Average Error: 0.0 → 0.0
Time: 1.8s
Precision: 64
\[5 \le a \le 10 \land 0.0 \le b \le 10^{-3}\]
\[\left(a + b\right) \cdot \left(a + b\right)\]
↓
\[{a}^{2} + \left(2 \cdot \left(a \cdot b\right) + {b}^{2}\right)\]
\left(a + b\right) \cdot \left(a + b\right)
↓
{a}^{2} + \left(2 \cdot \left(a \cdot b\right) + {b}^{2}\right)
double f(double a, double b) {
        double r79453 = a;
        double r79454 = b;
        double r79455 = r79453 + r79454;
        double r79456 = r79455 * r79455;
        return r79456;
}
↓
double f(double a, double b) {
        double r79457 = a;
        double r79458 = 2.0;
        double r79459 = pow(r79457, r79458);
        double r79460 = b;
        double r79461 = r79457 * r79460;
        double r79462 = r79458 * r79461;
        double r79463 = pow(r79460, r79458);
        double r79464 = r79462 + r79463;
        double r79465 = r79459 + r79464;
        return r79465;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

  3. Applied flip3-+0.3

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

  4. Applied flip3-+0.5

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

  5. Applied frac-times0.8

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

  6. Simplified0.8

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

  7. Taylor expanded around 0 0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2020021 
(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)))