Average Error: 9.9 → 0.1
Time: 28.0s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.995295059253408442856425608624704182148 \lor \neg \left(x \le 120.6357054631158973734272876754403114319\right):\\ \;\;\;\;\left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + \frac{\frac{2}{x \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{\sqrt{x + 1}}}{\sqrt{x + 1}} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \end{array}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.995295059253408442856425608624704182148 \lor \neg \left(x \le 120.6357054631158973734272876754403114319\right):\\
\;\;\;\;\left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + \frac{\frac{2}{x \cdot x}}{x}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{\frac{1}{\sqrt{x + 1}}}{\sqrt{x + 1}} - \frac{2}{x}\right) + \frac{1}{x - 1}\\

\end{array}
double f(double x) {
        double r227205 = 1.0;
        double r227206 = x;
        double r227207 = r227206 + r227205;
        double r227208 = r227205 / r227207;
        double r227209 = 2.0;
        double r227210 = r227209 / r227206;
        double r227211 = r227208 - r227210;
        double r227212 = r227206 - r227205;
        double r227213 = r227205 / r227212;
        double r227214 = r227211 + r227213;
        return r227214;
}
↓
double f(double x) {
        double r227215 = x;
        double r227216 = -0.9952950592534084;
        bool r227217 = r227215 <= r227216;
        double r227218 = 120.6357054631159;
        bool r227219 = r227215 <= r227218;
        double r227220 = !r227219;
        bool r227221 = r227217 || r227220;
        double r227222 = 2.0;
        double r227223 = 7.0;
        double r227224 = pow(r227215, r227223);
        double r227225 = r227222 / r227224;
        double r227226 = 5.0;
        double r227227 = pow(r227215, r227226);
        double r227228 = r227222 / r227227;
        double r227229 = r227225 + r227228;
        double r227230 = r227215 * r227215;
        double r227231 = r227222 / r227230;
        double r227232 = r227231 / r227215;
        double r227233 = r227229 + r227232;
        double r227234 = 1.0;
        double r227235 = r227215 + r227234;
        double r227236 = sqrt(r227235);
        double r227237 = r227234 / r227236;
        double r227238 = r227237 / r227236;
        double r227239 = r227222 / r227215;
        double r227240 = r227238 - r227239;
        double r227241 = r227215 - r227234;
        double r227242 = r227234 / r227241;
        double r227243 = r227240 + r227242;
        double r227244 = r227221 ? r227233 : r227243;
        return r227244;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Split input into 2 regimes

  2. if x < -0.9952950592534084 or 120.6357054631159 < x

    1. Initial program 20.2

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

    3. Applied add-sqr-sqrt50.2

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

    4. Applied associate-/r*52.4

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

    5. Taylor expanded around inf 0.8

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{7}} + \left(2 \cdot \frac{1}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{3}}\right)}\]

    6. Simplified0.8

      \[\leadsto \color{blue}{\left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + \frac{2}{{x}^{3}}}\]
    7. Using strategy rm

    8. Applied unpow30.8

      \[\leadsto \left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + \frac{2}{\color{blue}{\left(x \cdot x\right) \cdot x}}\]

    9. Applied associate-/r*0.3

      \[\leadsto \left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + \color{blue}{\frac{\frac{2}{x \cdot x}}{x}}\]

    if -0.9952950592534084 < x < 120.6357054631159

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.0

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

    4. Applied associate-/r*0.0

      \[\leadsto \left(\color{blue}{\frac{\frac{1}{\sqrt{x + 1}}}{\sqrt{x + 1}}} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.995295059253408442856425608624704182148 \lor \neg \left(x \le 120.6357054631158973734272876754403114319\right):\\ \;\;\;\;\left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + \frac{\frac{2}{x \cdot x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{\sqrt{x + 1}}}{\sqrt{x + 1}} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 +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))))