Average Error: 36.8 → 13.0
Time: 26.8s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos x}} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\log \left(e^{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}\right)}{\cos x}} - \frac{\sin x}{\cos x}\right)\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}{\cos x}} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\log \left(e^{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \sin x}\right)}{\cos x}} - \frac{\sin x}{\cos x}\right)
double f(double x, double eps) {
        double r6303031 = x;
        double r6303032 = eps;
        double r6303033 = r6303031 + r6303032;
        double r6303034 = tan(r6303033);
        double r6303035 = tan(r6303031);
        double r6303036 = r6303034 - r6303035;
        return r6303036;
}
↓
double f(double x, double eps) {
        double r6303037 = eps;
        double r6303038 = sin(r6303037);
        double r6303039 = cos(r6303037);
        double r6303040 = r6303038 / r6303039;
        double r6303041 = 1.0;
        double r6303042 = x;
        double r6303043 = sin(r6303042);
        double r6303044 = r6303040 * r6303043;
        double r6303045 = cos(r6303042);
        double r6303046 = r6303044 / r6303045;
        double r6303047 = r6303041 - r6303046;
        double r6303048 = r6303040 / r6303047;
        double r6303049 = r6303043 / r6303045;
        double r6303050 = exp(r6303044);
        double r6303051 = log(r6303050);
        double r6303052 = r6303051 / r6303045;
        double r6303053 = r6303041 - r6303052;
        double r6303054 = r6303049 / r6303053;
        double r6303055 = r6303054 - r6303049;
        double r6303056 = r6303048 + r6303055;
        return r6303056;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.8
Target15.1
Herbie13.0
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.8

    \[\tan \left(x + \varepsilon\right) - \tan x\]
  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 21.9

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

  5. Simplified12.9

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

  7. Applied add-log-exp13.0

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

  8. Final simplification13.0

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

Reproduce

herbie shell --seed 2019172 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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