Average Error: 0.0 → 0
Time: 2.2s
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 r58414 = a;
        double r58415 = b;
        double r58416 = r58414 + r58415;
        double r58417 = r58416 * r58416;
        return r58417;
}
↓
double f(double a, double b) {
        double r58418 = a;
        double r58419 = 2.0;
        double r58420 = b;
        double r58421 = r58418 * r58420;
        double r58422 = pow(r58420, r58419);
        double r58423 = fma(r58419, r58421, r58422);
        double r58424 = fma(r58418, r58418, r58423);
        return r58424;
}

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. Simplified0.3

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

  7. Simplified0.3

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

  8. Taylor expanded around 0 0.0

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

  9. Simplified0

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

  10. Final simplification0

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

Reproduce

herbie shell --seed 2020035 +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)))