Average Error: 0.0 → 0.0
Time: 3.8s
Precision: 64
\[-\log \left(\frac{1}{x} - 1\right)\]
↓
\[-\left(\log \left(\sqrt{\sqrt{\frac{1}{x}} + \sqrt{1}} \cdot \sqrt{\sqrt{\frac{1}{x}} - \sqrt{1}}\right) + \log \left(\sqrt{\frac{1}{x} - 1}\right)\right)\]
-\log \left(\frac{1}{x} - 1\right)
↓
-\left(\log \left(\sqrt{\sqrt{\frac{1}{x}} + \sqrt{1}} \cdot \sqrt{\sqrt{\frac{1}{x}} - \sqrt{1}}\right) + \log \left(\sqrt{\frac{1}{x} - 1}\right)\right)
double f(double x) {
        double r33107 = 1.0;
        double r33108 = x;
        double r33109 = r33107 / r33108;
        double r33110 = r33109 - r33107;
        double r33111 = log(r33110);
        double r33112 = -r33111;
        return r33112;
}
↓
double f(double x) {
        double r33113 = 1.0;
        double r33114 = x;
        double r33115 = r33113 / r33114;
        double r33116 = sqrt(r33115);
        double r33117 = sqrt(r33113);
        double r33118 = r33116 + r33117;
        double r33119 = sqrt(r33118);
        double r33120 = r33116 - r33117;
        double r33121 = sqrt(r33120);
        double r33122 = r33119 * r33121;
        double r33123 = log(r33122);
        double r33124 = r33115 - r33113;
        double r33125 = sqrt(r33124);
        double r33126 = log(r33125);
        double r33127 = r33123 + r33126;
        double r33128 = -r33127;
        return r33128;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[-\log \left(\frac{1}{x} - 1\right)\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt0.0

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

  4. Applied log-prod0.0

    \[\leadsto -\color{blue}{\left(\log \left(\sqrt{\frac{1}{x} - 1}\right) + \log \left(\sqrt{\frac{1}{x} - 1}\right)\right)}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt0.0

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

  7. Applied add-sqr-sqrt0.0

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

  8. Applied difference-of-squares0.0

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

  9. Applied sqrt-prod0.0

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

  10. Final simplification0.0

    \[\leadsto -\left(\log \left(\sqrt{\sqrt{\frac{1}{x}} + \sqrt{1}} \cdot \sqrt{\sqrt{\frac{1}{x}} - \sqrt{1}}\right) + \log \left(\sqrt{\frac{1}{x} - 1}\right)\right)\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (x)
  :name "neg log"
  :precision binary64
  (- (log (- (/ 1 x) 1))))