Average Error: 0.0 → 0.0
Time: 2.5min
Precision: binary64
Cost: 1856
\[-\log \left(\frac{1}{x} - 1\right)\]
↓
\[-\left(\log \left(1 + e^{-\log \left(\sqrt{x}\right)}\right) + \left(\log \left(\sqrt{-1 + \frac{1}{\sqrt{x}}}\right) + \log \left(e^{\log \left(\sqrt{-1 + \frac{1}{\sqrt{x}}}\right)}\right)\right)\right)\]
-\log \left(\frac{1}{x} - 1\right)
↓
-\left(\log \left(1 + e^{-\log \left(\sqrt{x}\right)}\right) + \left(\log \left(\sqrt{-1 + \frac{1}{\sqrt{x}}}\right) + \log \left(e^{\log \left(\sqrt{-1 + \frac{1}{\sqrt{x}}}\right)}\right)\right)\right)
(FPCore (x) :precision binary64 (- (log (- (/ 1.0 x) 1.0))))
↓
(FPCore (x)
 :precision binary64
 (-
  (+
   (log (+ 1.0 (exp (- (log (sqrt x))))))
   (+
    (log (sqrt (+ -1.0 (/ 1.0 (sqrt x)))))
    (log (exp (log (sqrt (+ -1.0 (/ 1.0 (sqrt x)))))))))))
double code(double x) {
	return -log((1.0 / x) - 1.0);
}
↓
double code(double x) {
	return -(log(1.0 + exp(-log(sqrt(x)))) + (log(sqrt(-1.0 + (1.0 / sqrt(x)))) + log(exp(log(sqrt(-1.0 + (1.0 / sqrt(x))))))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs
Alternative 1
Accuracy0.8
Cost2560
\[-\left(\log \left(1 + {\left(e^{\sqrt[3]{-\log \left(\sqrt{x}\right)} \cdot \sqrt[3]{-\log \left(\sqrt{x}\right)}}\right)}^{\left(\sqrt[3]{-\log \left(\sqrt{x}\right)}\right)}\right) + \left(\log \left(\sqrt{-1 + \frac{1}{\sqrt{x}}}\right) + \log \left(\sqrt{-1 + \frac{1}{\sqrt{x}}}\right)\right)\right)\]

Derivation

  1. Initial program 0.0

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

  3. Applied add-sqr-sqrt_binary640.0

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

  4. Applied add-sqr-sqrt_binary640.0

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

  5. Applied times-frac_binary640.0

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

  6. Applied difference-of-sqr-1_binary640.0

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

  7. Applied log-prod_binary640.0

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

  8. Simplified0.0

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

  9. Simplified0.0

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

  11. Applied add-exp-log_binary640.0

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

  12. Applied 1-exp_binary640.0

    \[\leadsto -\left(\log \left(1 + \frac{\color{blue}{e^{0}}}{e^{\log \left(\sqrt{x}\right)}}\right) + \log \left(\frac{1}{\sqrt{x}} - 1\right)\right)\]

  13. Applied div-exp_binary640.0

    \[\leadsto -\left(\log \left(1 + \color{blue}{e^{0 - \log \left(\sqrt{x}\right)}}\right) + \log \left(\frac{1}{\sqrt{x}} - 1\right)\right)\]

  14. Simplified0.0

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

  16. Applied add-sqr-sqrt_binary640.0

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

  17. Applied log-prod_binary640.0

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

  18. Simplified0.0

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

  19. Simplified0.0

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

  21. Applied add-exp-log_binary640.0

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

  22. Final simplification0.0

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

Reproduce

herbie shell --seed 2020322 
(FPCore (x)
  :name "neg log"
  :precision binary64
  (- (log (- (/ 1.0 x) 1.0))))