Average Error: 0.3 → 0.3
Time: 21.3s
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 r1108379 = 1.0;
        double r1108380 = x;
        double r1108381 = tan(r1108380);
        double r1108382 = r1108381 * r1108381;
        double r1108383 = r1108379 - r1108382;
        double r1108384 = r1108379 + r1108382;
        double r1108385 = r1108383 / r1108384;
        return r1108385;
}
↓
double f(double x) {
        double r1108386 = 1.0;
        double r1108387 = x;
        double r1108388 = tan(r1108387);
        double r1108389 = sin(r1108387);
        double r1108390 = r1108388 * r1108389;
        double r1108391 = cos(r1108387);
        double r1108392 = r1108390 / r1108391;
        double r1108393 = r1108386 - r1108392;
        double r1108394 = r1108386 + r1108392;
        double r1108395 = r1108393 / r1108394;
        return r1108395;
}

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 2019170 
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))