2tan (problem 3.3.2)

Average Error: 36.7 → 12.9

Time: 30.0s

Precision: 64

Math

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

↓

\[\left(\mathsf{fma}\left(\frac{-1}{\cos x}, \sin x, \frac{\sin x}{\cos x}\right) + \mathsf{fma}\left(\frac{\frac{\sin x}{\sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, \frac{-\sin \varepsilon}{\cos \varepsilon}, 1\right)}}}{\sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, \frac{-\sin \varepsilon}{\cos \varepsilon}, 1\right)}}, \frac{\frac{1}{\cos x}}{\sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, \frac{-\sin \varepsilon}{\cos \varepsilon}, 1\right)}}, -\frac{\sin x}{\cos x}\right)\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{\mathsf{fma}\left(\frac{\sin x}{\cos x}, \frac{-\sin \varepsilon}{\cos \varepsilon}, 1\right)}\]

C

double f(double x, double eps) {
        double r173285 = x;
        double r173286 = eps;
        double r173287 = r173285 + r173286;
        double r173288 = tan(r173287);
        double r173289 = tan(r173285);
        double r173290 = r173288 - r173289;
        return r173290;
}

↓

double f(double x, double eps) {
        double r173291 = -1.0;
        double r173292 = x;
        double r173293 = cos(r173292);
        double r173294 = r173291 / r173293;
        double r173295 = sin(r173292);
        double r173296 = r173295 / r173293;
        double r173297 = fma(r173294, r173295, r173296);
        double r173298 = eps;
        double r173299 = sin(r173298);
        double r173300 = -r173299;
        double r173301 = cos(r173298);
        double r173302 = r173300 / r173301;
        double r173303 = 1.0;
        double r173304 = fma(r173296, r173302, r173303);
        double r173305 = cbrt(r173304);
        double r173306 = r173295 / r173305;
        double r173307 = r173306 / r173305;
        double r173308 = r173303 / r173293;
        double r173309 = r173308 / r173305;
        double r173310 = -r173296;
        double r173311 = fma(r173307, r173309, r173310);
        double r173312 = r173297 + r173311;
        double r173313 = r173299 / r173301;
        double r173314 = r173313 / r173304;
        double r173315 = r173312 + r173314;
        return r173315;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.7
Target	14.9
Herbie	12.9

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

Derivation

1. Initial program 36.7

   \[\tan \left(x + \varepsilon\right) - \tan x\]
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. Simplified 21.7

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

5. Simplified 21.7

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

6. Taylor expanded around inf 21.8

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

7. Simplified 12.7

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

9. Applied add-cube-cbrt 21.0

   \[\leadsto \left(\frac{\frac{\sin x}{\cos x}}{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)} - \color{blue}{\left(\sqrt[3]{\frac{\sin x}{\cos x}} \cdot \sqrt[3]{\frac{\sin x}{\cos x}}\right) \cdot \sqrt[3]{\frac{\sin x}{\cos x}}}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)}\]

10. Applied add-cube-cbrt 21.0

    \[\leadsto \left(\frac{\frac{\sin x}{\cos x}}{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)}}} - \left(\sqrt[3]{\frac{\sin x}{\cos x}} \cdot \sqrt[3]{\frac{\sin x}{\cos x}}\right) \cdot \sqrt[3]{\frac{\sin x}{\cos x}}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)}\]

11. Applied div-inv 21.1

    \[\leadsto \left(\frac{\color{blue}{\sin x \cdot \frac{1}{\cos x}}}{\left(\sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)}} - \left(\sqrt[3]{\frac{\sin x}{\cos x}} \cdot \sqrt[3]{\frac{\sin x}{\cos x}}\right) \cdot \sqrt[3]{\frac{\sin x}{\cos x}}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)}\]

12. Applied times-frac 21.1

    \[\leadsto \left(\color{blue}{\frac{\sin x}{\sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)}} \cdot \frac{\frac{1}{\cos x}}{\sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)}}} - \left(\sqrt[3]{\frac{\sin x}{\cos x}} \cdot \sqrt[3]{\frac{\sin x}{\cos x}}\right) \cdot \sqrt[3]{\frac{\sin x}{\cos x}}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)}\]

13. Applied prod-diff 22.4

    \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{\sin x}{\sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)}}, \frac{\frac{1}{\cos x}}{\sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)}}, -\sqrt[3]{\frac{\sin x}{\cos x}} \cdot \left(\sqrt[3]{\frac{\sin x}{\cos x}} \cdot \sqrt[3]{\frac{\sin x}{\cos x}}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\frac{\sin x}{\cos x}}, \sqrt[3]{\frac{\sin x}{\cos x}} \cdot \sqrt[3]{\frac{\sin x}{\cos x}}, \sqrt[3]{\frac{\sin x}{\cos x}} \cdot \left(\sqrt[3]{\frac{\sin x}{\cos x}} \cdot \sqrt[3]{\frac{\sin x}{\cos x}}\right)\right)\right)} + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)}\]

14. Simplified 22.2

    \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{\frac{\sin x}{\sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, \frac{-\sin \varepsilon}{\cos \varepsilon}, 1\right)}}}{\sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, \frac{-\sin \varepsilon}{\cos \varepsilon}, 1\right)}}, \frac{\frac{1}{\cos x}}{\sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, \frac{-\sin \varepsilon}{\cos \varepsilon}, 1\right)}}, \frac{-\sin x}{\cos x}\right)} + \mathsf{fma}\left(-\sqrt[3]{\frac{\sin x}{\cos x}}, \sqrt[3]{\frac{\sin x}{\cos x}} \cdot \sqrt[3]{\frac{\sin x}{\cos x}}, \sqrt[3]{\frac{\sin x}{\cos x}} \cdot \left(\sqrt[3]{\frac{\sin x}{\cos x}} \cdot \sqrt[3]{\frac{\sin x}{\cos x}}\right)\right)\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)}\]

15. Simplified 12.9

    \[\leadsto \left(\mathsf{fma}\left(\frac{\frac{\sin x}{\sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, \frac{-\sin \varepsilon}{\cos \varepsilon}, 1\right)}}}{\sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, \frac{-\sin \varepsilon}{\cos \varepsilon}, 1\right)}}, \frac{\frac{1}{\cos x}}{\sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, \frac{-\sin \varepsilon}{\cos \varepsilon}, 1\right)}}, \frac{-\sin x}{\cos x}\right) + \color{blue}{\mathsf{fma}\left(\frac{-1}{\cos x}, \sin x, \frac{\sin x}{\cos x}\right)}\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)}\]

16. Final simplification 12.9

    \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{\cos x}, \sin x, \frac{\sin x}{\cos x}\right) + \mathsf{fma}\left(\frac{\frac{\sin x}{\sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, \frac{-\sin \varepsilon}{\cos \varepsilon}, 1\right)}}}{\sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, \frac{-\sin \varepsilon}{\cos \varepsilon}, 1\right)}}, \frac{\frac{1}{\cos x}}{\sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, \frac{-\sin \varepsilon}{\cos \varepsilon}, 1\right)}}, -\frac{\sin x}{\cos x}\right)\right) + \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{\mathsf{fma}\left(\frac{\sin x}{\cos x}, \frac{-\sin \varepsilon}{\cos \varepsilon}, 1\right)}\]

Reproduce

herbie shell --seed 2019195 +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)))