Average Error: 36.6 → 0.3

Time: 2.3min

Precision: binary64

Cost: 131522

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.406472418576731 \cdot 10^{-05}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.903921111721757 \cdot 10^{-06}:\\ \;\;\;\;\left(\left(\varepsilon + {\varepsilon}^{3} \cdot \left(0.3333333333333333 + \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right) + \left(1.3333333333333333 \cdot \left({\varepsilon}^{3} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;\varepsilon \leq -4.406472418576731 \cdot 10^{-05}:\\
\;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x\\

\mathbf{elif}\;\varepsilon \leq 4.903921111721757 \cdot 10^{-06}:\\
\;\;\;\;\left(\left(\varepsilon + {\varepsilon}^{3} \cdot \left(0.3333333333333333 + \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right) + \left(1.3333333333333333 \cdot \left({\varepsilon}^{3} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\

\end{array}

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

↓

(FPCore (x eps)
 :precision binary64
 (if (<= eps -4.406472418576731e-05)
   (-
    (*
     (/
      (+ (tan x) (tan eps))
      (- 1.0 (* (* (tan x) (tan eps)) (* (tan x) (tan eps)))))
     (+ 1.0 (* (tan x) (tan eps))))
    (tan x))
   (if (<= eps 4.903921111721757e-06)
     (+
      (+
       (+
        eps
        (*
         (pow eps 3.0)
         (+ 0.3333333333333333 (/ (pow (sin x) 4.0) (pow (cos x) 4.0)))))
       (*
        (* eps eps)
        (+ (/ (sin x) (cos x)) (/ (pow (sin x) 3.0) (pow (cos x) 3.0)))))
      (+
       (*
        1.3333333333333333
        (* (pow eps 3.0) (/ (pow (sin x) 2.0) (pow (cos x) 2.0))))
       (* eps (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))))
     (- (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (tan eps)))) (tan x)))))

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

↓

double code(double x, double eps) {
	double tmp;
	if (eps <= -4.406472418576731e-05) {
		tmp = (((tan(x) + tan(eps)) / (1.0 - ((tan(x) * tan(eps)) * (tan(x) * tan(eps))))) * (1.0 + (tan(x) * tan(eps)))) - tan(x);
	} else if (eps <= 4.903921111721757e-06) {
		tmp = ((eps + (pow(eps, 3.0) * (0.3333333333333333 + (pow(sin(x), 4.0) / pow(cos(x), 4.0))))) + ((eps * eps) * ((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(x), 3.0))))) + ((1.3333333333333333 * (pow(eps, 3.0) * (pow(sin(x), 2.0) / pow(cos(x), 2.0)))) + (eps * (pow(sin(x), 2.0) / pow(cos(x), 2.0))));
	} else {
		tmp = ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	}
	return tmp;
}

Error

The X axis uses an exponential scale — Bits error versus `x`

Try it out

In
Out

Enter valid numbers for all inputs

Target

Original	36.6
Target	15.0
Herbie	0.3

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

Alternatives

Alternative 1
Error	0.3
Cost	66114

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.877889322174695 \cdot 10^{-07}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.1106641687023677 \cdot 10^{-07}:\\ \;\;\;\;\varepsilon + \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right) + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array}\]

Alternative 2
Error	0.3
Cost	59650

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.3224501117002462 \cdot 10^{-07}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.1106641687023677 \cdot 10^{-07}:\\ \;\;\;\;\left(\varepsilon + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + {\left(\frac{\sin x}{\cos x}\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array}\]

Alternative 3
Error	0.4
Cost	59265

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.735739348028543 \cdot 10^{-09}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.948638247358394 \cdot 10^{-09}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array}\]

Alternative 4
Error	0.4
Cost	33032

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -5.947394389597907 \cdot 10^{-09} \lor \neg \left(\varepsilon \leq 4.347937839294806 \cdot 10^{-09}\right):\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \end{array}\]

Alternative 5
Error	13.7
Cost	33032

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.7861050372932166 \cdot 10^{-05} \lor \neg \left(\varepsilon \leq 4.750141807257479 \cdot 10^{-06}\right):\\ \;\;\;\;\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \end{array}\]

Alternative 6
Error	14.3
Cost	26504

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.1161599242012276 \cdot 10^{-05} \lor \neg \left(\varepsilon \leq 3.6673023185841255 \cdot 10^{-06}\right):\\ \;\;\;\;\tan \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + \frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}}\\ \end{array}\]

Alternative 7
Error	14.3
Cost	26504

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.1686865321513423 \cdot 10^{-07} \lor \neg \left(\varepsilon \leq 1.903094475028712 \cdot 10^{-06}\right):\\ \;\;\;\;\tan \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \end{array}\]

