Average Error: 0.0 → 0
Time: 1.1s
Precision: 64
\[5 \le a \le 10 \land 0.0 \le b \le 10^{-3}\]
\[\left(a + b\right) \cdot \left(a + b\right)\]
↓
\[\mathsf{fma}\left(a, a, \mathsf{fma}\left(2, a \cdot b, {b}^{2}\right)\right)\]
\left(a + b\right) \cdot \left(a + b\right)
↓
\mathsf{fma}\left(a, a, \mathsf{fma}\left(2, a \cdot b, {b}^{2}\right)\right)
double f(double a, double b) {
        double r70758 = a;
        double r70759 = b;
        double r70760 = r70758 + r70759;
        double r70761 = r70760 * r70760;
        return r70761;
}
↓
double f(double a, double b) {
        double r70762 = a;
        double r70763 = 2.0;
        double r70764 = b;
        double r70765 = r70762 * r70764;
        double r70766 = pow(r70764, r70763);
        double r70767 = fma(r70763, r70765, r70766);
        double r70768 = fma(r70762, r70762, r70767);
        return r70768;
}

Error

Bits error versus a

Bits error versus b

Target

Original0.0
Target0.0
Herbie0
\[\left(\left(b \cdot a + b \cdot b\right) + b \cdot a\right) + a \cdot a\]

Derivation

  1. Initial program 0.0

    \[\left(a + b\right) \cdot \left(a + b\right)\]
  2. Using strategy rm

  3. Applied distribute-lft-in0.0

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

  4. Simplified0.0

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

  5. Simplified0.0

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

  6. Taylor expanded around 0 0.0

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

  7. Simplified0

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

  8. Final simplification0

    \[\leadsto \mathsf{fma}\left(a, a, \mathsf{fma}\left(2, a \cdot b, {b}^{2}\right)\right)\]

Reproduce

herbie shell --seed 2020003 +o rules:numerics
(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)))