2tan (problem 3.3.2)

Average Error: 37.1 → 0.3

Time: 28.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00018789018785287954:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{{\left({\left(\tan x \cdot \tan \varepsilon\right)}^{3}\right)}^{3}}} \cdot \left(1 + \left(\tan x \cdot \tan \varepsilon + \sqrt[3]{\left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) \cdot \left(\left(\tan x \cdot \left(\tan x \cdot \tan x\right)\right) \cdot \left(\tan \varepsilon \cdot \left(\tan \varepsilon \cdot \tan \varepsilon\right)\right)\right)}\right)\right) - \tan x\\ \mathbf{elif}\;\varepsilon \leq 0.00023798155281232094:\\ \;\;\;\;1.3333333333333333 \cdot \frac{{\sin x}^{2} \cdot {\varepsilon}^{3}}{{\cos x}^{2}} + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(\frac{{\sin x}^{5} \cdot {\varepsilon}^{4}}{{\cos x}^{5}} + \left(1.6666666666666667 \cdot \frac{{\varepsilon}^{4} \cdot {\sin x}^{3}}{{\cos x}^{3}} + \left(\left(\left(\varepsilon + {\varepsilon}^{3} \cdot \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} + 0.3333333333333333\right)\right) + 0.6666666666666666 \cdot \left({\varepsilon}^{4} \cdot \frac{\sin x}{\cos x}\right)\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 + \left(\tan x \cdot \tan \varepsilon + \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) - \tan x\\ \end{array}\]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.00018789018785287954)
   (-
    (*
     (/
      (+ (tan x) (tan eps))
      (- 1.0 (cbrt (pow (pow (* (tan x) (tan eps)) 3.0) 3.0))))
     (+
      1.0
      (+
       (* (tan x) (tan eps))
       (cbrt
        (*
         (*
          (* (tan x) (tan eps))
          (* (* (tan x) (tan eps)) (* (tan x) (tan eps))))
         (*
          (* (tan x) (* (tan x) (tan x)))
          (* (tan eps) (* (tan eps) (tan eps)))))))))
    (tan x))
   (if (<= eps 0.00023798155281232094)
     (+
      (*
       1.3333333333333333
       (/ (* (pow (sin x) 2.0) (pow eps 3.0)) (pow (cos x) 2.0)))
      (+
       (/ (* eps (pow (sin x) 2.0)) (pow (cos x) 2.0))
       (+
        (/ (* (pow (sin x) 5.0) (pow eps 4.0)) (pow (cos x) 5.0))
        (+
         (*
          1.6666666666666667
          (/ (* (pow eps 4.0) (pow (sin x) 3.0)) (pow (cos x) 3.0)))
         (+
          (+
           (+
            eps
            (*
             (pow eps 3.0)
             (+ (/ (pow (sin x) 4.0) (pow (cos x) 4.0)) 0.3333333333333333)))
           (* 0.6666666666666666 (* (pow eps 4.0) (/ (sin x) (cos x)))))
          (*
           (* eps eps)
           (+
            (/ (sin x) (cos x))
            (/ (pow (sin x) 3.0) (pow (cos x) 3.0)))))))))
     (-
      (*
       (/ (+ (tan x) (tan eps)) (- 1.0 (pow (* (tan x) (tan eps)) 3.0)))
       (+
        1.0
        (+
         (* (tan x) (tan eps))
         (* (* (tan x) (tan eps)) (* (tan x) (tan eps))))))
      (tan x)))))

C

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

↓

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.00018789018785287954) {
		tmp = (((tan(x) + tan(eps)) / (1.0 - cbrt(pow(pow((tan(x) * tan(eps)), 3.0), 3.0)))) * (1.0 + ((tan(x) * tan(eps)) + cbrt(((tan(x) * tan(eps)) * ((tan(x) * tan(eps)) * (tan(x) * tan(eps)))) * ((tan(x) * (tan(x) * tan(x))) * (tan(eps) * (tan(eps) * tan(eps)))))))) - tan(x);
	} else if (eps <= 0.00023798155281232094) {
		tmp = (1.3333333333333333 * ((pow(sin(x), 2.0) * pow(eps, 3.0)) / pow(cos(x), 2.0))) + (((eps * pow(sin(x), 2.0)) / pow(cos(x), 2.0)) + (((pow(sin(x), 5.0) * pow(eps, 4.0)) / pow(cos(x), 5.0)) + ((1.6666666666666667 * ((pow(eps, 4.0) * pow(sin(x), 3.0)) / pow(cos(x), 3.0))) + (((eps + (pow(eps, 3.0) * ((pow(sin(x), 4.0) / pow(cos(x), 4.0)) + 0.3333333333333333))) + (0.6666666666666666 * (pow(eps, 4.0) * (sin(x) / cos(x))))) + ((eps * eps) * ((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(x), 3.0))))))));
	} else {
		tmp = (((tan(x) + tan(eps)) / (1.0 - pow((tan(x) * tan(eps)), 3.0))) * (1.0 + ((tan(x) * tan(eps)) + ((tan(x) * tan(eps)) * (tan(x) * tan(eps)))))) - tan(x);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	15.6
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if eps < -1.878901878528795e-4

