Average Error: 0.0 → 0.0
Time: 1.2s
Precision: 64
\[5 \le a \le 10 \land 0.0 \le b \le 10^{-3}\]
\[\left(a + b\right) \cdot \left(a + b\right)\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(2, a, b\right), b, {a}^{2}\right)\]
\left(a + b\right) \cdot \left(a + b\right)
↓
\mathsf{fma}\left(\mathsf{fma}\left(2, a, b\right), b, {a}^{2}\right)
double f(double a, double b) {
        double r100582 = a;
        double r100583 = b;
        double r100584 = r100582 + r100583;
        double r100585 = r100584 * r100584;
        return r100585;
}
↓
double f(double a, double b) {
        double r100586 = 2.0;
        double r100587 = a;
        double r100588 = b;
        double r100589 = fma(r100586, r100587, r100588);
        double r100590 = pow(r100587, r100586);
        double r100591 = fma(r100589, r100588, r100590);
        return r100591;
}

Error

Bits error versus a

Bits error versus b

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 add-cbrt-cube0.5

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

  4. Applied add-cbrt-cube0.7

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

  5. Applied cbrt-unprod0.5

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

  6. Simplified0.5

    \[\leadsto \sqrt[3]{\color{blue}{{\left(a + b\right)}^{6}}}\]

  7. Taylor expanded around 0 0.0

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

  8. Simplified0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2020064 +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)))