Average Error: 9.7 → 0.1
Time: 8.6s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{1}{\left(x + 1\right) \cdot x} \cdot \frac{2}{x - 1}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{1}{\left(x + 1\right) \cdot x} \cdot \frac{2}{x - 1}
double f(double x) {
        double r162066 = 1.0;
        double r162067 = x;
        double r162068 = r162067 + r162066;
        double r162069 = r162066 / r162068;
        double r162070 = 2.0;
        double r162071 = r162070 / r162067;
        double r162072 = r162069 - r162071;
        double r162073 = r162067 - r162066;
        double r162074 = r162066 / r162073;
        double r162075 = r162072 + r162074;
        return r162075;
}
↓
double f(double x) {
        double r162076 = 1.0;
        double r162077 = x;
        double r162078 = 1.0;
        double r162079 = r162077 + r162078;
        double r162080 = r162079 * r162077;
        double r162081 = r162076 / r162080;
        double r162082 = 2.0;
        double r162083 = r162077 - r162078;
        double r162084 = r162082 / r162083;
        double r162085 = r162081 * r162084;
        return r162085;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.7
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.7

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

  3. Applied frac-sub25.5

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

  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. Simplified25.4

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

  6. Taylor expanded around 0 0.3

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

  8. Applied *-un-lft-identity0.3

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

  9. Applied times-frac0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(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))))