Average Error: 0.3 → 0.4
Time: 22.0s
Precision: 64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{\log \left(e^{1 - \frac{\sin x}{\cos x}}\right) \cdot \left(1 + \frac{\sin x}{\cos x}\right)}{1 + \frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{\log \left(e^{1 - \frac{\sin x}{\cos x}}\right) \cdot \left(1 + \frac{\sin x}{\cos x}\right)}{1 + \frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}}
double f(double x) {
        double r543444 = 1.0;
        double r543445 = x;
        double r543446 = tan(r543445);
        double r543447 = r543446 * r543446;
        double r543448 = r543444 - r543447;
        double r543449 = r543444 + r543447;
        double r543450 = r543448 / r543449;
        return r543450;
}
↓
double f(double x) {
        double r543451 = 1.0;
        double r543452 = x;
        double r543453 = sin(r543452);
        double r543454 = cos(r543452);
        double r543455 = r543453 / r543454;
        double r543456 = r543451 - r543455;
        double r543457 = exp(r543456);
        double r543458 = log(r543457);
        double r543459 = r543451 + r543455;
        double r543460 = r543458 * r543459;
        double r543461 = r543455 * r543455;
        double r543462 = r543451 + r543461;
        double r543463 = r543460 / r543462;
        return r543463;
}

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. 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-squares0.3

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

  5. Taylor expanded around inf 0.5

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

  6. Simplified0.4

    \[\leadsto \color{blue}{\frac{\left(1 + \frac{\sin x}{\cos x}\right) \cdot \left(1 - \frac{\sin x}{\cos x}\right)}{1 + \frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}}}\]
  7. Using strategy rm

  8. Applied add-log-exp0.4

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

  9. Applied add-log-exp0.4

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

  10. Applied diff-log0.5

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

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019141 
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))