Average Error: 0.0 → 0.0
Time: 5.0s
Precision: 64
\[-\log \left(\frac{1}{x} - 1\right)\]
↓
\[-\log \left(\mathsf{fma}\left(\frac{\sqrt{1}}{\sqrt{x}}, \frac{\sqrt{1}}{\sqrt{x}}, -1\right)\right)\]
-\log \left(\frac{1}{x} - 1\right)
↓
-\log \left(\mathsf{fma}\left(\frac{\sqrt{1}}{\sqrt{x}}, \frac{\sqrt{1}}{\sqrt{x}}, -1\right)\right)
double code(double x) {
	return -log(((1.0 / x) - 1.0));
}
↓
double code(double x) {
	return -log(fma((sqrt(1.0) / sqrt(x)), (sqrt(1.0) / sqrt(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-sqrt0.0

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

  4. Applied add-sqr-sqrt0.0

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

  5. Applied times-frac0.0

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

  6. Applied fma-neg0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (x)
  :name "neg log"
  :precision binary64
  (- (log (- (/ 1 x) 1))))