qlog (example 3.10)

Average Error: 61.2 → 59.3

Time: 9.5s

Precision: 64

\[-1 \lt x \land x \lt 1\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -5.524707695762122406128297926502901213277 \cdot 10^{-17}:\\ \;\;\;\;\frac{\log \left({1}^{3} - {x}^{3}\right) - \sqrt[3]{{\left(\sqrt[3]{\log \left(x \cdot \left(x + 1\right) + {1}^{2}\right) \cdot \log \left(x \cdot \left(x + 1\right) + {1}^{2}\right)} \cdot \sqrt[3]{\log \left(x \cdot \left(x + 1\right) + {1}^{2}\right)}\right)}^{3}}}{\log \left(1 + x\right)}\\ \mathbf{elif}\;x \le 1.134036592157405800381078481820496753738 \cdot 10^{-16}:\\ \;\;\;\;\frac{\log \left({1}^{3} - {x}^{3}\right) - \sqrt[3]{{\left(\log \left(x \cdot \left(x + 1\right) + {1}^{2}\right)\right)}^{3}}}{\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left({1}^{3} - {x}^{3}\right) - \sqrt[3]{{\left(\sqrt[3]{\log \left(x \cdot \left(x + 1\right) + {1}^{2}\right) \cdot \log \left(x \cdot \left(x + 1\right) + {1}^{2}\right)} \cdot \sqrt[3]{\log \left(x \cdot \left(x + 1\right) + {1}^{2}\right)}\right)}^{3}}}{e^{\log \left(\log \left(1 + x\right)\right)}}\\ \end{array}\]

C

double f(double x) {
        double r97015 = 1.0;
        double r97016 = x;
        double r97017 = r97015 - r97016;
        double r97018 = log(r97017);
        double r97019 = r97015 + r97016;
        double r97020 = log(r97019);
        double r97021 = r97018 / r97020;
        return r97021;
}

↓

double f(double x) {
        double r97022 = x;
        double r97023 = -5.5247076957621224e-17;
        bool r97024 = r97022 <= r97023;
        double r97025 = 1.0;
        double r97026 = 3.0;
        double r97027 = pow(r97025, r97026);
        double r97028 = pow(r97022, r97026);
        double r97029 = r97027 - r97028;
        double r97030 = log(r97029);
        double r97031 = r97022 + r97025;
        double r97032 = r97022 * r97031;
        double r97033 = 2.0;
        double r97034 = pow(r97025, r97033);
        double r97035 = r97032 + r97034;
        double r97036 = log(r97035);
        double r97037 = r97036 * r97036;
        double r97038 = cbrt(r97037);
        double r97039 = cbrt(r97036);
        double r97040 = r97038 * r97039;
        double r97041 = pow(r97040, r97026);
        double r97042 = cbrt(r97041);
        double r97043 = r97030 - r97042;
        double r97044 = r97025 + r97022;
        double r97045 = log(r97044);
        double r97046 = r97043 / r97045;
        double r97047 = 1.1340365921574058e-16;
        bool r97048 = r97022 <= r97047;
        double r97049 = pow(r97036, r97026);
        double r97050 = cbrt(r97049);
        double r97051 = r97030 - r97050;
        double r97052 = r97025 * r97022;
        double r97053 = log(r97025);
        double r97054 = r97052 + r97053;
        double r97055 = 0.5;
        double r97056 = pow(r97022, r97033);
        double r97057 = r97056 / r97034;
        double r97058 = r97055 * r97057;
        double r97059 = r97054 - r97058;
        double r97060 = r97051 / r97059;
        double r97061 = log(r97045);
        double r97062 = exp(r97061);
        double r97063 = r97043 / r97062;
        double r97064 = r97048 ? r97060 : r97063;
        double r97065 = r97024 ? r97046 : r97064;
        return r97065;
}

Error

Bits error versus x

Target

Original	61.2
Target	0.2
Herbie	59.3

\[-\left(\left(\left(1 + x\right) + \frac{x \cdot x}{2}\right) + 0.4166666666666666851703837437526090070605 \cdot {x}^{3}\right)\]

Derivation

1. Split input into 3 regimes

2. if x < -5.5247076957621224e-17

   1. Initial program 14.7

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

   3. Applied flip3-- 14.8

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

   4. Applied log-div 12.9

      \[\leadsto \frac{\color{blue}{\log \left({1}^{3} - {x}^{3}\right) - \log \left(1 \cdot 1 + \left(x \cdot x + 1 \cdot x\right)\right)}}{\log \left(1 + x\right)}\]
   5. Using strategy rm

   6. Applied add-cbrt-cube 12.9

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

   7. Simplified 12.9

      \[\leadsto \frac{\log \left({1}^{3} - {x}^{3}\right) - \sqrt[3]{\color{blue}{{\left(\log \left(x \cdot \left(x + 1\right) + {1}^{2}\right)\right)}^{3}}}}{\log \left(1 + x\right)}\]
   8. Using strategy rm

   9. Applied add-cube-cbrt 12.8

      \[\leadsto \frac{\log \left({1}^{3} - {x}^{3}\right) - \sqrt[3]{{\color{blue}{\left(\left(\sqrt[3]{\log \left(x \cdot \left(x + 1\right) + {1}^{2}\right)} \cdot \sqrt[3]{\log \left(x \cdot \left(x + 1\right) + {1}^{2}\right)}\right) \cdot \sqrt[3]{\log \left(x \cdot \left(x + 1\right) + {1}^{2}\right)}\right)}}^{3}}}{\log \left(1 + x\right)}\]

