Average Error: 0.0 → 0.0
Time: 2.9s
Precision: binary64
\[5 \leq a \land a \leq 10 \land 0 \leq b \land b \leq 0.001\]
\[\left(a + b\right) \cdot \left(a + b\right)\]
↓
\[b \cdot b + \left(a \cdot a + \left(b \cdot a\right) \cdot 2\right)\]
\left(a + b\right) \cdot \left(a + b\right)
↓
b \cdot b + \left(a \cdot a + \left(b \cdot a\right) \cdot 2\right)
(FPCore (a b) :precision binary64 (* (+ a b) (+ a b)))
↓
(FPCore (a b) :precision binary64 (+ (* b b) (+ (* a a) (* (* b a) 2.0))))
double code(double a, double b) {
	return (a + b) * (a + b);
}
↓
double code(double a, double b) {
	return (b * b) + ((a * a) + ((b * a) * 2.0));
}

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-+_binary64_17360.0

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

  4. Applied flip3-+_binary64_17650.3

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

  5. Applied frac-times_binary64_17720.3

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

  6. Simplified0.3

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

  7. Taylor expanded around 0 0.0

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

  8. Simplified0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2020303 
(FPCore (a b)
  :name "Expression 4, p15"
  :precision binary64
  :pre (and (<= 5.0 a 10.0) (<= 0.0 b 0.001))

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

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