Average Error: 0.0 → 0.0
Time: 1.3s
Precision: 64
\[5 \le a \le 10 \land 0.0 \le b \le 10^{-3}\]
\[\left(a + b\right) \cdot \left(a + b\right)\]
↓
\[a \cdot \left(b \cdot 2 + a\right) + {b}^{2}\]
\left(a + b\right) \cdot \left(a + b\right)
↓
a \cdot \left(b \cdot 2 + a\right) + {b}^{2}
double f(double a, double b) {
        double r96512 = a;
        double r96513 = b;
        double r96514 = r96512 + r96513;
        double r96515 = r96514 * r96514;
        return r96515;
}
↓
double f(double a, double b) {
        double r96516 = a;
        double r96517 = b;
        double r96518 = 2.0;
        double r96519 = r96517 * r96518;
        double r96520 = r96519 + r96516;
        double r96521 = r96516 * r96520;
        double r96522 = pow(r96517, r96518);
        double r96523 = r96521 + r96522;
        return r96523;
}

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 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. Using strategy rm

  8. Applied associate-+r+0.0

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2020056 
(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)))