2tan (problem 3.3.2)

Average Error: 37.5 → 15.7

Time: 7.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.3047993601266133 \cdot 10^{-36}:\\ \;\;\;\;\frac{\cos x \cdot \left(\left(\tan x + \tan \varepsilon\right) \cdot \left(1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) - \sin x \cdot \left(\left(1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)\right)}{\cos x \cdot \left(\left(1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)\right)}\\ \mathbf{elif}\;\varepsilon \leq 1.3314514851199195 \cdot 10^{-51}:\\ \;\;\;\;x \cdot \left(\varepsilon \cdot \varepsilon\right) + \left(\varepsilon + \varepsilon \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x\\ \end{array}\]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (if (<= eps -2.3047993601266133e-36)
   (/
    (-
     (*
      (cos x)
      (*
       (+ (tan x) (tan eps))
       (- 1.0 (* (* (tan x) (tan eps)) (* (tan x) (tan eps))))))
     (*
      (sin x)
      (*
       (- 1.0 (* (* (tan x) (tan eps)) (* (tan x) (tan eps))))
       (- 1.0 (* (tan x) (tan eps))))))
    (*
     (cos x)
     (*
      (- 1.0 (* (* (tan x) (tan eps)) (* (tan x) (tan eps))))
      (- 1.0 (* (tan x) (tan eps))))))
   (if (<= eps 1.3314514851199195e-51)
     (+ (* x (* eps eps)) (+ eps (* eps (* x x))))
     (-
      (*
       (/
        (+ (tan x) (tan eps))
        (- 1.0 (* (* (tan x) (tan x)) (* (tan eps) (tan eps)))))
       (+ 1.0 (* (tan x) (tan eps))))
      (tan x)))))

C

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

↓

double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.3047993601266133e-36)) {
		tmp = (((double) (((double) (((double) cos(x)) * ((double) (((double) (((double) tan(x)) + ((double) tan(eps)))) * ((double) (1.0 - ((double) (((double) (((double) tan(x)) * ((double) tan(eps)))) * ((double) (((double) tan(x)) * ((double) tan(eps)))))))))))) - ((double) (((double) sin(x)) * ((double) (((double) (1.0 - ((double) (((double) (((double) tan(x)) * ((double) tan(eps)))) * ((double) (((double) tan(x)) * ((double) tan(eps)))))))) * ((double) (1.0 - ((double) (((double) tan(x)) * ((double) tan(eps)))))))))))) / ((double) (((double) cos(x)) * ((double) (((double) (1.0 - ((double) (((double) (((double) tan(x)) * ((double) tan(eps)))) * ((double) (((double) tan(x)) * ((double) tan(eps)))))))) * ((double) (1.0 - ((double) (((double) tan(x)) * ((double) tan(eps)))))))))));
	} else {
		double tmp_1;
		if ((eps <= 1.3314514851199195e-51)) {
			tmp_1 = ((double) (((double) (x * ((double) (eps * eps)))) + ((double) (eps + ((double) (eps * ((double) (x * x))))))));
		} else {
			tmp_1 = ((double) (((double) ((((double) (((double) tan(x)) + ((double) tan(eps)))) / ((double) (1.0 - ((double) (((double) (((double) tan(x)) * ((double) tan(x)))) * ((double) (((double) tan(eps)) * ((double) tan(eps))))))))) * ((double) (1.0 + ((double) (((double) tan(x)) * ((double) tan(eps)))))))) - ((double) tan(x))));
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.5
Target	15.4
Herbie	15.7

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

Derivation

1. Split input into 3 regimes

2. if eps < -2.3047993601266133e-36

   1. Initial program 30.4

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

   3. Applied tan-sum_binary64 2.8

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

   5. Applied flip--_binary64 2.9

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

   6. Applied associate-/r/_binary64 2.9

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

   7. Simplified 2.9

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

   9. Applied tan-quot_binary64 2.9

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

   10. Applied flip-+_binary64 2.9

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

   11. Applied frac-times_binary64 2.9

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

   12. Applied frac-sub_binary64 2.9

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

   13. Simplified 2.9

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

   14. Simplified 2.9

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

   if -2.3047993601266133e-36 < eps < 1.3314514851199195e-51

   1. Initial program 46.8

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

   3. Applied tan-sum_binary64 46.8

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

   5. Applied flip--_binary64 46.8

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

   6. Applied associate-/r/_binary64 46.8

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

   7. Simplified 46.8

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

   8. Taylor expanded around 0 31.8

      \[\leadsto \color{blue}{x \cdot {\varepsilon}^{2} + \left(\varepsilon + {x}^{2} \cdot \varepsilon\right)}\]

   9. Simplified 31.8

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

   if 1.3314514851199195e-51 < eps

   1. Initial program 30.5

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

   3. Applied tan-sum_binary64 4.1

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

   5. Applied flip--_binary64 4.1

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

   6. Applied associate-/r/_binary64 4.1

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

   7. Simplified 4.1

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

   9. Applied swap-sqr_binary64 4.1

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

4. Final simplification 15.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.3047993601266133 \cdot 10^{-36}:\\ \;\;\;\;\frac{\cos x \cdot \left(\left(\tan x + \tan \varepsilon\right) \cdot \left(1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) - \sin x \cdot \left(\left(1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)\right)}{\cos x \cdot \left(\left(1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right) \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)\right)}\\ \mathbf{elif}\;\varepsilon \leq 1.3314514851199195 \cdot 10^{-51}:\\ \;\;\;\;x \cdot \left(\varepsilon \cdot \varepsilon\right) + \left(\varepsilon + \varepsilon \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x\\ \end{array}\]

Reproduce

herbie shell --seed 2020210 
(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)))