Average Error: 0.3 → 0.4
Time: 5.5s
Precision: 64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{\frac{\left(\sqrt{1} - \tan x\right) \cdot \left(1 + \left(-\tan x \cdot \tan x\right)\right)}{\sqrt{1} - \tan x}}{1 + \tan x \cdot \tan x}\]
\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}
↓
\frac{\frac{\left(\sqrt{1} - \tan x\right) \cdot \left(1 + \left(-\tan x \cdot \tan x\right)\right)}{\sqrt{1} - \tan x}}{1 + \tan x \cdot \tan x}
double f(double x) {
        double r12487 = 1.0;
        double r12488 = x;
        double r12489 = tan(r12488);
        double r12490 = r12489 * r12489;
        double r12491 = r12487 - r12490;
        double r12492 = r12487 + r12490;
        double r12493 = r12491 / r12492;
        return r12493;
}
↓
double f(double x) {
        double r12494 = 1.0;
        double r12495 = sqrt(r12494);
        double r12496 = x;
        double r12497 = tan(r12496);
        double r12498 = r12495 - r12497;
        double r12499 = r12497 * r12497;
        double r12500 = -r12499;
        double r12501 = r12494 + r12500;
        double r12502 = r12498 * r12501;
        double r12503 = r12502 / r12498;
        double r12504 = r12494 + r12499;
        double r12505 = r12503 / r12504;
        return r12505;
}

Error

Bits error versus x

Try it out

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 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

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

  7. Applied add-log-exp0.4

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

  8. Applied diff-log0.5

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

  9. Simplified0.4

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

  11. Applied flip-+0.4

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

  12. Applied associate-*l/0.4

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

  13. Simplified0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))