Average Error: 36.3 → 12.5
Time: 54.5s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon}}{\cos x}} + \frac{\left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon}}{\cos x}} - \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon}}{\cos x}}\right)}{\frac{\sin x}{\cos x} + \frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon}}{\cos x}}}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon}}{\cos x}} + \frac{\left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon}}{\cos x}} - \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon}}{\cos x}}\right)}{\frac{\sin x}{\cos x} + \frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon}}{\cos x}}}
double f(double x, double eps) {
        double r6758589 = x;
        double r6758590 = eps;
        double r6758591 = r6758589 + r6758590;
        double r6758592 = tan(r6758591);
        double r6758593 = tan(r6758589);
        double r6758594 = r6758592 - r6758593;
        return r6758594;
}
↓
double f(double x, double eps) {
        double r6758595 = eps;
        double r6758596 = sin(r6758595);
        double r6758597 = cos(r6758595);
        double r6758598 = r6758596 / r6758597;
        double r6758599 = 1.0;
        double r6758600 = x;
        double r6758601 = sin(r6758600);
        double r6758602 = r6758601 / r6758597;
        double r6758603 = r6758596 * r6758602;
        double r6758604 = cos(r6758600);
        double r6758605 = r6758603 / r6758604;
        double r6758606 = r6758599 - r6758605;
        double r6758607 = r6758598 / r6758606;
        double r6758608 = r6758601 / r6758604;
        double r6758609 = r6758608 / r6758606;
        double r6758610 = r6758609 - r6758608;
        double r6758611 = r6758608 + r6758609;
        double r6758612 = r6758610 * r6758611;
        double r6758613 = r6758612 / r6758611;
        double r6758614 = r6758607 + r6758613;
        return r6758614;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.3
Target15.3
Herbie12.5
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.3

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

  3. Applied tan-sum21.0

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

  4. Taylor expanded around inf 21.1

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

  5. Simplified12.5

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

  7. Applied flip--12.5

    \[\leadsto \color{blue}{\frac{\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon}}{\cos x}} \cdot \frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon}}{\cos x}} - \frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}}{\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon}}{\cos x}} + \frac{\sin x}{\cos x}}} + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon \cdot \frac{\sin x}{\cos \varepsilon}}{\cos x}}\]
  8. Using strategy rm

  9. Applied difference-of-squares12.5

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

  10. Final simplification12.5

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

Reproduce

herbie shell --seed 2019165 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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