Average Error: 9.9 → 0.1
Time: 14.8s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{2}{x \cdot x - 1 \cdot 1}}{\frac{x + 1}{x + 1} \cdot x}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{2}{x \cdot x - 1 \cdot 1}}{\frac{x + 1}{x + 1} \cdot x}
double f(double x) {
        double r110996 = 1.0;
        double r110997 = x;
        double r110998 = r110997 + r110996;
        double r110999 = r110996 / r110998;
        double r111000 = 2.0;
        double r111001 = r111000 / r110997;
        double r111002 = r110999 - r111001;
        double r111003 = r110997 - r110996;
        double r111004 = r110996 / r111003;
        double r111005 = r111002 + r111004;
        return r111005;
}
↓
double f(double x) {
        double r111006 = 2.0;
        double r111007 = x;
        double r111008 = r111007 * r111007;
        double r111009 = 1.0;
        double r111010 = r111009 * r111009;
        double r111011 = r111008 - r111010;
        double r111012 = r111006 / r111011;
        double r111013 = r111007 + r111009;
        double r111014 = r111013 / r111013;
        double r111015 = r111014 * r111007;
        double r111016 = r111012 / r111015;
        return r111016;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.9
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.9

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

  2. Simplified9.9

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

  4. Applied frac-sub25.7

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

  5. Applied frac-add25.3

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

  6. Simplified25.3

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

  7. Taylor expanded around 0 0.3

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

  9. Applied associate-/r*0.1

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

  11. Applied flip--0.1

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

  12. Applied associate-/r/0.1

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

  13. Applied associate-/l*0.1

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

  14. Simplified0.1

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

  15. Final simplification0.1

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

Reproduce

herbie shell --seed 2019174 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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