Average Error: 0.3 → 0.4
Time: 4.6s
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 r12216 = 1.0;
        double r12217 = x;
        double r12218 = tan(r12217);
        double r12219 = r12218 * r12218;
        double r12220 = r12216 - r12219;
        double r12221 = r12216 + r12219;
        double r12222 = r12220 / r12221;
        return r12222;
}
↓
double f(double x) {
        double r12223 = 1.0;
        double r12224 = sqrt(r12223);
        double r12225 = x;
        double r12226 = tan(r12225);
        double r12227 = r12224 + r12226;
        double r12228 = r12224 - r12226;
        double r12229 = r12226 * r12226;
        double r12230 = r12223 + r12229;
        double r12231 = r12228 / r12230;
        double r12232 = r12227 * r12231;
        return r12232;
}

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-squares0.3

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

  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 2020059 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))