2tan (problem 3.3.2)

Average Error: 36.8 → 0.4

Time: 12.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ t_1 := \tan x \cdot \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -3.511738504762205 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t_0}{1 - {t_1}^{3}}, \mathsf{fma}\left(t_1, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), 1\right), -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 8.542991026995156 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - t_1} - \tan x\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))) (t_1 (* (tan x) (tan eps))))
   (if (<= eps -3.511738504762205e-9)
     (fma
      (/ t_0 (- 1.0 (pow t_1 3.0)))
      (fma t_1 (fma (tan x) (tan eps) 1.0) 1.0)
      (- (tan x)))
     (if (<= eps 8.542991026995156e-11)
       (fma eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0)) eps)
       (- (/ t_0 (- 1.0 t_1)) (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 tmp;
	if (eps <= -3.511738504762205e-9) {
		tmp = fma((t_0 / (1.0 - pow(t_1, 3.0))), fma(t_1, fma(tan(x), tan(eps), 1.0), 1.0), -tan(x));
	} else if (eps <= 8.542991026995156e-11) {
		tmp = fma(eps, (pow(sin(x), 2.0) / pow(cos(x), 2.0)), eps);
	} else {
		tmp = (t_0 / (1.0 - t_1)) - tan(x);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.8
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 < -3.511738504762205e-9

   1. Initial program 29.7

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

   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 add-cube-cbrt_binary64 0.8

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

   4. Applied flip3--_binary64 0.8

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

   5. Applied associate-/r/_binary64 0.8

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

   6. Applied prod-diff_binary64 0.8

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

   7. Simplified 0.5

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

   8. Simplified 0.5

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

   9. Applied pow1_binary64 0.5

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

   if -3.511738504762205e-9 < eps < 8.54299102699515649e-11

   1. Initial program 44.6

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

   2. Taylor expanded in eps around 0 0.4

      \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

   3. Simplified 0.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon\right)} \]

   if 8.54299102699515649e-11 < eps

   1. Initial program 29.6

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

   2. Applied tan-sum_binary64 0.6

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

   3. Applied *-un-lft-identity_binary64 0.6

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

   4. Applied cancel-sign-sub-inv_binary64 0.6

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.511738504762205 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - {\left(\tan x \cdot \tan \varepsilon\right)}^{3}}, \mathsf{fma}\left(\tan x \cdot \tan \varepsilon, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), 1\right), -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 8.542991026995156 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array} \]

Reproduce

herbie shell --seed 2022024 
(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)))