Average Error: 9.5 → 0.1
Time: 38.7s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[2 \cdot \frac{\frac{1}{x}}{x \cdot x - 1}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
2 \cdot \frac{\frac{1}{x}}{x \cdot x - 1}
double f(double x) {
        double r3625224 = 1.0;
        double r3625225 = x;
        double r3625226 = r3625225 + r3625224;
        double r3625227 = r3625224 / r3625226;
        double r3625228 = 2.0;
        double r3625229 = r3625228 / r3625225;
        double r3625230 = r3625227 - r3625229;
        double r3625231 = r3625225 - r3625224;
        double r3625232 = r3625224 / r3625231;
        double r3625233 = r3625230 + r3625232;
        return r3625233;
}
↓
double f(double x) {
        double r3625234 = 2.0;
        double r3625235 = 1.0;
        double r3625236 = x;
        double r3625237 = r3625235 / r3625236;
        double r3625238 = r3625236 * r3625236;
        double r3625239 = r3625238 - r3625235;
        double r3625240 = r3625237 / r3625239;
        double r3625241 = r3625234 * r3625240;
        return r3625241;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.5
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.5

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

  3. Applied frac-sub25.6

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

  4. Applied frac-add24.9

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

  5. 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 div-inv0.3

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

  8. Simplified0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2019149 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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