Average Error: 0.0 → 0.0
Time: 1.3s
Precision: 64
\[5 \le a \le 10 \land 0.0 \le b \le 10^{-3}\]
\[\left(a + b\right) \cdot \left(a + b\right)\]
↓
\[\mathsf{fma}\left(\sqrt{{a}^{2}}, \sqrt{{a}^{2}}, a \cdot b\right) + b \cdot \left(a + b\right)\]
\left(a + b\right) \cdot \left(a + b\right)
↓
\mathsf{fma}\left(\sqrt{{a}^{2}}, \sqrt{{a}^{2}}, a \cdot b\right) + b \cdot \left(a + b\right)
double f(double a, double b) {
        double r98840 = a;
        double r98841 = b;
        double r98842 = r98840 + r98841;
        double r98843 = r98842 * r98842;
        return r98843;
}
↓
double f(double a, double b) {
        double r98844 = a;
        double r98845 = 2.0;
        double r98846 = pow(r98844, r98845);
        double r98847 = sqrt(r98846);
        double r98848 = b;
        double r98849 = r98844 * r98848;
        double r98850 = fma(r98847, r98847, r98849);
        double r98851 = r98844 + r98848;
        double r98852 = r98848 * r98851;
        double r98853 = r98850 + r98852;
        return r98853;
}

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 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. Using strategy rm

  7. Applied distribute-lft-in0.0

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

  8. Simplified0.0

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

  10. Applied add-sqr-sqrt0.0

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

  11. Applied fma-def0.0

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

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2020060 +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)))