Average Error: 9.5 → 0.1
Time: 7.6s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{\frac{2}{x + 1}}{x}}{x - 1}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{\frac{2}{x + 1}}{x}}{x - 1}
double f(double x) {
        double r121304 = 1.0;
        double r121305 = x;
        double r121306 = r121305 + r121304;
        double r121307 = r121304 / r121306;
        double r121308 = 2.0;
        double r121309 = r121308 / r121305;
        double r121310 = r121307 - r121309;
        double r121311 = r121305 - r121304;
        double r121312 = r121304 / r121311;
        double r121313 = r121310 + r121312;
        return r121313;
}
↓
double f(double x) {
        double r121314 = 2.0;
        double r121315 = x;
        double r121316 = 1.0;
        double r121317 = r121315 + r121316;
        double r121318 = r121314 / r121317;
        double r121319 = r121318 / r121315;
        double r121320 = r121315 - r121316;
        double r121321 = r121319 / r121320;
        return r121321;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.5
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.5

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

  3. Applied frac-sub26.0

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

  4. Applied frac-add25.5

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

  5. Taylor expanded around 0 0.3

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

  7. Applied associate-/r*0.1

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

  9. Applied associate-/r*0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2020047 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

  :herbie-target
  (/ 2 (* x (- (* x x) 1)))

  (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))))