Average Error: 10.2 → 0.1
Time: 9.3s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{-1}{\left(x + 1\right) \cdot x} \cdot \frac{2}{1 - x}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{-1}{\left(x + 1\right) \cdot x} \cdot \frac{2}{1 - x}
double f(double x) {
        double r124765 = 1.0;
        double r124766 = x;
        double r124767 = r124766 + r124765;
        double r124768 = r124765 / r124767;
        double r124769 = 2.0;
        double r124770 = r124769 / r124766;
        double r124771 = r124768 - r124770;
        double r124772 = r124766 - r124765;
        double r124773 = r124765 / r124772;
        double r124774 = r124771 + r124773;
        return r124774;
}
↓
double f(double x) {
        double r124775 = -1.0;
        double r124776 = x;
        double r124777 = 1.0;
        double r124778 = r124776 + r124777;
        double r124779 = r124778 * r124776;
        double r124780 = r124775 / r124779;
        double r124781 = 2.0;
        double r124782 = r124777 - r124776;
        double r124783 = r124781 / r124782;
        double r124784 = r124780 * r124783;
        return r124784;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.2
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 10.2

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

  3. Applied frac-2neg10.2

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

  4. Simplified10.2

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

  6. Applied frac-sub26.4

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

  7. Applied frac-add25.8

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

  8. Simplified25.8

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

  9. Taylor expanded around 0 0.3

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

  11. Applied neg-mul-10.3

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

  12. Applied times-frac0.1

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

  13. Final simplification0.1

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

Reproduce

herbie shell --seed 2020043 
(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))))