Average Error: 0.3 → 0.3
Time: 21.5s
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 r846588 = 1.0;
        double r846589 = x;
        double r846590 = tan(r846589);
        double r846591 = r846590 * r846590;
        double r846592 = r846588 - r846591;
        double r846593 = r846588 + r846591;
        double r846594 = r846592 / r846593;
        return r846594;
}
↓
double f(double x) {
        double r846595 = 1.0;
        double r846596 = x;
        double r846597 = tan(r846596);
        double r846598 = sin(r846596);
        double r846599 = r846597 * r846598;
        double r846600 = cos(r846596);
        double r846601 = r846599 / r846600;
        double r846602 = r846595 - r846601;
        double r846603 = r846595 + r846601;
        double r846604 = r846602 / r846603;
        return r846604;
}

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 2019172 
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))