Average Error: 0.3 → 0.4
Time: 20.3s
Precision: 64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\left(\sqrt{1} + \tan x\right) \cdot \frac{\sqrt{1} - \tan x}{1 + \tan x \cdot \tan x}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\left(\sqrt{1} + \tan x\right) \cdot \frac{\sqrt{1} - \tan x}{1 + \tan x \cdot \tan x}
double f(double x) {
        double r745188 = 1.0;
        double r745189 = x;
        double r745190 = tan(r745189);
        double r745191 = r745190 * r745190;
        double r745192 = r745188 - r745191;
        double r745193 = r745188 + r745191;
        double r745194 = r745192 / r745193;
        return r745194;
}
↓
double f(double x) {
        double r745195 = 1.0;
        double r745196 = sqrt(r745195);
        double r745197 = x;
        double r745198 = tan(r745197);
        double r745199 = r745196 + r745198;
        double r745200 = r745196 - r745198;
        double r745201 = r745198 * r745198;
        double r745202 = r745195 + r745201;
        double r745203 = r745200 / r745202;
        double r745204 = r745199 * r745203;
        return r745204;
}

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

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019192 
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))