Average Error: 0.0 → 0.0
Time: 2.1s
Precision: 64
\[5 \le a \le 10 \land 0.0 \le b \le 10^{-3}\]
\[\left(a + b\right) \cdot \left(a + b\right)\]
↓
\[\left(a \cdot \left(b \cdot 2\right) + {a}^{2}\right) + {b}^{2}\]
\left(a + b\right) \cdot \left(a + b\right)
↓
\left(a \cdot \left(b \cdot 2\right) + {a}^{2}\right) + {b}^{2}
double f(double a, double b) {
        double r84042 = a;
        double r84043 = b;
        double r84044 = r84042 + r84043;
        double r84045 = r84044 * r84044;
        return r84045;
}
↓
double f(double a, double b) {
        double r84046 = a;
        double r84047 = b;
        double r84048 = 2.0;
        double r84049 = r84047 * r84048;
        double r84050 = r84046 * r84049;
        double r84051 = pow(r84046, r84048);
        double r84052 = r84050 + r84051;
        double r84053 = pow(r84047, r84048);
        double r84054 = r84052 + r84053;
        return r84054;
}

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 flip3-+0.3

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

  5. Applied frac-times0.3

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

  6. Simplified0.3

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

  7. Simplified0.3

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

  8. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{{a}^{2} + \left(2 \cdot \left(a \cdot b\right) + {b}^{2}\right)}\]
  9. Using strategy rm

  10. Applied associate-+r+0.0

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

  11. Simplified0.0

    \[\leadsto \color{blue}{a \cdot \left(b \cdot 2 + a\right)} + {b}^{2}\]
  12. Using strategy rm

  13. Applied distribute-lft-in0.0

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

  14. Simplified0.0

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

  15. Final simplification0.0

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

Reproduce

herbie shell --seed 2020018 
(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)))