Average Error: 0.1 → 0
Time: 2.3s
Precision: 64
\[\left(x \cdot x\right) \cdot x\]
↓
\[{x}^{3}\]
double f(double x) {
        double r60650965 = x;
        double r60650966 = r60650965 * r60650965;
        double r60650967 = r60650966 * r60650965;
        return r60650967;
}
↓
double f(double x) {
        double r60650968 = x;
        double r60650969 = 3.0;
        double r60650970 = pow(r60650968, r60650969);
        return r60650970;
}


\left(x \cdot x\right) \cdot x
↓
{x}^{3}

Error

Bits error versus x

Target

Original0.1
Target0
Herbie0
\[{x}^{3}\]

Derivation

  1. Initial program 0.1

    \[\left(x \cdot x\right) \cdot x\]
  2. Using strategy rm

  3. Applied pow10.1

    \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{{x}^{1}}\]

  4. Applied pow10.1

    \[\leadsto \left(x \cdot \color{blue}{{x}^{1}}\right) \cdot {x}^{1}\]

  5. Applied pow10.1

    \[\leadsto \left(\color{blue}{{x}^{1}} \cdot {x}^{1}\right) \cdot {x}^{1}\]

  6. Applied pow-sqr0.1

    \[\leadsto \color{blue}{{x}^{\left(2 \cdot 1\right)}} \cdot {x}^{1}\]

  7. Applied pow-prod-up0

    \[\leadsto \color{blue}{{x}^{\left(2 \cdot 1 + 1\right)}}\]

  8. Simplified0

    \[\leadsto {x}^{\color{blue}{3}}\]

  9. Final simplification0

    \[\leadsto {x}^{3}\]

Reproduce

herbie shell --seed 2019102 +o rules:numerics
(FPCore (x)
  :name "math.cube on real"

  :herbie-target
  (pow x 3)

  (* (* x x) x))