Expression 4, p15

Average Error: 0.0 → 0.0

Time: 1.8s

Precision: 64

\[5 \le a \le 10 \land 0.0 \le b \le 10^{-3}\]

Math

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

↓

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

C

double f(double a, double b) {
        double r99373 = a;
        double r99374 = b;
        double r99375 = r99373 + r99374;
        double r99376 = r99375 * r99375;
        return r99376;
}

↓

double f(double a, double b) {
        double r99377 = a;
        double r99378 = 2.0;
        double r99379 = pow(r99377, r99378);
        double r99380 = b;
        double r99381 = r99377 * r99380;
        double r99382 = r99378 * r99381;
        double r99383 = pow(r99380, r99378);
        double r99384 = r99382 + r99383;
        double r99385 = r99379 + r99384;
        return r99385;
}

Error

Bits error versus a

Bits error versus b

Target

Original	0.0
Target	0.0
Herbie	0.0

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

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 associate-*r/ 0.5

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

5. Taylor expanded around 0 0.0

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

6. Final simplification 0.0

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

Reproduce

herbie shell --seed 2020081 
(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)))