2tan (problem 3.3.2)

Average Error: 37.3 → 0.4

Time: 13.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -9.830087725539631 \cdot 10^{-5}:\\ \;\;\;\;\begin{array}{l} t_1 := \tan x \cdot \tan \varepsilon\\ \mathsf{fma}\left(\frac{t_0}{1 - {t_1}^{3}}, 1 + \left(t_1 \cdot t_1 + \log \left({\left(e^{\tan x}\right)}^{\tan \varepsilon}\right)\right), -\tan x\right) \end{array}\\ \mathbf{elif}\;\varepsilon \leq 0.0002654352846451175:\\ \;\;\;\;\begin{array}{l} t_2 := {\sin x}^{2}\\ t_3 := {\cos x}^{2}\\ t_4 := \frac{\cos x}{\sin x}\\ \left(\varepsilon + \left(\frac{{\varepsilon}^{3}}{{t_4}^{4}} + \mathsf{fma}\left(1.6666666666666667, \frac{{\varepsilon}^{4}}{{t_4}^{3}}, \mathsf{fma}\left(\frac{{\varepsilon}^{4}}{{\cos x}^{5}}, {\sin x}^{5}, \mathsf{fma}\left(1.3333333333333333, \frac{{\varepsilon}^{3} \cdot t_2}{t_3}, \mathsf{fma}\left(\varepsilon, \frac{t_2}{t_3}, \mathsf{fma}\left(0.3333333333333333, {\varepsilon}^{3}, 0.6666666666666666 \cdot \left({\varepsilon}^{4} \cdot \frac{\sin x}{\cos x}\right)\right)\right)\right)\right)\right)\right)\right) + \frac{\varepsilon \cdot \varepsilon}{\cos x} \cdot \left(\sin x + \frac{{\sin x}^{3}}{t_3}\right) \end{array}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\frac{\sin x \cdot \sin \varepsilon}{\cos x}}{\cos \varepsilon}} - \tan x\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -9.830087725539631e-5)
     (let* ((t_1 (* (tan x) (tan eps))))
       (fma
        (/ t_0 (- 1.0 (pow t_1 3.0)))
        (+ 1.0 (+ (* t_1 t_1) (log (pow (exp (tan x)) (tan eps)))))
        (- (tan x))))
     (if (<= eps 0.0002654352846451175)
       (let* ((t_2 (pow (sin x) 2.0))
              (t_3 (pow (cos x) 2.0))
              (t_4 (/ (cos x) (sin x))))
         (+
          (+
           eps
           (+
            (/ (pow eps 3.0) (pow t_4 4.0))
            (fma
             1.6666666666666667
             (/ (pow eps 4.0) (pow t_4 3.0))
             (fma
              (/ (pow eps 4.0) (pow (cos x) 5.0))
              (pow (sin x) 5.0)
              (fma
               1.3333333333333333
               (/ (* (pow eps 3.0) t_2) t_3)
               (fma
                eps
                (/ t_2 t_3)
                (fma
                 0.3333333333333333
                 (pow eps 3.0)
                 (*
                  0.6666666666666666
                  (* (pow eps 4.0) (/ (sin x) (cos x)))))))))))
          (* (/ (* eps eps) (cos x)) (+ (sin x) (/ (pow (sin x) 3.0) t_3)))))
       (-
        (/ t_0 (- 1.0 (/ (/ (* (sin x) (sin eps)) (cos x)) (cos eps))))
        (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 tmp;
	if (eps <= -9.830087725539631e-5) {
		double t_1_1 = tan(x) * tan(eps);
		tmp = fma((t_0 / (1.0 - pow(t_1_1, 3.0))), (1.0 + ((t_1_1 * t_1_1) + log(pow(exp(tan(x)), tan(eps))))), -tan(x));
	} else if (eps <= 0.0002654352846451175) {
		double t_2 = pow(sin(x), 2.0);
		double t_3 = pow(cos(x), 2.0);
		double t_4 = cos(x) / sin(x);
		tmp = (eps + ((pow(eps, 3.0) / pow(t_4, 4.0)) + fma(1.6666666666666667, (pow(eps, 4.0) / pow(t_4, 3.0)), fma((pow(eps, 4.0) / pow(cos(x), 5.0)), pow(sin(x), 5.0), fma(1.3333333333333333, ((pow(eps, 3.0) * t_2) / t_3), fma(eps, (t_2 / t_3), fma(0.3333333333333333, pow(eps, 3.0), (0.6666666666666666 * (pow(eps, 4.0) * (sin(x) / cos(x))))))))))) + (((eps * eps) / cos(x)) * (sin(x) + (pow(sin(x), 3.0) / t_3)));
	} else {
		tmp = (t_0 / (1.0 - (((sin(x) * sin(eps)) / cos(x)) / cos(eps)))) - tan(x);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.3
Target	15.2
Herbie	0.4

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

Derivation

1. Split input into 3 regimes

2. if eps < -9.830087725539631e-5

   1. Initial program 30.2

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

   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 flip3--_binary64 0.4

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

   4. Applied associate-/r/_binary64 0.4

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

   5. Applied fma-neg_binary64 0.4

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

   6. Applied add-log-exp_binary64 0.6

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

   7. Simplified 0.6

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

   if -9.830087725539631e-5 < eps < 2.6543528464511749e-4

   1. Initial program 45.3

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

   2. Taylor expanded in eps around 0 0.2

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

   3. Simplified 0.2

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

   if 2.6543528464511749e-4 < eps

   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 tan-quot_binary64 0.4

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

   4. Applied tan-quot_binary64 0.4

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

   5. Applied frac-times_binary64 0.4

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

   6. Applied associate-/r*_binary64 0.4

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -9.830087725539631 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, 1 + \left(\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right) + \log \left({\left(e^{\tan x}\right)}^{\tan \varepsilon}\right)\right), -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0002654352846451175:\\ \;\;\;\;\left(\varepsilon + \left(\frac{{\varepsilon}^{3}}{{\left(\frac{\cos x}{\sin x}\right)}^{4}} + \mathsf{fma}\left(1.6666666666666667, \frac{{\varepsilon}^{4}}{{\left(\frac{\cos x}{\sin x}\right)}^{3}}, \mathsf{fma}\left(\frac{{\varepsilon}^{4}}{{\cos x}^{5}}, {\sin x}^{5}, \mathsf{fma}\left(1.3333333333333333, \frac{{\varepsilon}^{3} \cdot {\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(0.3333333333333333, {\varepsilon}^{3}, 0.6666666666666666 \cdot \left({\varepsilon}^{4} \cdot \frac{\sin x}{\cos x}\right)\right)\right)\right)\right)\right)\right)\right) + \frac{\varepsilon \cdot \varepsilon}{\cos x} \cdot \left(\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\frac{\sin x \cdot \sin \varepsilon}{\cos x}}{\cos \varepsilon}} - \tan x\\ \end{array} \]

Reproduce

herbie shell --seed 2022088 
(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)))