2tan (problem 3.3.2)

Average Error: 37.5 → 0.3

Time: 21.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := \tan x \cdot \tan \varepsilon\\ t_2 := 1 + t_1\\ t_3 := \tan \varepsilon \cdot \left(\tan x \cdot t_1\right)\\ \mathbf{if}\;\varepsilon \leq -4.798181894578827 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_0}{\frac{1 - t_3}{t_2}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 5.704032184871116 \cdot 10^{-5}:\\ \;\;\;\;\begin{array}{l} t_4 := {\sin x}^{2}\\ t_5 := {\cos x}^{2}\\ \left(\frac{{\varepsilon}^{3}}{{\left(\frac{\cos x}{\sin x}\right)}^{4}} + \left(\mathsf{fma}\left(1.3333333333333333, \frac{{\varepsilon}^{3} \cdot t_4}{t_5}, {\varepsilon}^{3} \cdot 0.3333333333333333\right) + \mathsf{fma}\left(\varepsilon, \frac{t_4}{t_5}, \varepsilon\right)\right)\right) + \frac{\varepsilon \cdot \varepsilon}{\cos x} \cdot \left(\sin x + \frac{{\sin x}^{3}}{t_5}\right) \end{array}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_6 := \sqrt{t_3}\\ \frac{t_0}{\frac{1 - t_6 \cdot t_6}{t_2}} - \tan x \end{array}\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps)))
        (t_1 (* (tan x) (tan eps)))
        (t_2 (+ 1.0 t_1))
        (t_3 (* (tan eps) (* (tan x) t_1))))
   (if (<= eps -4.798181894578827e-5)
     (- (/ t_0 (/ (- 1.0 t_3) t_2)) (tan x))
     (if (<= eps 5.704032184871116e-5)
       (let* ((t_4 (pow (sin x) 2.0)) (t_5 (pow (cos x) 2.0)))
         (+
          (+
           (/ (pow eps 3.0) (pow (/ (cos x) (sin x)) 4.0))
           (+
            (fma
             1.3333333333333333
             (/ (* (pow eps 3.0) t_4) t_5)
             (* (pow eps 3.0) 0.3333333333333333))
            (fma eps (/ t_4 t_5) eps)))
          (* (/ (* eps eps) (cos x)) (+ (sin x) (/ (pow (sin x) 3.0) t_5)))))
       (let* ((t_6 (sqrt t_3)))
         (- (/ t_0 (/ (- 1.0 (* t_6 t_6)) t_2)) (tan x)))))))

C

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

↓

double code(double x, double eps) {
	double t_0 = tan(x) + tan(eps);
	double t_1 = tan(x) * tan(eps);
	double t_2 = 1.0 + t_1;
	double t_3 = tan(eps) * (tan(x) * t_1);
	double tmp;
	if (eps <= -4.798181894578827e-5) {
		tmp = (t_0 / ((1.0 - t_3) / t_2)) - tan(x);
	} else if (eps <= 5.704032184871116e-5) {
		double t_4 = pow(sin(x), 2.0);
		double t_5 = pow(cos(x), 2.0);
		tmp = ((pow(eps, 3.0) / pow((cos(x) / sin(x)), 4.0)) + (fma(1.3333333333333333, ((pow(eps, 3.0) * t_4) / t_5), (pow(eps, 3.0) * 0.3333333333333333)) + fma(eps, (t_4 / t_5), eps))) + (((eps * eps) / cos(x)) * (sin(x) + (pow(sin(x), 3.0) / t_5)));
	} else {
		double t_6 = sqrt(t_3);
		tmp = (t_0 / ((1.0 - (t_6 * t_6)) / t_2)) - tan(x);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.5
Target	15.0
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if eps < -4.7981818945788272e-5

   1. Initial program 29.2

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

   2. Applied tan-sum_binary64 0.3

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

   3. Applied flip--_binary64 0.4

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

   4. Applied associate-*r*_binary64 0.4

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

   5. Applied pow1_binary64 0.4

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

   6. Applied pow1_binary64 0.4

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

   7. Applied pow-prod-down_binary64 0.4

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

   if -4.7981818945788272e-5 < eps < 5.7040321848711158e-5

   1. Initial program 45.6

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

   2. Taylor expanded in eps around 0 0.2

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

   3. Simplified 0.2

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

   if 5.7040321848711158e-5 < eps

   1. Initial program 30.1

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

   2. Applied tan-sum_binary64 0.4

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

   3. Applied flip--_binary64 0.4

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

   4. Applied associate-*r*_binary64 0.4

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

   5. Applied add-sqr-sqrt_binary64 0.4

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.798181894578827 \cdot 10^{-5}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{\frac{1 - \tan \varepsilon \cdot \left(\tan x \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}{1 + \tan x \cdot \tan \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 5.704032184871116 \cdot 10^{-5}:\\ \;\;\;\;\left(\frac{{\varepsilon}^{3}}{{\left(\frac{\cos x}{\sin x}\right)}^{4}} + \left(\mathsf{fma}\left(1.3333333333333333, \frac{{\varepsilon}^{3} \cdot {\sin x}^{2}}{{\cos x}^{2}}, {\varepsilon}^{3} \cdot 0.3333333333333333\right) + \mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon\right)\right)\right) + \frac{\varepsilon \cdot \varepsilon}{\cos x} \cdot \left(\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{\frac{1 - \sqrt{\tan \varepsilon \cdot \left(\tan x \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)} \cdot \sqrt{\tan \varepsilon \cdot \left(\tan x \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}}{1 + \tan x \cdot \tan \varepsilon}} - \tan x\\ \end{array} \]

Reproduce

herbie shell --seed 2022077 
(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)))