Trigonometry B

Average Error: 0.3 → 0.3

Time: 2.1m

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r5014025 = 1.0;
        double r5014026 = x;
        double r5014027 = tan(r5014026);
        double r5014028 = r5014027 * r5014027;
        double r5014029 = r5014025 - r5014028;
        double r5014030 = r5014025 + r5014028;
        double r5014031 = r5014029 / r5014030;
        return r5014031;
}

↓

double f(double x) {
        double r5014032 = 1.0;
        double r5014033 = x;
        double r5014034 = tan(r5014033);
        double r5014035 = r5014034 * r5014034;
        double r5014036 = r5014032 - r5014035;
        double r5014037 = fma(r5014034, r5014034, r5014032);
        double r5014038 = r5014036 / r5014037;
        return r5014038;
}

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

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

4. Applied difference-of-squares 0.4

   \[\leadsto \frac{\color{blue}{\left(1 + \tan x\right) \cdot \left(1 - \tan x\right)}}{1 + \tan x \cdot \tan x}\]
5. Using strategy rm

6. Applied flip-+ 0.3

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

7. Applied associate-*l/ 0.4

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

8. Applied associate-/l/ 0.4

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

9. Simplified 0.4

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

11. Applied associate-/l* 0.3

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

12. Simplified 0.3

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

13. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))