Average Error: 0.3 → 0.4
Time: 19.9s
Precision: 64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{\left(1 + \tan x\right) \cdot \left(1 - \tan x\right)}{1 + \tan x \cdot \tan x}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{\left(1 + \tan x\right) \cdot \left(1 - \tan x\right)}{1 + \tan x \cdot \tan x}
double f(double x) {
        double r335636 = 1.0;
        double r335637 = x;
        double r335638 = tan(r335637);
        double r335639 = r335638 * r335638;
        double r335640 = r335636 - r335639;
        double r335641 = r335636 + r335639;
        double r335642 = r335640 / r335641;
        return r335642;
}
↓
double f(double x) {
        double r335643 = 1.0;
        double r335644 = x;
        double r335645 = tan(r335644);
        double r335646 = r335643 + r335645;
        double r335647 = r335643 - r335645;
        double r335648 = r335646 * r335647;
        double r335649 = r335645 * r335645;
        double r335650 = r335643 + r335649;
        double r335651 = r335648 / r335650;
        return r335651;
}

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

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

  4. Applied difference-of-squares0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2019155 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))