Average Error: 37.1 → 13.1
Time: 1.4m
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\sin \varepsilon}{1 - \frac{\frac{\sin x}{\cos x} \cdot \sin \varepsilon}{\cos \varepsilon}}}{\cos \varepsilon} - \left(\sin x - \frac{\sin x}{1 - \frac{\frac{\sin x}{\cos x} \cdot \sin \varepsilon}{\cos \varepsilon}}\right) \cdot \frac{1}{\cos x}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\sin \varepsilon}{1 - \frac{\frac{\sin x}{\cos x} \cdot \sin \varepsilon}{\cos \varepsilon}}}{\cos \varepsilon} - \left(\sin x - \frac{\sin x}{1 - \frac{\frac{\sin x}{\cos x} \cdot \sin \varepsilon}{\cos \varepsilon}}\right) \cdot \frac{1}{\cos x}
double f(double x, double eps) {
        double r4611030 = x;
        double r4611031 = eps;
        double r4611032 = r4611030 + r4611031;
        double r4611033 = tan(r4611032);
        double r4611034 = tan(r4611030);
        double r4611035 = r4611033 - r4611034;
        return r4611035;
}
↓
double f(double x, double eps) {
        double r4611036 = eps;
        double r4611037 = sin(r4611036);
        double r4611038 = 1.0;
        double r4611039 = x;
        double r4611040 = sin(r4611039);
        double r4611041 = cos(r4611039);
        double r4611042 = r4611040 / r4611041;
        double r4611043 = r4611042 * r4611037;
        double r4611044 = cos(r4611036);
        double r4611045 = r4611043 / r4611044;
        double r4611046 = r4611038 - r4611045;
        double r4611047 = r4611037 / r4611046;
        double r4611048 = r4611047 / r4611044;
        double r4611049 = r4611040 / r4611046;
        double r4611050 = r4611040 - r4611049;
        double r4611051 = r4611038 / r4611041;
        double r4611052 = r4611050 * r4611051;
        double r4611053 = r4611048 - r4611052;
        return r4611053;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.1
Target15.0
Herbie13.1
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.1

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

  3. Applied tan-sum21.9

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

  4. Taylor expanded around inf 22.1

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

  5. Simplified13.1

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

  7. Applied div-inv13.9

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

  8. Applied div-inv13.1

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

  9. Applied distribute-rgt-out--13.1

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

  10. Final simplification13.1

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

Reproduce

herbie shell --seed 2019200 +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)))