Average Error: 0.3 → 0.4
Time: 1.1m
Precision: 64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\left(1 + \tan x\right) \cdot \frac{1 - \tan x}{\tan x \cdot \tan x + 1}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\left(1 + \tan x\right) \cdot \frac{1 - \tan x}{\tan x \cdot \tan x + 1}
double f(double x) {
        double r521258 = 1.0;
        double r521259 = x;
        double r521260 = tan(r521259);
        double r521261 = r521260 * r521260;
        double r521262 = r521258 - r521261;
        double r521263 = r521258 + r521261;
        double r521264 = r521262 / r521263;
        return r521264;
}
↓
double f(double x) {
        double r521265 = 1.0;
        double r521266 = x;
        double r521267 = tan(r521266);
        double r521268 = r521265 + r521267;
        double r521269 = r521265 - r521267;
        double r521270 = r521267 * r521267;
        double r521271 = r521270 + r521265;
        double r521272 = r521269 / r521271;
        double r521273 = r521268 * r521272;
        return r521273;
}

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 *-un-lft-identity0.3

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

  4. Applied add-sqr-sqrt0.3

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

  5. Applied difference-of-squares0.4

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

  6. Applied times-frac0.4

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

  7. Simplified0.4

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

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019152 
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))