Average Error: 0.3 → 0.4
Time: 16.1s
Precision: 64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1 - \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}}{1 + \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{1 - \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}}{1 + \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}}
double f(double x) {
        double r20045 = 1.0;
        double r20046 = x;
        double r20047 = tan(r20046);
        double r20048 = r20047 * r20047;
        double r20049 = r20045 - r20048;
        double r20050 = r20045 + r20048;
        double r20051 = r20049 / r20050;
        return r20051;
}
↓
double f(double x) {
        double r20052 = 1.0;
        double r20053 = x;
        double r20054 = sin(r20053);
        double r20055 = 2.0;
        double r20056 = pow(r20054, r20055);
        double r20057 = cos(r20053);
        double r20058 = pow(r20057, r20055);
        double r20059 = r20056 / r20058;
        double r20060 = r20052 - r20059;
        double r20061 = r20052 + r20059;
        double r20062 = r20060 / r20061;
        return r20062;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.3

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

  2. Simplified0.3

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-\tan x, \tan x, 1\right)}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}}\]

  3. Taylor expanded around inf 0.4

    \[\leadsto \color{blue}{\frac{1 - \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}}{\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} + 1}}\]

  4. Simplified0.4

    \[\leadsto \color{blue}{\frac{1 - \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}}{1 + \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}}}\]

  5. Final simplification0.4

    \[\leadsto \frac{1 - \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}}{1 + \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))