Average Error: 10.1 → 0.3
Time: 16.0s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{2}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{2}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}
double f(double x) {
        double r4623370 = 1.0;
        double r4623371 = x;
        double r4623372 = r4623371 + r4623370;
        double r4623373 = r4623370 / r4623372;
        double r4623374 = 2.0;
        double r4623375 = r4623374 / r4623371;
        double r4623376 = r4623373 - r4623375;
        double r4623377 = r4623371 - r4623370;
        double r4623378 = r4623370 / r4623377;
        double r4623379 = r4623376 + r4623378;
        return r4623379;
}
↓
double f(double x) {
        double r4623380 = 2.0;
        double r4623381 = x;
        double r4623382 = 1.0;
        double r4623383 = r4623381 + r4623382;
        double r4623384 = r4623383 * r4623381;
        double r4623385 = r4623381 - r4623382;
        double r4623386 = r4623384 * r4623385;
        double r4623387 = r4623380 / r4623386;
        return r4623387;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 10.1

    \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
  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-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 0 0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019158 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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