Average Error: 0.3 → 0.4
Time: 25.9s
Precision: 64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1 - \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}}{\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} + 1}\]
\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}}}{\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} + 1}
double f(double x) {
        double r22514 = 1.0;
        double r22515 = x;
        double r22516 = tan(r22515);
        double r22517 = r22516 * r22516;
        double r22518 = r22514 - r22517;
        double r22519 = r22514 + r22517;
        double r22520 = r22518 / r22519;
        return r22520;
}
↓
double f(double x) {
        double r22521 = 1.0;
        double r22522 = x;
        double r22523 = sin(r22522);
        double r22524 = 2.0;
        double r22525 = pow(r22523, r22524);
        double r22526 = cos(r22522);
        double r22527 = pow(r22526, r22524);
        double r22528 = r22525 / r22527;
        double r22529 = r22521 - r22528;
        double r22530 = r22528 + r22521;
        double r22531 = r22529 / r22530;
        return r22531;
}

Error

Bits error versus x

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Your Program's Arguments

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

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

  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. Final simplification0.4

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

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))