2tan (problem 3.3.2)

Average Error: 36.8 → 12.9

Time: 39.2s

Precision: 64

Math

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

↓

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

C

double f(double x, double eps) {
        double r4183032 = x;
        double r4183033 = eps;
        double r4183034 = r4183032 + r4183033;
        double r4183035 = tan(r4183034);
        double r4183036 = tan(r4183032);
        double r4183037 = r4183035 - r4183036;
        return r4183037;
}

↓

double f(double x, double eps) {
        double r4183038 = x;
        double r4183039 = sin(r4183038);
        double r4183040 = cos(r4183038);
        double r4183041 = r4183039 / r4183040;
        double r4183042 = 1.0;
        double r4183043 = eps;
        double r4183044 = cos(r4183043);
        double r4183045 = r4183044 * r4183040;
        double r4183046 = sin(r4183043);
        double r4183047 = r4183045 / r4183046;
        double r4183048 = r4183039 / r4183047;
        double r4183049 = r4183048 * r4183048;
        double r4183050 = r4183042 - r4183049;
        double r4183051 = r4183041 / r4183050;
        double r4183052 = r4183042 + r4183048;
        double r4183053 = -r4183039;
        double r4183054 = r4183053 / r4183040;
        double r4183055 = fma(r4183051, r4183052, r4183054);
        double r4183056 = r4183046 / r4183044;
        double r4183057 = fma(r4183054, r4183056, r4183042);
        double r4183058 = r4183056 / r4183057;
        double r4183059 = r4183055 + r4183058;
        return r4183059;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.8
Target	15.1
Herbie	12.9

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

Derivation

1. Initial program 36.8

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

3. Applied tan-sum 21.7

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

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. Simplified 12.9

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

6. Taylor expanded around inf 12.9

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

7. Simplified 12.9

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

9. Applied flip-- 13.0

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

10. Applied associate-/r/ 13.0

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

11. Applied fma-neg 12.9

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

12. Final simplification 12.9

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

Reproduce

herbie shell --seed 2019172 +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)))