Average Error: 9.9 → 0.2
Time: 5.6s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{2}{{x}^{3} - 1 \cdot x}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{2}{{x}^{3} - 1 \cdot x}
double f(double x) {
        double r136936 = 1.0;
        double r136937 = x;
        double r136938 = r136937 + r136936;
        double r136939 = r136936 / r136938;
        double r136940 = 2.0;
        double r136941 = r136940 / r136937;
        double r136942 = r136939 - r136941;
        double r136943 = r136937 - r136936;
        double r136944 = r136936 / r136943;
        double r136945 = r136942 + r136944;
        return r136945;
}
↓
double f(double x) {
        double r136946 = 2.0;
        double r136947 = x;
        double r136948 = 3.0;
        double r136949 = pow(r136947, r136948);
        double r136950 = 1.0;
        double r136951 = r136950 * r136947;
        double r136952 = r136949 - r136951;
        double r136953 = r136946 / r136952;
        return r136953;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 9.9

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

  3. Applied frac-sub26.2

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

  4. Applied frac-add25.9

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

  5. Simplified26.4

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

  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. Taylor expanded around 0 0.2

    \[\leadsto \frac{2}{\color{blue}{{x}^{3} - 1 \cdot x}}\]

  8. Final simplification0.2

    \[\leadsto \frac{2}{{x}^{3} - 1 \cdot x}\]

Reproduce

herbie shell --seed 2020024 +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))))