Average Error: 0.0 → 0.0
Time: 1.0min
Precision: binary64
Cost: 26368
\[-\log \left(\frac{1}{x} - 1\right)\]
↓
\[-\left(\log \left(\sqrt{\frac{1}{x} - 1}\right) + \log \left(\sqrt{\frac{1}{x} - 1}\right)\right)\]
-\log \left(\frac{1}{x} - 1\right)
↓
-\left(\log \left(\sqrt{\frac{1}{x} - 1}\right) + \log \left(\sqrt{\frac{1}{x} - 1}\right)\right)
(FPCore (x) :precision binary64 (- (log (- (/ 1.0 x) 1.0))))
↓
(FPCore (x)
 :precision binary64
 (- (+ (log (sqrt (- (/ 1.0 x) 1.0))) (log (sqrt (- (/ 1.0 x) 1.0))))))
double code(double x) {
	return -log((1.0 / x) - 1.0);
}
↓
double code(double x) {
	return -(log(sqrt((1.0 / x) - 1.0)) + log(sqrt((1.0 / x) - 1.0)));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error0.0
Cost6784
\[-\log \left(\frac{1}{x} - 1\right)\]
Alternative 2
Error0.6
Cost6592
\[x + \log x\]
Alternative 3
Error1.2
Cost6464
\[\log x\]
Alternative 4
Error55.0
Cost64
\[-1\]
Alternative 5
Error62.0
Cost64
\[0\]
Alternative 6
Error63.0
Cost64
\[1\]

Error

Time

Derivation

  1. Initial program 0.0

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

  3. Applied add-sqr-sqrt_binary640.0

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

  4. Applied log-prod_binary640.0

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

  6. Applied *-un-lft-identity_binary640.0

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

  7. Applied sqrt-prod_binary640.0

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

  8. Simplified0.0

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2021040 
(FPCore (x)
  :name "neg log"
  :precision binary64
  (- (log (- (/ 1.0 x) 1.0))))