Average Error: 0.3 → 0.3
Time: 22.0s
Precision: 64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1 - \frac{\tan x \cdot \sin x}{\cos x}}{1 + \frac{\tan x \cdot \sin x}{\cos x}}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{1 - \frac{\tan x \cdot \sin x}{\cos x}}{1 + \frac{\tan x \cdot \sin x}{\cos x}}
double f(double x) {
        double r623477 = 1.0;
        double r623478 = x;
        double r623479 = tan(r623478);
        double r623480 = r623479 * r623479;
        double r623481 = r623477 - r623480;
        double r623482 = r623477 + r623480;
        double r623483 = r623481 / r623482;
        return r623483;
}
↓
double f(double x) {
        double r623484 = 1.0;
        double r623485 = x;
        double r623486 = tan(r623485);
        double r623487 = sin(r623485);
        double r623488 = r623486 * r623487;
        double r623489 = cos(r623485);
        double r623490 = r623488 / r623489;
        double r623491 = r623484 - r623490;
        double r623492 = r623484 + r623490;
        double r623493 = r623491 / r623492;
        return r623493;
}

Error

Bits error versus x

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Results

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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 tan-quot0.4

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

  4. Applied associate-*r/0.4

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

  6. Applied tan-quot0.4

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

  7. Applied associate-*r/0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))