Average Error: 43.6 → 12.4
Time: 31.1s
Precision: 64
Internal Precision: 2368
\[\frac{1 - {\left(1 + x\right)}^{\left(-n\right)}}{x}\]
\[\begin{array}{l} \mathbf{if}\;x \le -4.4976252341662096 \cdot 10^{-11} \lor \neg \left(x \le 1.6080273876425427 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{1 - {\left({\left(1 + x\right)}^{\left(-n\right)}\right)}^{3}}{x \cdot \left(\left({\left(1 + x\right)}^{\left(-n\right)} \cdot {\left(1 + x\right)}^{\left(-n\right)} + {\left(1 + x\right)}^{\left(-n\right)}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot \left(n \cdot x\right)}\right) + n\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes
  2. if x < -4.4976252341662096e-11 or 1.6080273876425427e-15 < x

    1. Initial program 16.4

      \[\frac{1 - {\left(1 + x\right)}^{\left(-n\right)}}{x}\]
    2. Initial simplification16.4

      \[\leadsto \frac{1 - {\left(x + 1\right)}^{\left(-n\right)}}{x}\]
    3. Using strategy rm
    4. Applied flip3--16.5

      \[\leadsto \frac{\color{blue}{\frac{{1}^{3} - {\left({\left(x + 1\right)}^{\left(-n\right)}\right)}^{3}}{1 \cdot 1 + \left({\left(x + 1\right)}^{\left(-n\right)} \cdot {\left(x + 1\right)}^{\left(-n\right)} + 1 \cdot {\left(x + 1\right)}^{\left(-n\right)}\right)}}}{x}\]
    5. Applied associate-/l/16.5

      \[\leadsto \color{blue}{\frac{{1}^{3} - {\left({\left(x + 1\right)}^{\left(-n\right)}\right)}^{3}}{x \cdot \left(1 \cdot 1 + \left({\left(x + 1\right)}^{\left(-n\right)} \cdot {\left(x + 1\right)}^{\left(-n\right)} + 1 \cdot {\left(x + 1\right)}^{\left(-n\right)}\right)\right)}}\]
    6. Simplified16.5

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

    if -4.4976252341662096e-11 < x < 1.6080273876425427e-15

    1. Initial program 61.1

      \[\frac{1 - {\left(1 + x\right)}^{\left(-n\right)}}{x}\]
    2. Initial simplification61.1

      \[\leadsto \frac{1 - {\left(x + 1\right)}^{\left(-n\right)}}{x}\]
    3. Taylor expanded around 0 9.3

      \[\leadsto \color{blue}{\left(n + \frac{1}{3} \cdot \left(n \cdot {x}^{2}\right)\right) - \frac{1}{2} \cdot \left(n \cdot x\right)}\]
    4. Simplified9.3

      \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot \left(n \cdot x\right) + n}\]
    5. Using strategy rm
    6. Applied add-log-exp9.7

      \[\leadsto \color{blue}{\log \left(e^{\left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot \left(n \cdot x\right)}\right)} + n\]
  3. Recombined 2 regimes into one program.
  4. Final simplification12.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.4976252341662096 \cdot 10^{-11} \lor \neg \left(x \le 1.6080273876425427 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{1 - {\left({\left(1 + x\right)}^{\left(-n\right)}\right)}^{3}}{x \cdot \left(\left({\left(1 + x\right)}^{\left(-n\right)} \cdot {\left(1 + x\right)}^{\left(-n\right)} + {\left(1 + x\right)}^{\left(-n\right)}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot \left(n \cdot x\right)}\right) + n\\ \end{array}\]

Runtime

Time bar (total: 31.1s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)' 
(FPCore (x n)
  :name "(1-(1+x)^-n)/x"
  (/ (- 1 (pow (+ 1 x) (- n))) x))