Trigonometry B

Average Error: 0.3 → 0.4

Time: 23.5s

Precision: 64

Math

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

↓

\[\frac{1}{\frac{1}{\sqrt{1} - \tan x}} \cdot \frac{\tan x + \sqrt{1}}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}\]

C

double f(double x) {
        double r859247 = 1.0;
        double r859248 = x;
        double r859249 = tan(r859248);
        double r859250 = r859249 * r859249;
        double r859251 = r859247 - r859250;
        double r859252 = r859247 + r859250;
        double r859253 = r859251 / r859252;
        return r859253;
}

↓

double f(double x) {
        double r859254 = 1.0;
        double r859255 = 1.0;
        double r859256 = sqrt(r859255);
        double r859257 = x;
        double r859258 = tan(r859257);
        double r859259 = r859256 - r859258;
        double r859260 = r859254 / r859259;
        double r859261 = r859254 / r859260;
        double r859262 = r859258 + r859256;
        double r859263 = fma(r859258, r859258, r859255);
        double r859264 = r859262 / r859263;
        double r859265 = r859261 * r859264;
        return r859265;
}

Error

Bits error versus x

Derivation

1. Initial program 0.3

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

3. Applied *-un-lft-identity 0.3

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

4. Applied add-sqr-sqrt 0.3

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

5. Applied difference-of-squares 0.4

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

6. Applied times-frac 0.4

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

7. Simplified 0.4

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

8. Simplified 0.4

   \[\leadsto \left(\sqrt{1} + \tan x\right) \cdot \color{blue}{\frac{\sqrt{1} - \tan x}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}}\]
9. Using strategy rm

10. Applied clear-num 0.4

    \[\leadsto \left(\sqrt{1} + \tan x\right) \cdot \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\tan x, \tan x, 1\right)}{\sqrt{1} - \tan x}}}\]
11. Using strategy rm

12. Applied div-inv 0.4

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

13. Applied *-un-lft-identity 0.4

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

14. Applied times-frac 0.4

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

15. Applied associate-*r* 0.4

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

16. Simplified 0.4

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

17. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))