2tan (problem 3.3.2)

Average Error: 36.5 → 0.3

Time: 11.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := -\tan x\\ \mathbf{if}\;\varepsilon \leq -2.3680288234376444 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, t_1\right)\\ \mathbf{elif}\;\varepsilon \leq 5.9543033066741405 \cdot 10^{-5}:\\ \;\;\;\;\begin{array}{l} t_2 := {\sin x}^{2}\\ t_3 := {\cos x}^{2}\\ \left(\frac{{\varepsilon}^{3}}{{\left(\frac{\cos x}{\sin x}\right)}^{4}} + \left(\mathsf{fma}\left(1.3333333333333333, \frac{{\varepsilon}^{3} \cdot t_2}{t_3}, {\varepsilon}^{3} \cdot 0.3333333333333333\right) + \mathsf{fma}\left(\varepsilon, \frac{t_2}{t_3}, \varepsilon\right)\right)\right) + \frac{\varepsilon \cdot \varepsilon}{\cos x} \cdot \left(\sin x + \frac{{\sin x}^{3}}{t_3}\right) \end{array}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_4 := \cos x \cdot \cos \varepsilon\\ t_5 := \sin x \cdot \sin \varepsilon\\ \mathsf{fma}\left(\frac{t_0}{1 - \frac{t_5 \cdot t_5}{t_4 \cdot t_4}}, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), t_1\right) \end{array}\\ \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))) (t_1 (- (tan x))))
   (if (<= eps -2.3680288234376444e-5)
     (fma t_0 (/ 1.0 (- 1.0 (* (tan x) (tan eps)))) t_1)
     (if (<= eps 5.9543033066741405e-5)
       (let* ((t_2 (pow (sin x) 2.0)) (t_3 (pow (cos x) 2.0)))
         (+
          (+
           (/ (pow eps 3.0) (pow (/ (cos x) (sin x)) 4.0))
           (+
            (fma
             1.3333333333333333
             (/ (* (pow eps 3.0) t_2) t_3)
             (* (pow eps 3.0) 0.3333333333333333))
            (fma eps (/ t_2 t_3) eps)))
          (* (/ (* eps eps) (cos x)) (+ (sin x) (/ (pow (sin x) 3.0) t_3)))))
       (let* ((t_4 (* (cos x) (cos eps))) (t_5 (* (sin x) (sin eps))))
         (fma
          (/ t_0 (- 1.0 (/ (* t_5 t_5) (* t_4 t_4))))
          (fma (tan x) (tan eps) 1.0)
          t_1))))))

C

double code(double x, double eps) {
	return tan(x + eps) - tan(x);
}

↓

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double t_1 = -tan(x);
	double tmp;
	if (eps <= -2.3680288234376444e-5) {
		tmp = fma(t_0, (1.0 / (1.0 - (tan(x) * tan(eps)))), t_1);
	} else if (eps <= 5.9543033066741405e-5) {
		double t_2 = pow(sin(x), 2.0);
		double t_3 = pow(cos(x), 2.0);
		tmp = ((pow(eps, 3.0) / pow((cos(x) / sin(x)), 4.0)) + (fma(1.3333333333333333, ((pow(eps, 3.0) * t_2) / t_3), (pow(eps, 3.0) * 0.3333333333333333)) + fma(eps, (t_2 / t_3), eps))) + (((eps * eps) / cos(x)) * (sin(x) + (pow(sin(x), 3.0) / t_3)));
	} else {
		double t_4 = cos(x) * cos(eps);
		double t_5 = sin(x) * sin(eps);
		tmp = fma((t_0 / (1.0 - ((t_5 * t_5) / (t_4 * t_4)))), fma(tan(x), tan(eps), 1.0), t_1);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.5
Target	15.1
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if eps < -2.36802882343764439e-5

