Average Error: 15.0 → 0.0
Time: 1.6m
Precision: 64
Internal Precision: 576
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -490.84465569610137 \lor \neg \left(x \le 14683.278761548403\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\sqrt[3]{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1} \cdot \sqrt[3]{{\left(x \cdot x\right)}^{3} + {1}^{3}}} \cdot \left(\sqrt[3]{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 - x \cdot x\right)} \cdot \sqrt[3]{x \cdot x - 1}\right)}{\sqrt[3]{x \cdot x + 1}}\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.0
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -490.84465569610137 or 14683.278761548403 < x

    1. Initial program 30.1

      \[\frac{x}{x \cdot x + 1}\]

    2. Initial simplification30.1

      \[\leadsto \frac{x}{x \cdot x + 1}\]

    3. Taylor expanded around -inf 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}}\]

    if -490.84465569610137 < x < 14683.278761548403

    1. Initial program 0.0

      \[\frac{x}{x \cdot x + 1}\]

    2. Initial simplification0.0

      \[\leadsto \frac{x}{x \cdot x + 1}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt0.1

      \[\leadsto \frac{x}{\color{blue}{\left(\sqrt[3]{x \cdot x + 1} \cdot \sqrt[3]{x \cdot x + 1}\right) \cdot \sqrt[3]{x \cdot x + 1}}}\]

    5. Applied associate-/r*0.1

      \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt[3]{x \cdot x + 1} \cdot \sqrt[3]{x \cdot x + 1}}}{\sqrt[3]{x \cdot x + 1}}}\]
    6. Using strategy rm

    7. Applied flip3-+0.0

      \[\leadsto \frac{\frac{x}{\sqrt[3]{x \cdot x + 1} \cdot \sqrt[3]{\color{blue}{\frac{{\left(x \cdot x\right)}^{3} + {1}^{3}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 \cdot 1 - \left(x \cdot x\right) \cdot 1\right)}}}}}{\sqrt[3]{x \cdot x + 1}}\]

    8. Applied cbrt-div0.0

      \[\leadsto \frac{\frac{x}{\sqrt[3]{x \cdot x + 1} \cdot \color{blue}{\frac{\sqrt[3]{{\left(x \cdot x\right)}^{3} + {1}^{3}}}{\sqrt[3]{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 \cdot 1 - \left(x \cdot x\right) \cdot 1\right)}}}}}{\sqrt[3]{x \cdot x + 1}}\]

    9. Applied flip-+0.0

      \[\leadsto \frac{\frac{x}{\sqrt[3]{\color{blue}{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot 1}{x \cdot x - 1}}} \cdot \frac{\sqrt[3]{{\left(x \cdot x\right)}^{3} + {1}^{3}}}{\sqrt[3]{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 \cdot 1 - \left(x \cdot x\right) \cdot 1\right)}}}}{\sqrt[3]{x \cdot x + 1}}\]

    10. Applied cbrt-div0.0

      \[\leadsto \frac{\frac{x}{\color{blue}{\frac{\sqrt[3]{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot 1}}{\sqrt[3]{x \cdot x - 1}}} \cdot \frac{\sqrt[3]{{\left(x \cdot x\right)}^{3} + {1}^{3}}}{\sqrt[3]{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 \cdot 1 - \left(x \cdot x\right) \cdot 1\right)}}}}{\sqrt[3]{x \cdot x + 1}}\]

    11. Applied frac-times0.0

      \[\leadsto \frac{\frac{x}{\color{blue}{\frac{\sqrt[3]{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot 1} \cdot \sqrt[3]{{\left(x \cdot x\right)}^{3} + {1}^{3}}}{\sqrt[3]{x \cdot x - 1} \cdot \sqrt[3]{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 \cdot 1 - \left(x \cdot x\right) \cdot 1\right)}}}}}{\sqrt[3]{x \cdot x + 1}}\]

    12. Applied associate-/r/0.0

      \[\leadsto \frac{\color{blue}{\frac{x}{\sqrt[3]{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot 1} \cdot \sqrt[3]{{\left(x \cdot x\right)}^{3} + {1}^{3}}} \cdot \left(\sqrt[3]{x \cdot x - 1} \cdot \sqrt[3]{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 \cdot 1 - \left(x \cdot x\right) \cdot 1\right)}\right)}}{\sqrt[3]{x \cdot x + 1}}\]

    13. Simplified0.0

      \[\leadsto \frac{\frac{x}{\sqrt[3]{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot 1} \cdot \sqrt[3]{{\left(x \cdot x\right)}^{3} + {1}^{3}}} \cdot \color{blue}{\left(\sqrt[3]{x \cdot x - 1} \cdot \sqrt[3]{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 - x \cdot x\right)}\right)}}{\sqrt[3]{x \cdot x + 1}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -490.84465569610137 \lor \neg \left(x \le 14683.278761548403\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\sqrt[3]{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1} \cdot \sqrt[3]{{\left(x \cdot x\right)}^{3} + {1}^{3}}} \cdot \left(\sqrt[3]{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 - x \cdot x\right)} \cdot \sqrt[3]{x \cdot x - 1}\right)}{\sqrt[3]{x \cdot x + 1}}\\ \end{array}\]

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2018234 
(FPCore (x)
  :name "x / (x^2 + 1)"

  :herbie-target
  (/ 1 (+ x (/ 1 x)))

  (/ x (+ (* x x) 1)))