Alternative 8
Error	26.6
Cost	6464

\[\tan \varepsilon\]

Alternative 9
Error	41.9
Cost	1218

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.3558768098808724:\\ \;\;\;\;-1\\ \mathbf{elif}\;\varepsilon \leq 3.3880205997539494 \cdot 10^{+35}:\\ \;\;\;\;\varepsilon + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

Alternative 10
Error	42.0
Cost	706

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.3558768098808724:\\ \;\;\;\;-1\\ \mathbf{elif}\;\varepsilon \leq 5.605199843625231 \cdot 10^{+38}:\\ \;\;\;\;\varepsilon\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

Alternative 11
Error	59.6
Cost	64

\[-1\]

Derivation

Split input into 3 regimes
if eps < -4.40647241857673128e-5
1. Initial program 28.6
  \[\tan \left(x + \varepsilon\right) - \tan x\]
2. Using strategy rm
3. Applied tan-sum_binary64_19180.3
  \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
4. Using strategy rm
5. Applied flip--_binary64_17580.4
  \[\leadsto \frac{\tan x + \tan \varepsilon}{\color{blue}{\frac{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}{1 + \tan x \cdot \tan \varepsilon}}} - \tan x\]
6. Applied associate-/r/_binary64_17290.4
  \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right)} - \tan x\]
7. Simplified0.4
  \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x\]
8. Simplified0.4
  \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x}\]
if -4.40647241857673128e-5 < eps < 4.9039211117217574e-6
1. Initial program 44.2
  \[\tan \left(x + \varepsilon\right) - \tan x\]
2. Using strategy rm
3. Applied tan-sum_binary64_191843.7
  \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
4. Taylor expanded around 0 0.2
  \[\leadsto \color{blue}{\frac{\sin x \cdot {\varepsilon}^{2}}{\cos x} + \left(1.3333333333333333 \cdot \frac{{\sin x}^{2} \cdot {\varepsilon}^{3}}{{\cos x}^{2}} + \left(\frac{{\sin x}^{2} \cdot \varepsilon}{{\cos x}^{2}} + \left(0.3333333333333333 \cdot {\varepsilon}^{3} + \left(\frac{{\sin x}^{3} \cdot {\varepsilon}^{2}}{{\cos x}^{3}} + \left(\varepsilon + \frac{{\sin x}^{4} \cdot {\varepsilon}^{3}}{{\cos x}^{4}}\right)\right)\right)\right)\right)}\]
5. Simplified0.2
  \[\leadsto \color{blue}{\left(\frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \varepsilon + 1.3333333333333333 \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot {\varepsilon}^{3}\right)\right) + \left(\left(\varepsilon + {\varepsilon}^{3} \cdot \left(\frac{{\sin x}^{4}}{{\cos x}^{4}} + 0.3333333333333333\right)\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{{\sin x}^{3}}{{\cos x}^{3}} + \frac{\sin x}{\cos x}\right)\right)}\]
6. Simplified0.2
  \[\leadsto \color{blue}{\left(\left(\varepsilon + {\varepsilon}^{3} \cdot \left(0.3333333333333333 + \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right) + \left(1.3333333333333333 \cdot \left({\varepsilon}^{3} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)}\]
if 4.9039211117217574e-6 < eps
1. Initial program 29.9
  \[\tan \left(x + \varepsilon\right) - \tan x\]
2. Using strategy rm
3. Applied tan-sum_binary64_19180.4
  \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
4. Using strategy rm
5. Applied pow1_binary64_18440.4
  \[\leadsto \frac{\color{blue}{{\left(\tan x + \tan \varepsilon\right)}^{1}}}{1 - \tan x \cdot \tan \varepsilon} - \tan x\]
6. Using strategy rm
7. Applied *-un-lft-identity_binary64_17830.4
  \[\leadsto \color{blue}{1 \cdot \frac{{\left(\tan x + \tan \varepsilon\right)}^{1}}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
8. Simplified0.4
  \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x}\]
Recombined 3 regimes into one program.
Final simplification0.3
\[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.406472418576731 \cdot 10^{-05}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.903921111721757 \cdot 10^{-06}:\\ \;\;\;\;\left(\left(\varepsilon + {\varepsilon}^{3} \cdot \left(0.3333333333333333 + \frac{{\sin x}^{4}}{{\cos x}^{4}}\right)\right) + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right) + \left(1.3333333333333333 \cdot \left({\varepsilon}^{3} \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) + \varepsilon \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array}\]

Reproduce

herbie shell --seed 2021014 
(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)))

Error

Try it out

Target

Alternatives

Error

Derivation

`if eps < -4.40647241857673128e-5`

`if -4.40647241857673128e-5 < eps < 4.9039211117217574e-6`

`if 4.9039211117217574e-6 < eps`

Reproduce