   1. Initial program 29.2

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

   2. Applied tan-sum_binary64 0.4

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

   3. Applied div-inv_binary64 0.4

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

   4. Applied fma-neg_binary64 0.4

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

   if -2.36802882343764439e-5 < eps < 5.95430330667414047e-5

   1. Initial program 44.2

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

   2. Taylor expanded in eps around 0 0.2

      \[\leadsto \color{blue}{\frac{{\varepsilon}^{2} \cdot \sin x}{\cos x} + \left(\frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}} + \left(\frac{{\varepsilon}^{3} \cdot {\sin x}^{4}}{{\cos x}^{4}} + \left(\varepsilon + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(0.3333333333333333 \cdot {\varepsilon}^{3} + 1.3333333333333333 \cdot \frac{{\varepsilon}^{3} \cdot {\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right)\right)\right)} \]

   3. Simplified 0.2

      \[\leadsto \color{blue}{\left(\frac{{\varepsilon}^{3}}{{\left(\frac{\cos x}{\sin x}\right)}^{4}} + \left(\mathsf{fma}\left(1.3333333333333333, \frac{{\sin x}^{2} \cdot {\varepsilon}^{3}}{{\cos x}^{2}}, 0.3333333333333333 \cdot {\varepsilon}^{3}\right) + \mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon\right)\right)\right) + \frac{\varepsilon \cdot \varepsilon}{\cos x} \cdot \left(\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}\right)} \]

   if 5.95430330667414047e-5 < eps

   1. Initial program 29.3

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

   2. Applied tan-sum_binary64 0.3

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

   3. Applied add-cube-cbrt_binary64 0.7

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

   4. Applied flip--_binary64 0.7

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

   5. Applied associate-/r/_binary64 0.7

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

   6. Applied prod-diff_binary64 0.7

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

   7. Simplified 0.4

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

   8. Simplified 0.4

      \[\leadsto \mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), -\tan x\right) + \color{blue}{0} \]

   9. Applied tan-quot_binary64 0.4

      \[\leadsto \mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}}\right)}, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), -\tan x\right) + 0 \]

   10. Applied tan-quot_binary64 0.4

       \[\leadsto \mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\color{blue}{\frac{\sin x}{\cos x}} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), -\tan x\right) + 0 \]

   11. Applied frac-times_binary64 0.4

       \[\leadsto \mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \color{blue}{\frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}}}, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), -\tan x\right) + 0 \]

   12. Applied tan-quot_binary64 0.4

       \[\leadsto \mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}}\right) \cdot \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}}, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), -\tan x\right) + 0 \]

   13. Applied tan-quot_binary64 0.5

       \[\leadsto \mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \left(\color{blue}{\frac{\sin x}{\cos x}} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right) \cdot \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}}, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), -\tan x\right) + 0 \]

   14. Applied frac-times_binary64 0.5

       \[\leadsto \mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}} \cdot \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}}, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), -\tan x\right) + 0 \]

   15. Applied frac-times_binary64 0.5

       \[\leadsto \mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)}{\left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)}}}, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), -\tan x\right) + 0 \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.3680288234376444 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 5.9543033066741405 \cdot 10^{-5}:\\ \;\;\;\;\left(\frac{{\varepsilon}^{3}}{{\left(\frac{\cos x}{\sin x}\right)}^{4}} + \left(\mathsf{fma}\left(1.3333333333333333, \frac{{\varepsilon}^{3} \cdot {\sin x}^{2}}{{\cos x}^{2}}, {\varepsilon}^{3} \cdot 0.3333333333333333\right) + \mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon\right)\right)\right) + \frac{\varepsilon \cdot \varepsilon}{\cos x} \cdot \left(\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \frac{\left(\sin x \cdot \sin \varepsilon\right) \cdot \left(\sin x \cdot \sin \varepsilon\right)}{\left(\cos x \cdot \cos \varepsilon\right) \cdot \left(\cos x \cdot \cos \varepsilon\right)}}, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), -\tan x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021344 
(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)))