2tan (problem 3.3.2)

Average Error: 36.8 → 12.6

Time: 6.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.82287172944081 \cdot 10^{-11}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \log \left({\left(e^{\tan x}\right)}^{\left(\tan \varepsilon\right)}\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.298064167888159 \cdot 10^{-24}:\\ \;\;\;\;\varepsilon + \left(\varepsilon \cdot \log 1\right) \cdot \left(\varepsilon + x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \left(\tan x \cdot \sqrt[3]{{\left(\tan \varepsilon\right)}^{6}}\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x\\ \end{array}\]

C

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

↓

double code(double x, double eps) {
	double VAR;
	if ((eps <= -7.82287172944081e-11)) {
		VAR = ((double) ((((double) (((double) tan(x)) + ((double) tan(eps)))) / ((double) (1.0 - ((double) log(((double) pow(((double) exp(((double) tan(x)))), ((double) tan(eps))))))))) - ((double) tan(x))));
	} else {
		double VAR_1;
		if ((eps <= 1.298064167888159e-24)) {
			VAR_1 = ((double) (eps + ((double) (((double) (eps * ((double) log(1.0)))) * ((double) (eps + x))))));
		} else {
			VAR_1 = ((double) (((double) ((((double) (((double) tan(x)) + ((double) tan(eps)))) / ((double) (1.0 - ((double) (((double) tan(x)) * ((double) (((double) tan(x)) * ((double) cbrt(((double) pow(((double) tan(eps)), 6.0))))))))))) * ((double) (1.0 + ((double) (((double) tan(x)) * ((double) tan(eps)))))))) - ((double) tan(x))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.8
Target	15.2
Herbie	12.6

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

Derivation

1. Split input into 3 regimes

2. if eps < -7.82287172944080946e-11

   1. Initial program Error: 30.3 bits

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

   3. Applied tan-sum Error: 0.6 bits

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

   5. Applied add-log-exp Error: 0.7 bits

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

   6. Simplified Error: 0.7 bits

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

   if -7.82287172944080946e-11 < eps < 1.298064167888159e-24

   1. Initial program Error: 44.3 bits

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

   3. Applied tan-sum Error: 44.2 bits

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

   5. Applied add-log-exp Error: 44.2 bits

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

   6. Simplified Error: 44.2 bits

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

   7. Taylor expanded around 0 Error: 25.2 bits

      \[\leadsto \color{blue}{\log 1 \cdot {\varepsilon}^{2} + \left(\varepsilon + \log 1 \cdot \left(x \cdot \varepsilon\right)\right)}\]

   8. Simplified Error: 25.2 bits

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

   if 1.298064167888159e-24 < eps

   1. Initial program Error: 30.0 bits

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

   3. Applied tan-sum Error: 1.8 bits

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

   5. Applied flip-- Error: 1.9 bits

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

   6. Applied associate-/r/ Error: 1.9 bits

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

   7. Simplified Error: 1.9 bits

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

   9. Applied add-cbrt-cube Error: 1.9 bits

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

   10. Applied add-cbrt-cube Error: 2.0 bits

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

   11. Applied cbrt-unprod Error: 1.9 bits

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

   12. Simplified Error: 1.9 bits

       \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \left(\tan x \cdot \sqrt[3]{\color{blue}{{\left(\tan \varepsilon\right)}^{6}}}\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x\]
3. Recombined 3 regimes into one program.

4. Final simplification Error: 12.6 bits

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.82287172944081 \cdot 10^{-11}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \log \left({\left(e^{\tan x}\right)}^{\left(\tan \varepsilon\right)}\right)} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 1.298064167888159 \cdot 10^{-24}:\\ \;\;\;\;\varepsilon + \left(\varepsilon \cdot \log 1\right) \cdot \left(\varepsilon + x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \left(\tan x \cdot \sqrt[3]{{\left(\tan \varepsilon\right)}^{6}}\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x\\ \end{array}\]

Reproduce

herbie shell --seed 2020200 
(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)))