Average Error: 0.3 → 0.4
Time: 49.8s
Precision: 64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\log \left(e^{\frac{1 - \tan x \cdot \tan x}{\tan x \cdot \tan x + 1}}\right)\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\log \left(e^{\frac{1 - \tan x \cdot \tan x}{\tan x \cdot \tan x + 1}}\right)
double f(double x) {
        double r1798683 = 1.0;
        double r1798684 = x;
        double r1798685 = tan(r1798684);
        double r1798686 = r1798685 * r1798685;
        double r1798687 = r1798683 - r1798686;
        double r1798688 = r1798683 + r1798686;
        double r1798689 = r1798687 / r1798688;
        return r1798689;
}
↓
double f(double x) {
        double r1798690 = 1.0;
        double r1798691 = x;
        double r1798692 = tan(r1798691);
        double r1798693 = r1798692 * r1798692;
        double r1798694 = r1798690 - r1798693;
        double r1798695 = r1798693 + r1798690;
        double r1798696 = r1798694 / r1798695;
        double r1798697 = exp(r1798696);
        double r1798698 = log(r1798697);
        return r1798698;
}

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 add-log-exp0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019121 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))