Average Error: 9.6 → 0.1
Time: 2.5s
Precision: binary64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{2}{x}}{x \cdot x - 1 \cdot 1}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{2}{x}}{x \cdot x - 1 \cdot 1}
(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))
↓
(FPCore (x) :precision binary64 (/ (/ 2.0 x) (- (* x x) (* 1.0 1.0))))
double code(double x) {
	return ((double) (((double) ((1.0 / ((double) (x + 1.0))) - (2.0 / x))) + (1.0 / ((double) (x - 1.0)))));
}
↓
double code(double x) {
	return ((2.0 / x) / ((double) (((double) (x * x)) - ((double) (1.0 * 1.0)))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.6
Target0.2
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program Error: 9.6 bits

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

  3. Applied frac-subError: 25.8 bits

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

  4. Applied frac-addError: 25.3 bits

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

  5. SimplifiedError: 25.3 bits

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

  6. SimplifiedError: 25.3 bits

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

  7. Taylor expanded around 0 Error: 0.2 bits

    \[\leadsto \frac{\color{blue}{2}}{x \cdot \left(x \cdot x - 1 \cdot 1\right)}\]
  8. Using strategy rm

  9. Applied associate-/r*Error: 0.1 bits

    \[\leadsto \color{blue}{\frac{\frac{2}{x}}{x \cdot x - 1 \cdot 1}}\]

  10. Final simplificationError: 0.1 bits

    \[\leadsto \frac{\frac{2}{x}}{x \cdot x - 1 \cdot 1}\]

Reproduce

herbie shell --seed 2020204 
(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))))