   1. Initial program 30.3

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

   3. Applied tan-sum_binary64_2259 0.3

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

   5. Applied flip3--_binary64_2128 0.4

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

   6. Applied associate-/r/_binary64_2070 0.4

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

   7. Simplified 0.4

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

   9. Applied add-cbrt-cube_binary64_2160 0.4

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

   10. Simplified 0.4

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{\color{blue}{{\left({\left(\tan x \cdot \tan \varepsilon\right)}^{3}\right)}^{3}}}} \cdot \left(1 \cdot 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + 1 \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) - \tan x\]
   11. Using strategy rm

   12. Applied add-cbrt-cube_binary64_2160 0.4

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

   13. Applied add-cbrt-cube_binary64_2160 0.4

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

   14. Applied cbrt-unprod_binary64_2157 0.4

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

   15. Applied add-cbrt-cube_binary64_2160 0.4

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

   16. Applied cbrt-unprod_binary64_2157 0.4

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

   if -1.878901878528795e-4 < eps < 2.3798155281232094e-4

   1. Initial program 44.4

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

   2. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{1.3333333333333333 \cdot \frac{{\sin x}^{2} \cdot {\varepsilon}^{3}}{{\cos x}^{2}} + \left(\frac{{\sin x}^{2} \cdot \varepsilon}{{\cos x}^{2}} + \left(\frac{{\sin x}^{5} \cdot {\varepsilon}^{4}}{{\cos x}^{5}} + \left(1.6666666666666667 \cdot \frac{{\sin x}^{3} \cdot {\varepsilon}^{4}}{{\cos x}^{3}} + \left(\frac{\sin x \cdot {\varepsilon}^{2}}{\cos x} + \left(\frac{{\sin x}^{3} \cdot {\varepsilon}^{2}}{{\cos x}^{3}} + \left(0.3333333333333333 \cdot {\varepsilon}^{3} + \left(0.6666666666666666 \cdot \frac{\sin x \cdot {\varepsilon}^{4}}{\cos x} + \left(\varepsilon + \frac{{\sin x}^{4} \cdot {\varepsilon}^{3}}{{\cos x}^{4}}\right)\right)\right)\right)\right)\right)\right)\right)}\]

   3. Simplified 0.2

      \[\leadsto \color{blue}{1.3333333333333333 \cdot \frac{{\sin x}^{2} \cdot {\varepsilon}^{3}}{{\cos x}^{2}} + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(\frac{{\sin x}^{5} \cdot {\varepsilon}^{4}}{{\cos x}^{5}} + \left(1.6666666666666667 \cdot \frac{{\sin x}^{3} \cdot {\varepsilon}^{4}}{{\cos x}^{3}} + \left(\left(\left(\varepsilon + {\varepsilon}^{3} \cdot \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} + 0.3333333333333333\right)\right) + 0.6666666666666666 \cdot \left(\frac{\sin x}{\cos x} \cdot {\varepsilon}^{4}\right)\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x}{\cos x}\right)\right)\right)\right)\right)}\]

   if 2.3798155281232094e-4 < eps

   1. Initial program 29.9

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

   3. Applied tan-sum_binary64_2259 0.3

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

   5. Applied flip3--_binary64_2128 0.4

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

   6. Applied associate-/r/_binary64_2070 0.4

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00018789018785287954:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{{\left({\left(\tan x \cdot \tan \varepsilon\right)}^{3}\right)}^{3}}} \cdot \left(1 + \left(\tan x \cdot \tan \varepsilon + \sqrt[3]{\left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) \cdot \left(\left(\tan x \cdot \left(\tan x \cdot \tan x\right)\right) \cdot \left(\tan \varepsilon \cdot \left(\tan \varepsilon \cdot \tan \varepsilon\right)\right)\right)}\right)\right) - \tan x\\ \mathbf{elif}\;\varepsilon \leq 0.00023798155281232094:\\ \;\;\;\;1.3333333333333333 \cdot \frac{{\sin x}^{2} \cdot {\varepsilon}^{3}}{{\cos x}^{2}} + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(\frac{{\sin x}^{5} \cdot {\varepsilon}^{4}}{{\cos x}^{5}} + \left(1.6666666666666667 \cdot \frac{{\varepsilon}^{4} \cdot {\sin x}^{3}}{{\cos x}^{3}} + \left(\left(\left(\varepsilon + {\varepsilon}^{3} \cdot \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} + 0.3333333333333333\right)\right) + 0.6666666666666666 \cdot \left({\varepsilon}^{4} \cdot \frac{\sin x}{\cos x}\right)\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 + \left(\tan x \cdot \tan \varepsilon + \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right) - \tan x\\ \end{array}\]

Reproduce

herbie shell --seed 2021050 
(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)))