Average Error: 0.3 → 0.4
Time: 5.2s
Precision: 64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{\log \left(e^{\sqrt{1} + \tan x}\right) \cdot \left(\sqrt{1} - \tan x\right)}{1 + \tan x \cdot \tan x}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{\log \left(e^{\sqrt{1} + \tan x}\right) \cdot \left(\sqrt{1} - \tan x\right)}{1 + \tan x \cdot \tan x}
double f(double x) {
        double r12480 = 1.0;
        double r12481 = x;
        double r12482 = tan(r12481);
        double r12483 = r12482 * r12482;
        double r12484 = r12480 - r12483;
        double r12485 = r12480 + r12483;
        double r12486 = r12484 / r12485;
        return r12486;
}
↓
double f(double x) {
        double r12487 = 1.0;
        double r12488 = sqrt(r12487);
        double r12489 = x;
        double r12490 = tan(r12489);
        double r12491 = r12488 + r12490;
        double r12492 = exp(r12491);
        double r12493 = log(r12492);
        double r12494 = r12488 - r12490;
        double r12495 = r12493 * r12494;
        double r12496 = r12490 * r12490;
        double r12497 = r12487 + r12496;
        double r12498 = r12495 / r12497;
        return r12498;
}

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-sqr-sqrt0.3

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

  4. Applied difference-of-squares0.4

    \[\leadsto \frac{\color{blue}{\left(\sqrt{1} + \tan x\right) \cdot \left(\sqrt{1} - \tan x\right)}}{1 + \tan x \cdot \tan x}\]
  5. Using strategy rm

  6. Applied add-log-exp0.4

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

  7. Applied add-log-exp0.4

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

  8. Applied sum-log0.5

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2020057 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))