2tan (problem 3.3.2)

Average Error: 37.6 → 0.4

Time: 8.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := -\tan x\\ t_1 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.00028767276362063015:\\ \;\;\;\;\begin{array}{l} t_2 := \tan x \cdot \tan \varepsilon\\ \mathsf{fma}\left(t_1, \frac{1}{1 - {t_2}^{3}} \cdot \left(1 + \left(t_2 + t_2 \cdot t_2\right)\right), t_0\right) \end{array}\\ \mathbf{elif}\;\varepsilon \leq 4.0940859351585484 \cdot 10^{-13}:\\ \;\;\;\;\begin{array}{l} t_3 := {\cos x}^{2}\\ t_4 := {\sin x}^{3}\\ t_5 := {\sin x}^{2}\\ t_6 := {\cos x}^{3}\\ \frac{{\varepsilon}^{2} \cdot t_4}{t_6} + \left(\frac{{\varepsilon}^{2} \cdot \sin x}{\cos x} + \left(\varepsilon + \left(\frac{{\varepsilon}^{3} \cdot {\sin x}^{4}}{{\cos x}^{4}} + \left(1.6666666666666667 \cdot \frac{t_4 \cdot {\varepsilon}^{4}}{t_6} + \left(\frac{{\varepsilon}^{4} \cdot {\sin x}^{5}}{{\cos x}^{5}} + \left(1.3333333333333333 \cdot \frac{{\varepsilon}^{3} \cdot t_5}{t_3} + \left(\frac{\varepsilon \cdot t_5}{t_3} + \left(0.6666666666666666 \cdot \frac{\sin x \cdot {\varepsilon}^{4}}{\cos x} + {\varepsilon}^{3} \cdot 0.3333333333333333\right)\right)\right)\right)\right)\right)\right)\right) \end{array}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_1, \frac{1}{1 - \frac{\sqrt[3]{{\left(\tan \varepsilon \cdot \sin x\right)}^{3}}}{\cos x}}, t_0\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (tan x))) (t_1 (+ (tan x) (tan eps))))
   (if (<= eps -0.00028767276362063015)
     (let* ((t_2 (* (tan x) (tan eps))))
       (fma
        t_1
        (* (/ 1.0 (- 1.0 (pow t_2 3.0))) (+ 1.0 (+ t_2 (* t_2 t_2))))
        t_0))
     (if (<= eps 4.0940859351585484e-13)
       (let* ((t_3 (pow (cos x) 2.0))
              (t_4 (pow (sin x) 3.0))
              (t_5 (pow (sin x) 2.0))
              (t_6 (pow (cos x) 3.0)))
         (+
          (/ (* (pow eps 2.0) t_4) t_6)
          (+
           (/ (* (pow eps 2.0) (sin x)) (cos x))
           (+
            eps
            (+
             (/ (* (pow eps 3.0) (pow (sin x) 4.0)) (pow (cos x) 4.0))
             (+
              (* 1.6666666666666667 (/ (* t_4 (pow eps 4.0)) t_6))
              (+
               (/ (* (pow eps 4.0) (pow (sin x) 5.0)) (pow (cos x) 5.0))
               (+
                (* 1.3333333333333333 (/ (* (pow eps 3.0) t_5) t_3))
                (+
                 (/ (* eps t_5) t_3)
                 (+
                  (* 0.6666666666666666 (/ (* (sin x) (pow eps 4.0)) (cos x)))
                  (* (pow eps 3.0) 0.3333333333333333)))))))))))
       (fma
        t_1
        (/ 1.0 (- 1.0 (/ (cbrt (pow (* (tan eps) (sin x)) 3.0)) (cos x))))
        t_0)))))

C

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

↓

double code(double x, double eps) {
	double t_0 = -tan(x);
	double t_1 = tan(x) + tan(eps);
	double tmp;
	if (eps <= -0.00028767276362063015) {
		double t_2_1 = tan(x) * tan(eps);
		tmp = fma(t_1, ((1.0 / (1.0 - pow(t_2_1, 3.0))) * (1.0 + (t_2_1 + (t_2_1 * t_2_1)))), t_0);
	} else if (eps <= 4.0940859351585484e-13) {
		double t_3 = pow(cos(x), 2.0);
		double t_4 = pow(sin(x), 3.0);
		double t_5 = pow(sin(x), 2.0);
		double t_6 = pow(cos(x), 3.0);
		tmp = ((pow(eps, 2.0) * t_4) / t_6) + (((pow(eps, 2.0) * sin(x)) / cos(x)) + (eps + (((pow(eps, 3.0) * pow(sin(x), 4.0)) / pow(cos(x), 4.0)) + ((1.6666666666666667 * ((t_4 * pow(eps, 4.0)) / t_6)) + (((pow(eps, 4.0) * pow(sin(x), 5.0)) / pow(cos(x), 5.0)) + ((1.3333333333333333 * ((pow(eps, 3.0) * t_5) / t_3)) + (((eps * t_5) / t_3) + ((0.6666666666666666 * ((sin(x) * pow(eps, 4.0)) / cos(x))) + (pow(eps, 3.0) * 0.3333333333333333)))))))));
	} else {
		tmp = fma(t_1, (1.0 / (1.0 - (cbrt(pow((tan(eps) * sin(x)), 3.0)) / cos(x)))), t_0);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.6
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 < -2.8767276362063015e-4

   1. Initial program 29.8

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

   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 div-inv_binary64 0.3

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

   4. Applied fma-neg_binary64 0.3

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

   5. Applied flip3--_binary64 0.4

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

   6. Applied associate-/r/_binary64 0.4

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

   7. Simplified 0.4

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

   if -2.8767276362063015e-4 < eps < 4.0940859351585484e-13

   1. Initial program 45.6

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

   2. Taylor expanded in eps around 0 0.2

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

   if 4.0940859351585484e-13 < eps

   1. Initial program 30.2

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

   2. Applied tan-sum_binary64 0.8

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

   3. Applied div-inv_binary64 0.9

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

   4. Applied fma-neg_binary64 0.8

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

   5. Applied tan-quot_binary64 0.9

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

   6. Applied associate-*l/_binary64 0.9

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

   7. Applied add-cbrt-cube_binary64 0.9

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

   8. Simplified 0.9

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00028767276362063015:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}} \cdot \left(1 + \left(\tan x \cdot \tan \varepsilon + \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)\right), -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 4.0940859351585484 \cdot 10^{-13}:\\ \;\;\;\;\frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}} + \left(\frac{{\varepsilon}^{2} \cdot \sin x}{\cos x} + \left(\varepsilon + \left(\frac{{\varepsilon}^{3} \cdot {\sin x}^{4}}{{\cos x}^{4}} + \left(1.6666666666666667 \cdot \frac{{\sin x}^{3} \cdot {\varepsilon}^{4}}{{\cos x}^{3}} + \left(\frac{{\varepsilon}^{4} \cdot {\sin x}^{5}}{{\cos x}^{5}} + \left(1.3333333333333333 \cdot \frac{{\varepsilon}^{3} \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(0.6666666666666666 \cdot \frac{\sin x \cdot {\varepsilon}^{4}}{\cos x} + {\varepsilon}^{3} \cdot 0.3333333333333333\right)\right)\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \frac{\sqrt[3]{{\left(\tan \varepsilon \cdot \sin x\right)}^{3}}}{\cos x}}, -\tan x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022019 
(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)))