Average Error: 9.6 → 0.1
Time: 6.6s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{2}{x}}{\left(x - 1\right) \cdot \left(x + 1\right)}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{2}{x}}{\left(x - 1\right) \cdot \left(x + 1\right)}
double f(double x) {
        double r92374 = 1.0;
        double r92375 = x;
        double r92376 = r92375 + r92374;
        double r92377 = r92374 / r92376;
        double r92378 = 2.0;
        double r92379 = r92378 / r92375;
        double r92380 = r92377 - r92379;
        double r92381 = r92375 - r92374;
        double r92382 = r92374 / r92381;
        double r92383 = r92380 + r92382;
        return r92383;
}
↓
double f(double x) {
        double r92384 = 2.0;
        double r92385 = x;
        double r92386 = r92384 / r92385;
        double r92387 = 1.0;
        double r92388 = r92385 - r92387;
        double r92389 = r92385 + r92387;
        double r92390 = r92388 * r92389;
        double r92391 = r92386 / r92390;
        return r92391;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.6
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.6

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

  3. Applied frac-sub25.7

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

  4. Applied frac-add25.0

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

  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 *-un-lft-identity0.1

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

  10. Applied times-frac0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2019304 
(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))))