2tan (problem 3.3.2)

Average Error: 37.2 → 13.1

Time: 43.6s

Precision: 64

Math

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

↓

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

C

double f(double x, double eps) {
        double r2171014 = x;
        double r2171015 = eps;
        double r2171016 = r2171014 + r2171015;
        double r2171017 = tan(r2171016);
        double r2171018 = tan(r2171014);
        double r2171019 = r2171017 - r2171018;
        return r2171019;
}

↓

double f(double x, double eps) {
        double r2171020 = x;
        double r2171021 = sin(r2171020);
        double r2171022 = 1.0;
        double r2171023 = cos(r2171020);
        double r2171024 = r2171022 / r2171023;
        double r2171025 = r2171021 / r2171023;
        double r2171026 = eps;
        double r2171027 = sin(r2171026);
        double r2171028 = cos(r2171026);
        double r2171029 = r2171027 / r2171028;
        double r2171030 = r2171025 * r2171029;
        double r2171031 = r2171022 - r2171030;
        double r2171032 = r2171024 / r2171031;
        double r2171033 = -1.0;
        double r2171034 = r2171033 / r2171023;
        double r2171035 = r2171034 * r2171021;
        double r2171036 = fma(r2171021, r2171032, r2171035);
        double r2171037 = r2171024 * r2171021;
        double r2171038 = fma(r2171034, r2171021, r2171037);
        double r2171039 = r2171036 + r2171038;
        double r2171040 = r2171029 / r2171031;
        double r2171041 = r2171039 + r2171040;
        return r2171041;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.2
Target	14.9
Herbie	13.1

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

Derivation

1. Initial program 37.2

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

3. Applied tan-sum 22.1

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

4. Taylor expanded around -inf 22.3

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

5. Simplified 13.1

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

7. Applied div-inv 14.0

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

8. Applied *-un-lft-identity 14.0

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

9. Applied div-inv 13.1

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

10. Applied times-frac 13.1

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

11. Applied prod-diff 13.1

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

12. Final simplification 13.1

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

Reproduce

herbie shell --seed 2019130 +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)))