2tan (problem 3.3.2)

Average Error: 36.7 → 14.7

Time: 7.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -6.1493629123170849 \cdot 10^{-28}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right) \cdot \sin x}{\left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right) \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 9.19032739794848937 \cdot 10^{-24}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x \cdot \tan x - \tan \varepsilon \cdot \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} \cdot \frac{1}{\tan x - \tan \varepsilon} - \tan x\\ \end{array}\]

C

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

↓

double code(double x, double eps) {
	double VAR;
	if ((eps <= -6.149362912317085e-28)) {
		VAR = ((((tan(x) + tan(eps)) * cos(x)) - ((1.0 - ((sin(x) * sin(eps)) / (cos(x) * cos(eps)))) * sin(x))) / ((1.0 - ((sin(x) * sin(eps)) / (cos(x) * cos(eps)))) * cos(x)));
	} else {
		double VAR_1;
		if ((eps <= 9.19032739794849e-24)) {
			VAR_1 = (((eps * x) * (x + eps)) + eps);
		} else {
			VAR_1 = (((((tan(x) * tan(x)) - (tan(eps) * tan(eps))) / (1.0 - (tan(x) * tan(eps)))) * (1.0 / (tan(x) - tan(eps)))) - tan(x));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.7
Target	15.8
Herbie	14.7

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

Derivation

1. Split input into 3 regimes

2. if eps < -6.149362912317085e-28

   1. Initial program 30.5

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

   3. Applied tan-sum 2.1

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

   5. Applied tan-quot 2.1

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

   6. Applied tan-quot 2.1

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

   7. Applied frac-times 2.1

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

   9. Applied tan-quot 2.2

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

   10. Applied frac-sub 2.2

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

   if -6.149362912317085e-28 < eps < 9.19032739794849e-24

   1. Initial program 44.8

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

   2. Taylor expanded around 0 30.9

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

   3. Simplified 30.7

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

   if 9.19032739794849e-24 < eps

   1. Initial program 30.1

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

   3. Applied tan-sum 1.8

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

   5. Applied clear-num 1.9

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

   7. Applied flip-+ 2.0

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

   8. Applied associate-/r/ 2.1

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

   9. Applied add-cube-cbrt 2.1

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

   10. Applied times-frac 2.1

       \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x \cdot \tan x - \tan \varepsilon \cdot \tan \varepsilon}} \cdot \frac{\sqrt[3]{1}}{\tan x - \tan \varepsilon}} - \tan x\]

   11. Simplified 2.0

       \[\leadsto \color{blue}{\frac{\tan x \cdot \tan x - \tan \varepsilon \cdot \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} \cdot \frac{\sqrt[3]{1}}{\tan x - \tan \varepsilon} - \tan x\]

   12. Simplified 2.0

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

4. Final simplification 14.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -6.1493629123170849 \cdot 10^{-28}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right) \cdot \sin x}{\left(1 - \frac{\sin x \cdot \sin \varepsilon}{\cos x \cdot \cos \varepsilon}\right) \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 9.19032739794848937 \cdot 10^{-24}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot \left(x + \varepsilon\right) + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x \cdot \tan x - \tan \varepsilon \cdot \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} \cdot \frac{1}{\tan x - \tan \varepsilon} - \tan x\\ \end{array}\]

Reproduce

herbie shell --seed 2020102 
(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)))