Average Error: 9.4 → 0.1
Time: 9.6s
Precision: binary64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
↓
\[\frac{\frac{2}{x + 1}}{x \cdot \left(x - 1\right)} \]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{2}{x + 1}}{x \cdot \left(x - 1\right)}

(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))
↓
(FPCore (x) :precision binary64 (/ (/ 2.0 (+ x 1.0)) (* x (- x 1.0))))
double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}
↓
double code(double x) {
	return (2.0 / (x + 1.0)) / (x * (x - 1.0));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.4
Target0.2
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)} \]

Derivation

  1. Initial program 9.4

    \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

  2. Applied associate-+l-_binary649.4

    \[\leadsto \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]

  3. Applied frac-sub_binary6425.7

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

  4. Applied frac-sub_binary6425.3

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

  5. Taylor expanded in x around 0 0.2

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

  6. Applied associate-/r*_binary640.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2022024 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))