Average Error: 10.3 → 0.1
Time: 1.2m
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{2}{x + -1}}{\left(x + 1\right) \cdot x}\]
double f(double x) {
        double r11457292 = 1.0;
        double r11457293 = x;
        double r11457294 = r11457293 + r11457292;
        double r11457295 = r11457292 / r11457294;
        double r11457296 = 2.0;
        double r11457297 = r11457296 / r11457293;
        double r11457298 = r11457295 - r11457297;
        double r11457299 = r11457293 - r11457292;
        double r11457300 = r11457292 / r11457299;
        double r11457301 = r11457298 + r11457300;
        return r11457301;
}
↓
double f(double x) {
        double r11457302 = 2.0;
        double r11457303 = x;
        double r11457304 = -1.0;
        double r11457305 = r11457303 + r11457304;
        double r11457306 = r11457302 / r11457305;
        double r11457307 = 1.0;
        double r11457308 = r11457303 + r11457307;
        double r11457309 = r11457308 * r11457303;
        double r11457310 = r11457306 / r11457309;
        return r11457310;
}


\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{2}{x + -1}}{\left(x + 1\right) \cdot x}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 10.3

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

  3. Applied frac-sub25.8

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

  4. Applied frac-add25.1

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

  5. Taylor expanded around inf 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 add-sqr-sqrt1.0

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

  8. Applied times-frac0.5

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

  10. Applied associate-*l/0.7

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019101 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

  :herbie-target
  (/ 2 (* x (- (* x x) 1)))

  (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))))