Average Error: 0.0 → 0.0
Time: 2.3s
Precision: binary64
\[-\log \left(\frac{1}{x} - 1\right)\]
↓
\[-\log \left(\left(1 + \sqrt{\frac{1}{x}}\right) \cdot \left(\sqrt{\frac{1}{x}} + -1\right)\right)\]
-\log \left(\frac{1}{x} - 1\right)
↓
-\log \left(\left(1 + \sqrt{\frac{1}{x}}\right) \cdot \left(\sqrt{\frac{1}{x}} + -1\right)\right)
(FPCore (x) :precision binary64 (- (log (- (/ 1.0 x) 1.0))))
↓
(FPCore (x)
 :precision binary64
 (- (log (* (+ 1.0 (sqrt (/ 1.0 x))) (+ (sqrt (/ 1.0 x)) -1.0)))))
double code(double x) {
	return -log((1.0 / x) - 1.0);
}
↓
double code(double x) {
	return -log((1.0 + sqrt(1.0 / x)) * (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

Derivation

  1. Initial program 0.0

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

  3. Applied add-sqr-sqrt_binary640.0

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

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

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

  5. Simplified0.0

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020233 
(FPCore (x)
  :name "neg log"
  :precision binary64
  (- (log (- (/ 1.0 x) 1.0))))