Average Error: 10.0 → 0.1
Time: 16.1s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{2}{\left(x + 1\right) \cdot x}}{x - 1}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{2}{\left(x + 1\right) \cdot x}}{x - 1}
double f(double x) {
        double r1826202 = 1.0;
        double r1826203 = x;
        double r1826204 = r1826203 + r1826202;
        double r1826205 = r1826202 / r1826204;
        double r1826206 = 2.0;
        double r1826207 = r1826206 / r1826203;
        double r1826208 = r1826205 - r1826207;
        double r1826209 = r1826203 - r1826202;
        double r1826210 = r1826202 / r1826209;
        double r1826211 = r1826208 + r1826210;
        return r1826211;
}
↓
double f(double x) {
        double r1826212 = 2.0;
        double r1826213 = x;
        double r1826214 = 1.0;
        double r1826215 = r1826213 + r1826214;
        double r1826216 = r1826215 * r1826213;
        double r1826217 = r1826212 / r1826216;
        double r1826218 = r1826213 - r1826214;
        double r1826219 = r1826217 / r1826218;
        return r1826219;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original10.0
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 10.0

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

  3. Applied frac-sub26.3

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

  4. Applied frac-add25.6

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

  5. Simplified26.1

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

  6. Simplified26.1

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

  7. Taylor expanded around 0 0.3

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

  9. Applied associate-/r*0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019155 +o rules:numerics
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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