Average Error: 9.9 → 0.3
Time: 3.6m
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{2}{\mathsf{fma}\left(x, x, -1\right) \cdot x}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{2}{\mathsf{fma}\left(x, x, -1\right) \cdot x}
double f(double x) {
        double r15724146 = 1.0;
        double r15724147 = x;
        double r15724148 = r15724147 + r15724146;
        double r15724149 = r15724146 / r15724148;
        double r15724150 = 2.0;
        double r15724151 = r15724150 / r15724147;
        double r15724152 = r15724149 - r15724151;
        double r15724153 = r15724147 - r15724146;
        double r15724154 = r15724146 / r15724153;
        double r15724155 = r15724152 + r15724154;
        return r15724155;
}
↓
double f(double x) {
        double r15724156 = 2.0;
        double r15724157 = x;
        double r15724158 = -1.0;
        double r15724159 = fma(r15724157, r15724157, r15724158);
        double r15724160 = r15724159 * r15724157;
        double r15724161 = r15724156 / r15724160;
        return r15724161;
}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 9.9

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

  3. Applied frac-sub26.1

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

  4. Applied frac-add25.4

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

  5. Simplified25.8

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

  6. Simplified25.8

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

  7. Taylor expanded around inf 0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019120 +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))))