Average Error: 0.0 → 0
Time: 5.8s
Precision: 64
\[5 \le a \le 10 \land 0 \le b \le 0.001\]
\[\left(a + b\right) \cdot \left(a + b\right)\]
↓
\[\mathsf{fma}\left(a, a, \mathsf{fma}\left(b, b, \left(a \cdot 2\right) \cdot b\right)\right)\]
\left(a + b\right) \cdot \left(a + b\right)
↓
\mathsf{fma}\left(a, a, \mathsf{fma}\left(b, b, \left(a \cdot 2\right) \cdot b\right)\right)
double f(double a, double b) {
        double r1624058 = a;
        double r1624059 = b;
        double r1624060 = r1624058 + r1624059;
        double r1624061 = r1624060 * r1624060;
        return r1624061;
}
↓
double f(double a, double b) {
        double r1624062 = a;
        double r1624063 = b;
        double r1624064 = 2.0;
        double r1624065 = r1624062 * r1624064;
        double r1624066 = r1624065 * r1624063;
        double r1624067 = fma(r1624063, r1624063, r1624066);
        double r1624068 = fma(r1624062, r1624062, r1624067);
        return r1624068;
}

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 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. Simplified0.1

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

  7. Simplified0.1

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

  8. Taylor expanded around 0 0.0

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

  9. Simplified0

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

  10. Final simplification0

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

Reproduce

herbie shell --seed 2019155 +o rules:numerics
(FPCore (a b)
  :name "Expression 4, p15"
  :pre (and (<= 5 a 10) (<= 0 b 0.001))

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

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