2tan (problem 3.3.2)

Average Error: 36.8 → 0.6

Time: 11.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := \tan x \cdot \tan \varepsilon\\ t_2 := -\tan x\\ \mathbf{if}\;\varepsilon \leq -0.0002726995289715794:\\ \;\;\;\;\mathsf{fma}\left(t_0, \frac{1}{1 - \log \left(e^{t_1}\right)}, t_2\right)\\ \mathbf{elif}\;\varepsilon \leq 1.9282140589339793 \cdot 10^{-22}:\\ \;\;\;\;\begin{array}{l} t_3 := {\sin x}^{3}\\ t_4 := {\sin x}^{2}\\ t_5 := {\cos x}^{2}\\ t_6 := {\cos x}^{3}\\ \frac{{\varepsilon}^{2} \cdot t_3}{t_6} + \left(\frac{{\varepsilon}^{2} \cdot \sin x}{\cos x} + \left(\varepsilon + \left(\frac{{\varepsilon}^{3} \cdot {\sin x}^{4}}{{\cos x}^{4}} + \left(1.6666666666666667 \cdot \frac{t_3 \cdot {\varepsilon}^{4}}{t_6} + \left(\frac{{\varepsilon}^{4} \cdot {\sin x}^{5}}{{\cos x}^{5}} + \left(1.3333333333333333 \cdot \frac{{\varepsilon}^{3} \cdot t_4}{t_5} + \left(\frac{\varepsilon \cdot t_4}{t_5} + \left(0.6666666666666666 \cdot \frac{\sin x \cdot {\varepsilon}^{4}}{\cos x} + {\varepsilon}^{3} \cdot 0.3333333333333333\right)\right)\right)\right)\right)\right)\right)\right) \end{array}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t_0}{1 - \sqrt[3]{{\left({t_1}^{3}\right)}^{3}}}, \mathsf{fma}\left(t_1, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), 1\right), t_2\right)\\ \end{array} \]

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 (- (tan x))))
   (if (<= eps -0.0002726995289715794)
     (fma t_0 (/ 1.0 (- 1.0 (log (exp t_1)))) t_2)
     (if (<= eps 1.9282140589339793e-22)
       (let* ((t_3 (pow (sin x) 3.0))
              (t_4 (pow (sin x) 2.0))
              (t_5 (pow (cos x) 2.0))
              (t_6 (pow (cos x) 3.0)))
         (+
          (/ (* (pow eps 2.0) t_3) 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_3 (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_4) t_5))
                (+
                 (/ (* eps t_4) t_5)
                 (+
                  (* 0.6666666666666666 (/ (* (sin x) (pow eps 4.0)) (cos x)))
                  (* (pow eps 3.0) 0.3333333333333333)))))))))))
       (fma
        (/ t_0 (- 1.0 (cbrt (pow (pow t_1 3.0) 3.0))))
        (fma t_1 (fma (tan x) (tan eps) 1.0) 1.0)
        t_2)))))

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 = -tan(x);
	double tmp;
	if (eps <= -0.0002726995289715794) {
		tmp = fma(t_0, (1.0 / (1.0 - log(exp(t_1)))), t_2);
	} else if (eps <= 1.9282140589339793e-22) {
		double t_3 = pow(sin(x), 3.0);
		double t_4 = pow(sin(x), 2.0);
		double t_5 = pow(cos(x), 2.0);
		double t_6 = pow(cos(x), 3.0);
		tmp = ((pow(eps, 2.0) * t_3) / 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_3 * 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_4) / t_5)) + (((eps * t_4) / t_5) + ((0.6666666666666666 * ((sin(x) * pow(eps, 4.0)) / cos(x))) + (pow(eps, 3.0) * 0.3333333333333333)))))))));
	} else {
		tmp = fma((t_0 / (1.0 - cbrt(pow(pow(t_1, 3.0), 3.0)))), fma(t_1, fma(tan(x), tan(eps), 1.0), 1.0), t_2);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.8
Target	14.9
Herbie	0.6

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

Derivation

1. Split input into 3 regimes

2. if eps < -2.7269952897157937e-4

   1. Initial program 30.1

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

   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 0.4

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

   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 add-log-exp_binary64 0.4

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

   if -2.7269952897157937e-4 < eps < 1.92821405893397928e-22

   1. Initial program 44.8

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

   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 1.92821405893397928e-22 < eps

   1. Initial program 28.9

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

   2. Applied tan-sum_binary64 1.6

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

   3. Applied add-cube-cbrt_binary64 1.9

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

   4. Applied flip3--_binary64 1.9

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

   5. Applied associate-/r/_binary64 1.9

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

   6. Applied prod-diff_binary64 1.9

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

   7. Simplified 1.5

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

   8. Simplified 1.6

      \[\leadsto \mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, \mathsf{fma}\left(\tan x \cdot \tan \varepsilon, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), 1\right), -\tan x\right) + \color{blue}{0} \]

   9. Applied add-cbrt-cube_binary64 1.6

      \[\leadsto \mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\sqrt[3]{\left({\left(\tan x \cdot \tan \varepsilon\right)}^{3} \cdot {\left(\tan x \cdot \tan \varepsilon\right)}^{3}\right) \cdot {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}}}, \mathsf{fma}\left(\tan x \cdot \tan \varepsilon, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), 1\right), -\tan x\right) + 0 \]

   10. Simplified 1.6

       \[\leadsto \mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{\color{blue}{{\left({\left(\tan x \cdot \tan \varepsilon\right)}^{3}\right)}^{3}}}}, \mathsf{fma}\left(\tan x \cdot \tan \varepsilon, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), 1\right), -\tan x\right) + 0 \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0002726995289715794:\\ \;\;\;\;\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \log \left(e^{\tan x \cdot \tan \varepsilon}\right)}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 1.9282140589339793 \cdot 10^{-22}:\\ \;\;\;\;\frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}} + \left(\frac{{\varepsilon}^{2} \cdot \sin x}{\cos x} + \left(\varepsilon + \left(\frac{{\varepsilon}^{3} \cdot {\sin x}^{4}}{{\cos x}^{4}} + \left(1.6666666666666667 \cdot \frac{{\sin x}^{3} \cdot {\varepsilon}^{4}}{{\cos x}^{3}} + \left(\frac{{\varepsilon}^{4} \cdot {\sin x}^{5}}{{\cos x}^{5}} + \left(1.3333333333333333 \cdot \frac{{\varepsilon}^{3} \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(0.6666666666666666 \cdot \frac{\sin x \cdot {\varepsilon}^{4}}{\cos x} + {\varepsilon}^{3} \cdot 0.3333333333333333\right)\right)\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{{\left({\left(\tan x \cdot \tan \varepsilon\right)}^{3}\right)}^{3}}}, \mathsf{fma}\left(\tan x \cdot \tan \varepsilon, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), 1\right), -\tan x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021313 
(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)))