Prelude:atanh from fay-base-0.20.0.1

Average Error: 0.0 → 0.0

Time: 8.9s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r30892 = x;
        double r30893 = 1.0;
        double r30894 = r30892 + r30893;
        double r30895 = r30893 - r30892;
        double r30896 = r30894 / r30895;
        return r30896;
}

↓

double f(double x) {
        double r30897 = x;
        double r30898 = 1.0;
        double r30899 = r30897 + r30898;
        double r30900 = r30898 - r30897;
        double r30901 = r30899 / r30900;
        return r30901;
}

Error

Bits error versus x

Derivation

1. Initial program 0.0

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

2. Simplified 0.0

   \[\leadsto \color{blue}{\frac{1 + x}{1 - x}}\]
3. Using strategy rm

4. Applied clear-num 0.0

   \[\leadsto \color{blue}{\frac{1}{\frac{1 - x}{1 + x}}}\]
5. Using strategy rm

6. Applied *-un-lft-identity 0.0

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

7. Applied *-un-lft-identity 0.0

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

8. Applied times-frac 0.0

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

9. Applied add-cube-cbrt 0.0

   \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{1}{1} \cdot \frac{1 - x}{1 + x}}\]

10. Applied times-frac 0.0

    \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{1}{1}} \cdot \frac{\sqrt[3]{1}}{\frac{1 - x}{1 + x}}}\]

11. Simplified 0.0

    \[\leadsto \color{blue}{1} \cdot \frac{\sqrt[3]{1}}{\frac{1 - x}{1 + x}}\]

12. Simplified 0.0

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

13. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x)
  :name "Prelude:atanh from fay-base-0.20.0.1"
  (/ (+ x 1.0) (- 1.0 x)))