2tan (problem 3.3.2)

Average Error: 37.1 → 15.5

Time: 6.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -8.8270901166319843 \cdot 10^{-91}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \sin x}{\left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 1.47192102037964442 \cdot 10^{-77}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \varepsilon, \varepsilon + x, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \end{array}\]

C

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

↓

double code(double x, double eps) {
	double VAR;
	if ((eps <= -8.827090116631984e-91)) {
		VAR = ((((tan(x) + tan(eps)) * cos(x)) - ((1.0 - (tan(eps) * tan(x))) * sin(x))) / ((1.0 - (tan(eps) * tan(x))) * cos(x)));
	} else {
		double VAR_1;
		if ((eps <= 1.4719210203796444e-77)) {
			VAR_1 = fma((x * eps), (eps + x), eps);
		} else {
			VAR_1 = fma((tan(x) + tan(eps)), (-1.0 / fma(tan(x), tan(eps), -1.0)), -tan(x));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	15.4
Herbie	15.5

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

Derivation

1. Split input into 3 regimes

2. if eps < -8.827090116631984e-91

   1. Initial program 31.5

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

   3. Applied tan-sum 7.3

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

   4. Simplified 7.3

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

   6. Applied tan-quot 7.4

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

   7. Applied frac-sub 7.4

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

   if -8.827090116631984e-91 < eps < 1.4719210203796444e-77

   1. Initial program 47.7

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

   3. Applied tan-sum 47.7

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

   4. Simplified 47.7

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

   6. Applied div-inv 47.7

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

   7. Applied fma-neg 47.7

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

   8. Taylor expanded around 0 31.1

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

   9. Simplified 30.8

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

   if 1.4719210203796444e-77 < eps

   1. Initial program 30.6

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

   3. Applied tan-sum 6.1

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

   4. Simplified 6.1

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

   6. Applied div-inv 6.1

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

   7. Applied fma-neg 6.1

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

   9. Applied frac-2neg 6.1

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

   10. Simplified 6.1

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

4. Final simplification 15.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -8.8270901166319843 \cdot 10^{-91}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \sin x}{\left(1 - \tan \varepsilon \cdot \tan x\right) \cdot \cos x}\\ \mathbf{elif}\;\varepsilon \le 1.47192102037964442 \cdot 10^{-77}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \varepsilon, \varepsilon + x, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{-1}{\mathsf{fma}\left(\tan x, \tan \varepsilon, -1\right)}, -\tan x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(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)))