Average Error: 36.7 → 13.0
Time: 1.1m
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \left(\sqrt[3]{\frac{\sin \varepsilon}{\cos \varepsilon}} \cdot \sqrt[3]{\frac{\sin \varepsilon}{\cos \varepsilon}}\right)\right) \cdot \left(\sqrt[3]{\frac{1}{\sqrt[3]{\cos \varepsilon} \cdot \sqrt[3]{\cos \varepsilon}}} \cdot \sqrt[3]{\frac{\sin \varepsilon}{\sqrt[3]{\cos \varepsilon}}}\right)} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right)\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \left(\sqrt[3]{\frac{\sin \varepsilon}{\cos \varepsilon}} \cdot \sqrt[3]{\frac{\sin \varepsilon}{\cos \varepsilon}}\right)\right) \cdot \left(\sqrt[3]{\frac{1}{\sqrt[3]{\cos \varepsilon} \cdot \sqrt[3]{\cos \varepsilon}}} \cdot \sqrt[3]{\frac{\sin \varepsilon}{\sqrt[3]{\cos \varepsilon}}}\right)} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right)
double f(double x, double eps) {
        double r6918473 = x;
        double r6918474 = eps;
        double r6918475 = r6918473 + r6918474;
        double r6918476 = tan(r6918475);
        double r6918477 = tan(r6918473);
        double r6918478 = r6918476 - r6918477;
        return r6918478;
}
↓
double f(double x, double eps) {
        double r6918479 = eps;
        double r6918480 = sin(r6918479);
        double r6918481 = cos(r6918479);
        double r6918482 = r6918480 / r6918481;
        double r6918483 = 1.0;
        double r6918484 = x;
        double r6918485 = sin(r6918484);
        double r6918486 = cos(r6918484);
        double r6918487 = r6918485 / r6918486;
        double r6918488 = cbrt(r6918482);
        double r6918489 = r6918488 * r6918488;
        double r6918490 = r6918487 * r6918489;
        double r6918491 = cbrt(r6918481);
        double r6918492 = r6918491 * r6918491;
        double r6918493 = r6918483 / r6918492;
        double r6918494 = cbrt(r6918493);
        double r6918495 = r6918480 / r6918491;
        double r6918496 = cbrt(r6918495);
        double r6918497 = r6918494 * r6918496;
        double r6918498 = r6918490 * r6918497;
        double r6918499 = r6918483 - r6918498;
        double r6918500 = r6918482 / r6918499;
        double r6918501 = r6918487 * r6918482;
        double r6918502 = r6918483 - r6918501;
        double r6918503 = r6918487 / r6918502;
        double r6918504 = r6918503 - r6918487;
        double r6918505 = r6918500 + r6918504;
        return r6918505;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.7
Target14.9
Herbie13.0
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.7

    \[\tan \left(x + \varepsilon\right) - \tan x\]
  2. Using strategy rm

  3. Applied tan-sum21.7

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

  4. Taylor expanded around inf 21.8

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

  5. Simplified13.0

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

  7. Applied add-cube-cbrt13.1

    \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin x}{\cos x} \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{\sin \varepsilon}{\cos \varepsilon}} \cdot \sqrt[3]{\frac{\sin \varepsilon}{\cos \varepsilon}}\right) \cdot \sqrt[3]{\frac{\sin \varepsilon}{\cos \varepsilon}}\right)}} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right)\]

  8. Applied associate-*r*13.1

    \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \color{blue}{\left(\frac{\sin x}{\cos x} \cdot \left(\sqrt[3]{\frac{\sin \varepsilon}{\cos \varepsilon}} \cdot \sqrt[3]{\frac{\sin \varepsilon}{\cos \varepsilon}}\right)\right) \cdot \sqrt[3]{\frac{\sin \varepsilon}{\cos \varepsilon}}}} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right)\]
  9. Using strategy rm

  10. Applied add-cube-cbrt13.1

    \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \left(\sqrt[3]{\frac{\sin \varepsilon}{\cos \varepsilon}} \cdot \sqrt[3]{\frac{\sin \varepsilon}{\cos \varepsilon}}\right)\right) \cdot \sqrt[3]{\frac{\sin \varepsilon}{\color{blue}{\left(\sqrt[3]{\cos \varepsilon} \cdot \sqrt[3]{\cos \varepsilon}\right) \cdot \sqrt[3]{\cos \varepsilon}}}}} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right)\]

  11. Applied *-un-lft-identity13.1

    \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \left(\sqrt[3]{\frac{\sin \varepsilon}{\cos \varepsilon}} \cdot \sqrt[3]{\frac{\sin \varepsilon}{\cos \varepsilon}}\right)\right) \cdot \sqrt[3]{\frac{\color{blue}{1 \cdot \sin \varepsilon}}{\left(\sqrt[3]{\cos \varepsilon} \cdot \sqrt[3]{\cos \varepsilon}\right) \cdot \sqrt[3]{\cos \varepsilon}}}} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right)\]

  12. Applied times-frac13.0

    \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \left(\sqrt[3]{\frac{\sin \varepsilon}{\cos \varepsilon}} \cdot \sqrt[3]{\frac{\sin \varepsilon}{\cos \varepsilon}}\right)\right) \cdot \sqrt[3]{\color{blue}{\frac{1}{\sqrt[3]{\cos \varepsilon} \cdot \sqrt[3]{\cos \varepsilon}} \cdot \frac{\sin \varepsilon}{\sqrt[3]{\cos \varepsilon}}}}} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right)\]

  13. Applied cbrt-prod13.0

    \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \left(\sqrt[3]{\frac{\sin \varepsilon}{\cos \varepsilon}} \cdot \sqrt[3]{\frac{\sin \varepsilon}{\cos \varepsilon}}\right)\right) \cdot \color{blue}{\left(\sqrt[3]{\frac{1}{\sqrt[3]{\cos \varepsilon} \cdot \sqrt[3]{\cos \varepsilon}}} \cdot \sqrt[3]{\frac{\sin \varepsilon}{\sqrt[3]{\cos \varepsilon}}}\right)}} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right)\]

  14. Final simplification13.0

    \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \left(\frac{\sin x}{\cos x} \cdot \left(\sqrt[3]{\frac{\sin \varepsilon}{\cos \varepsilon}} \cdot \sqrt[3]{\frac{\sin \varepsilon}{\cos \varepsilon}}\right)\right) \cdot \left(\sqrt[3]{\frac{1}{\sqrt[3]{\cos \varepsilon} \cdot \sqrt[3]{\cos \varepsilon}}} \cdot \sqrt[3]{\frac{\sin \varepsilon}{\sqrt[3]{\cos \varepsilon}}}\right)} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}} - \frac{\sin x}{\cos x}\right)\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

  (- (tan (+ x eps)) (tan x)))