Average Error: 10.9 → 3.7
Time: 32.2s
Precision: 64
Internal Precision: 1344
\[\frac{1}{\sqrt{1 + \left(\left(-alphax\right) \cdot alphax\right) \cdot \log \left(1 - u\right)}}\]
\[\begin{array}{l} \mathbf{if}\;1 - u \le 1.0:\\ \;\;\;\;\frac{1}{\sqrt{1 - alphax \cdot \left(\left(\left(alphax \cdot u\right) \cdot u\right) \cdot \left(\left(-\frac{1}{2}\right) - \frac{1}{3} \cdot u\right) - alphax \cdot u\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 - \log \left(e^{alphax \cdot \left(\log \left(1 - u\right) \cdot alphax\right)}\right)}}\\ \end{array}\]

Error

Bits error versus alphax

Bits error versus u

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes
  2. if (- 1 u) < 1.0

    1. Initial program 14.5

      \[\frac{1}{\sqrt{1 + \left(\left(-alphax\right) \cdot alphax\right) \cdot \log \left(1 - u\right)}}\]
    2. Initial simplification14.5

      \[\leadsto \frac{1}{\sqrt{1 - \left(alphax \cdot alphax\right) \cdot \log \left(1 - u\right)}}\]
    3. Using strategy rm
    4. Applied associate-*l*14.2

      \[\leadsto \frac{1}{\sqrt{1 - \color{blue}{alphax \cdot \left(alphax \cdot \log \left(1 - u\right)\right)}}}\]
    5. Taylor expanded around 0 5.0

      \[\leadsto \frac{1}{\sqrt{1 - alphax \cdot \color{blue}{\left(-\left(\frac{1}{2} \cdot \left({u}^{2} \cdot alphax\right) + \left(\frac{1}{3} \cdot \left({u}^{3} \cdot alphax\right) + u \cdot alphax\right)\right)\right)}}}\]
    6. Simplified5.0

      \[\leadsto \frac{1}{\sqrt{1 - alphax \cdot \color{blue}{\left(\left(u \cdot \left(u \cdot alphax\right)\right) \cdot \left(\left(-\frac{1}{2}\right) - u \cdot \frac{1}{3}\right) - u \cdot alphax\right)}}}\]

    if 1.0 < (- 1 u)

    1. Initial program 0.0

      \[\frac{1}{\sqrt{1 + \left(\left(-alphax\right) \cdot alphax\right) \cdot \log \left(1 - u\right)}}\]
    2. Initial simplification0.0

      \[\leadsto \frac{1}{\sqrt{1 - \left(alphax \cdot alphax\right) \cdot \log \left(1 - u\right)}}\]
    3. Using strategy rm
    4. Applied associate-*l*0.0

      \[\leadsto \frac{1}{\sqrt{1 - \color{blue}{alphax \cdot \left(alphax \cdot \log \left(1 - u\right)\right)}}}\]
    5. Using strategy rm
    6. Applied add-log-exp0.0

      \[\leadsto \frac{1}{\sqrt{1 - \color{blue}{\log \left(e^{alphax \cdot \left(alphax \cdot \log \left(1 - u\right)\right)}\right)}}}\]
  3. Recombined 2 regimes into one program.
  4. Final simplification3.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u \le 1.0:\\ \;\;\;\;\frac{1}{\sqrt{1 - alphax \cdot \left(\left(\left(alphax \cdot u\right) \cdot u\right) \cdot \left(\left(-\frac{1}{2}\right) - \frac{1}{3} \cdot u\right) - alphax \cdot u\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 - \log \left(e^{alphax \cdot \left(\log \left(1 - u\right) \cdot alphax\right)}\right)}}\\ \end{array}\]

Runtime

Time bar (total: 32.2s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)' 
(FPCore (alphax u)
  :name "1 / sqrt(1 + -alphax * alphax * log(1 - u))"
  (/ 1 (sqrt (+ 1 (* (* (- alphax) alphax) (log (- 1 u)))))))