   10. Simplified 12.8

       \[\leadsto \frac{\log \left({1}^{3} - {x}^{3}\right) - \sqrt[3]{{\left(\color{blue}{\sqrt[3]{\log \left(x \cdot \left(x + 1\right) + {1}^{2}\right) \cdot \log \left(x \cdot \left(x + 1\right) + {1}^{2}\right)}} \cdot \sqrt[3]{\log \left(x \cdot \left(x + 1\right) + {1}^{2}\right)}\right)}^{3}}}{\log \left(1 + x\right)}\]

   if -5.5247076957621224e-17 < x < 1.1340365921574058e-16

   1. Initial program 64.0

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

   3. Applied flip3-- 64.0

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

   4. Applied log-div 64.0

      \[\leadsto \frac{\color{blue}{\log \left({1}^{3} - {x}^{3}\right) - \log \left(1 \cdot 1 + \left(x \cdot x + 1 \cdot x\right)\right)}}{\log \left(1 + x\right)}\]
   5. Using strategy rm

   6. Applied add-cbrt-cube 64.0

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

   7. Simplified 64.0

      \[\leadsto \frac{\log \left({1}^{3} - {x}^{3}\right) - \sqrt[3]{\color{blue}{{\left(\log \left(x \cdot \left(x + 1\right) + {1}^{2}\right)\right)}^{3}}}}{\log \left(1 + x\right)}\]

   8. Taylor expanded around 0 62.0

      \[\leadsto \frac{\log \left({1}^{3} - {x}^{3}\right) - \sqrt[3]{{\left(\log \left(x \cdot \left(x + 1\right) + {1}^{2}\right)\right)}^{3}}}{\color{blue}{\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}}\]

   if 1.1340365921574058e-16 < x

   1. Initial program 14.1

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

   3. Applied flip3-- 3.8

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

   4. Applied log-div 13.2

      \[\leadsto \frac{\color{blue}{\log \left({1}^{3} - {x}^{3}\right) - \log \left(1 \cdot 1 + \left(x \cdot x + 1 \cdot x\right)\right)}}{\log \left(1 + x\right)}\]
   5. Using strategy rm

   6. Applied add-cbrt-cube 13.1

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

   7. Simplified 13.1

      \[\leadsto \frac{\log \left({1}^{3} - {x}^{3}\right) - \sqrt[3]{\color{blue}{{\left(\log \left(x \cdot \left(x + 1\right) + {1}^{2}\right)\right)}^{3}}}}{\log \left(1 + x\right)}\]
   8. Using strategy rm

   9. Applied add-cube-cbrt 13.2

      \[\leadsto \frac{\log \left({1}^{3} - {x}^{3}\right) - \sqrt[3]{{\color{blue}{\left(\left(\sqrt[3]{\log \left(x \cdot \left(x + 1\right) + {1}^{2}\right)} \cdot \sqrt[3]{\log \left(x \cdot \left(x + 1\right) + {1}^{2}\right)}\right) \cdot \sqrt[3]{\log \left(x \cdot \left(x + 1\right) + {1}^{2}\right)}\right)}}^{3}}}{\log \left(1 + x\right)}\]

   10. Simplified 13.2

       \[\leadsto \frac{\log \left({1}^{3} - {x}^{3}\right) - \sqrt[3]{{\left(\color{blue}{\sqrt[3]{\log \left(x \cdot \left(x + 1\right) + {1}^{2}\right) \cdot \log \left(x \cdot \left(x + 1\right) + {1}^{2}\right)}} \cdot \sqrt[3]{\log \left(x \cdot \left(x + 1\right) + {1}^{2}\right)}\right)}^{3}}}{\log \left(1 + x\right)}\]
   11. Using strategy rm

   12. Applied add-exp-log 13.3

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

4. Final simplification 59.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.524707695762122406128297926502901213277 \cdot 10^{-17}:\\ \;\;\;\;\frac{\log \left({1}^{3} - {x}^{3}\right) - \sqrt[3]{{\left(\sqrt[3]{\log \left(x \cdot \left(x + 1\right) + {1}^{2}\right) \cdot \log \left(x \cdot \left(x + 1\right) + {1}^{2}\right)} \cdot \sqrt[3]{\log \left(x \cdot \left(x + 1\right) + {1}^{2}\right)}\right)}^{3}}}{\log \left(1 + x\right)}\\ \mathbf{elif}\;x \le 1.134036592157405800381078481820496753738 \cdot 10^{-16}:\\ \;\;\;\;\frac{\log \left({1}^{3} - {x}^{3}\right) - \sqrt[3]{{\left(\log \left(x \cdot \left(x + 1\right) + {1}^{2}\right)\right)}^{3}}}{\left(1 \cdot x + \log 1\right) - \frac{1}{2} \cdot \frac{{x}^{2}}{{1}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left({1}^{3} - {x}^{3}\right) - \sqrt[3]{{\left(\sqrt[3]{\log \left(x \cdot \left(x + 1\right) + {1}^{2}\right) \cdot \log \left(x \cdot \left(x + 1\right) + {1}^{2}\right)} \cdot \sqrt[3]{\log \left(x \cdot \left(x + 1\right) + {1}^{2}\right)}\right)}^{3}}}{e^{\log \left(\log \left(1 + x\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019346 
(FPCore (x)
  :name "qlog (example 3.10)"
  :precision binary64
  :pre (and (< -1 x) (< x 1))

  :herbie-target
  (- (+ (+ (+ 1 x) (/ (* x x) 2)) (* 0.4166666666666667 (pow x 3))))

  (/ (log (- 1 x)) (log (+ 1 x))))