2tan (problem 3.3.2)

Average Error: 36.9 → 0.5

Time: 26.4s

Precision: 64

Math

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

↓

\[\mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{\cos \varepsilon}, \frac{\sin \varepsilon}{{\left(\cos x\right)}^{2} \cdot \left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right)}, \frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right) \cdot \cos \varepsilon}\right) + \left(\left(\mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}, \frac{{\left(\sin \varepsilon\right)}^{3}}{\left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos \varepsilon\right)}^{3}}, \frac{\frac{\sin x}{\cos x}}{1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}}\right) + \frac{{\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2} \cdot \left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right)} \cdot \left(\frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \frac{\sin x}{\cos x}\right)\right) - \frac{\sin x}{\cos x}\right)\]

C

double f(double x, double eps) {
        double r112393 = x;
        double r112394 = eps;
        double r112395 = r112393 + r112394;
        double r112396 = tan(r112395);
        double r112397 = tan(r112393);
        double r112398 = r112396 - r112397;
        return r112398;
}

↓

double f(double x, double eps) {
        double r112399 = x;
        double r112400 = sin(r112399);
        double r112401 = 2.0;
        double r112402 = pow(r112400, r112401);
        double r112403 = eps;
        double r112404 = cos(r112403);
        double r112405 = r112402 / r112404;
        double r112406 = sin(r112403);
        double r112407 = cos(r112399);
        double r112408 = pow(r112407, r112401);
        double r112409 = 1.0;
        double r112410 = r112400 * r112406;
        double r112411 = 3.0;
        double r112412 = pow(r112410, r112411);
        double r112413 = r112407 * r112404;
        double r112414 = pow(r112413, r112411);
        double r112415 = r112412 / r112414;
        double r112416 = r112409 - r112415;
        double r112417 = r112408 * r112416;
        double r112418 = r112406 / r112417;
        double r112419 = r112416 * r112404;
        double r112420 = r112406 / r112419;
        double r112421 = fma(r112405, r112418, r112420);
        double r112422 = r112402 / r112408;
        double r112423 = pow(r112406, r112411);
        double r112424 = pow(r112404, r112411);
        double r112425 = r112416 * r112424;
        double r112426 = r112423 / r112425;
        double r112427 = r112400 / r112407;
        double r112428 = r112427 / r112416;
        double r112429 = fma(r112422, r112426, r112428);
        double r112430 = pow(r112406, r112401);
        double r112431 = pow(r112404, r112401);
        double r112432 = pow(r112407, r112411);
        double r112433 = r112432 * r112424;
        double r112434 = r112412 / r112433;
        double r112435 = r112409 - r112434;
        double r112436 = r112431 * r112435;
        double r112437 = r112430 / r112436;
        double r112438 = pow(r112400, r112411);
        double r112439 = r112438 / r112432;
        double r112440 = r112439 + r112427;
        double r112441 = r112437 * r112440;
        double r112442 = r112429 + r112441;
        double r112443 = r112442 - r112427;
        double r112444 = r112421 + r112443;
        return r112444;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.9
Target	14.5
Herbie	0.5

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

Derivation

1. Initial program 36.9

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

3. Applied tan-sum 22.3

   \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
4. Using strategy rm

5. Applied flip3-- 22.3

   \[\leadsto \frac{\tan x + \tan \varepsilon}{\color{blue}{\frac{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}{1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}}} - \tan x\]

6. Applied associate-/r/ 22.3

   \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{{1}^{3} - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right)} - \tan x\]

7. Applied fma-neg 22.3

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

8. Taylor expanded around -inf 22.5

   \[\leadsto \color{blue}{\left(\frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos \varepsilon \cdot \left(\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos x\right)}^{2}\right)} + \left(\frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \cos \varepsilon} + \left(\frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{3}}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \left({\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{3}\right)} + \left(\frac{\sin x}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \cos x} + \left(\frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2} \cdot \left(\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos x\right)}^{3}\right)} + \frac{\sin x \cdot {\left(\sin \varepsilon\right)}^{2}}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \left(\cos x \cdot {\left(\cos \varepsilon\right)}^{2}\right)}\right)\right)\right)\right)\right) - \frac{\sin x}{\cos x}}\]

9. Simplified 0.5

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{\cos \varepsilon}, \frac{\sin \varepsilon}{{\left(\cos x\right)}^{2} \cdot \left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right)}, \frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right) \cdot \cos \varepsilon}\right) + \left(\left(\mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}, \frac{{\left(\sin \varepsilon\right)}^{3}}{\left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos \varepsilon\right)}^{3}}, \frac{\frac{\sin x}{\cos x}}{1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}}\right) + \frac{{\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2} \cdot \left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right)} \cdot \left(\frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \frac{\sin x}{\cos x}\right)\right) - \frac{\sin x}{\cos x}\right)}\]
10. Using strategy rm

11. Applied unpow-prod-down 0.5

    \[\leadsto \mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{\cos \varepsilon}, \frac{\sin \varepsilon}{{\left(\cos x\right)}^{2} \cdot \left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right)}, \frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right) \cdot \cos \varepsilon}\right) + \left(\left(\mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}, \frac{{\left(\sin \varepsilon\right)}^{3}}{\left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos \varepsilon\right)}^{3}}, \frac{\frac{\sin x}{\cos x}}{1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}}\right) + \frac{{\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2} \cdot \left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{\color{blue}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}}\right)} \cdot \left(\frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \frac{\sin x}{\cos x}\right)\right) - \frac{\sin x}{\cos x}\right)\]

12. Final simplification 0.5

    \[\leadsto \mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{\cos \varepsilon}, \frac{\sin \varepsilon}{{\left(\cos x\right)}^{2} \cdot \left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right)}, \frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right) \cdot \cos \varepsilon}\right) + \left(\left(\mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}, \frac{{\left(\sin \varepsilon\right)}^{3}}{\left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos \varepsilon\right)}^{3}}, \frac{\frac{\sin x}{\cos x}}{1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3}}}\right) + \frac{{\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2} \cdot \left(1 - \frac{{\left(\sin x \cdot \sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right)} \cdot \left(\frac{{\left(\sin x\right)}^{3}}{{\left(\cos x\right)}^{3}} + \frac{\sin x}{\cos x}\right)\right) - \frac{\sin x}{\cos x}\right)\]

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64

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

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