) \cdot \left(a + b\right)
↓
(\left((2 \cdot a + b)_*\right) \cdot b + \left(a \cdot a\right))_*
double f(double a, double b) {
        double r7001100 = a;
        double r7001101 = b;
        double r7001102 = r7001100 + r7001101;
        double r7001103 = r7001102 * r7001102;
        return r7001103;
}
↓
double f(double a, double b) {
        double r7001104 = 2.0;
        double r7001105 = a;
        double r7001106 = b;
        double r7001107 = fma(r7001104, r7001105, r7001106);
        double r7001108 = r7001105 * r7001105;
        double r7001109 = fma(r7001107, r7001106, r7001108);
        return r7001109;
}

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 inf 0.0

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

  3. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto (\left((2 \cdot a + b)_*\right) \cdot b + \left(a \cdot a\right))_*\]

Reproduce

herbie shell --seed 2019119 +o rules:numerics
(FPCore (a b)
  :name "Expression 4, p15"
  :pre (and (<= 5 a 10) (<= 0 b 0.001))

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

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