Asymptote A

Average Error: 14.6 → 0.4

Time: 4.4m

Precision: 64

Math

\[\frac{1}{x + 1} - \frac{1}{x - 1}\]

↓

\[-2 \cdot \mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(\frac{1}{\mathsf{fma}\left(x, x, -1\right)}\right)\right)\right)\right)\]

C

double f(double x) {
        double r30253942 = 1.0;
        double r30253943 = x;
        double r30253944 = r30253943 + r30253942;
        double r30253945 = r30253942 / r30253944;
        double r30253946 = r30253943 - r30253942;
        double r30253947 = r30253942 / r30253946;
        double r30253948 = r30253945 - r30253947;
        return r30253948;
}

↓

double f(double x) {
        double r30253949 = -2.0;
        double r30253950 = 1.0;
        double r30253951 = x;
        double r30253952 = -1.0;
        double r30253953 = fma(r30253951, r30253951, r30253952);
        double r30253954 = r30253950 / r30253953;
        double r30253955 = expm1(r30253954);
        double r30253956 = log1p(r30253955);
        double r30253957 = r30253949 * r30253956;
        return r30253957;
}

Error

Bits error versus x

Derivation

1. Initial program 14.6

   \[\frac{1}{x + 1} - \frac{1}{x - 1}\]
2. Using strategy rm

3. Applied flip-- 28.9

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

4. Applied associate-/r/ 29.0

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

5. Applied flip-+ 14.6

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

6. Applied associate-/r/ 14.6

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

7. Applied distribute-lft-out-- 14.0

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

8. Simplified 0.4

   \[\leadsto \frac{1}{x \cdot x - 1 \cdot 1} \cdot \color{blue}{-2}\]
9. Using strategy rm

10. Applied log1p-expm1-u 0.4

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(\frac{1}{x \cdot x - 1 \cdot 1}\right)\right)\right)\right)} \cdot -2\]

11. Simplified 0.4

    \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(\mathsf{expm1}\left(\left(\frac{1}{\mathsf{fma}\left(x, x, -1\right)}\right)\right)\right)}\right) \cdot -2\]

12. Final simplification 0.4

    \[\leadsto -2 \cdot \mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(\frac{1}{\mathsf{fma}\left(x, x, -1\right)}\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x)
  :name "Asymptote A"
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))