Average Error: 37.3 → 15.1
Time: 8.4s
Precision: binary64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.7248288871182305 \cdot 10^{-62} \lor \neg \left(\varepsilon \leq 5.817203080719242 \cdot 10^{-77}\right):\\ \;\;\;\;\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} \cdot \frac{{\left(\sin \varepsilon\right)}^{3}}{{\left(\cos \varepsilon\right)}^{3} \cdot \left(1 - {\left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}\right)} + \left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} \cdot \frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - {\left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}\right)} + \left(\left(\frac{\sin x}{\cos x \cdot \left(1 - {\left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}\right)} + \left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - {\left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}\right)} + \frac{{\left(\sin \varepsilon\right)}^{2}}{\left(1 - {\left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}\right) \cdot {\left(\cos \varepsilon\right)}^{2}} \cdot \left(\frac{\sin x}{\cos x} + {\left(\frac{\sin x}{\cos x}\right)}^{3}\right)\right)\right) - \frac{\sin x}{\cos x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + x \cdot \left(\varepsilon \cdot \left(\varepsilon + x\right)\right)\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \leq -1.7248288871182305 \cdot 10^{-62} \lor \neg \left(\varepsilon \leq 5.817203080719242 \cdot 10^{-77}\right):\\
\;\;\;\;\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} \cdot \frac{{\left(\sin \varepsilon\right)}^{3}}{{\left(\cos \varepsilon\right)}^{3} \cdot \left(1 - {\left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}\right)} + \left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} \cdot \frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - {\left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}\right)} + \left(\left(\frac{\sin x}{\cos x \cdot \left(1 - {\left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}\right)} + \left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - {\left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}\right)} + \frac{{\left(\sin \varepsilon\right)}^{2}}{\left(1 - {\left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}\right) \cdot {\left(\cos \varepsilon\right)}^{2}} \cdot \left(\frac{\sin x}{\cos x} + {\left(\frac{\sin x}{\cos x}\right)}^{3}\right)\right)\right) - \frac{\sin x}{\cos x}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\varepsilon + x \cdot \left(\varepsilon \cdot \left(\varepsilon + x\right)\right)\\

\end{array}
(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))
↓
(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.7248288871182305e-62) (not (<= eps 5.817203080719242e-77)))
   (+
    (*
     (/ (pow (sin x) 2.0) (pow (cos x) 2.0))
     (/
      (pow (sin eps) 3.0)
      (*
       (pow (cos eps) 3.0)
       (- 1.0 (pow (* (/ (sin x) (cos x)) (/ (sin eps) (cos eps))) 3.0)))))
    (+
     (*
      (/ (pow (sin x) 2.0) (pow (cos x) 2.0))
      (/
       (sin eps)
       (*
        (cos eps)
        (- 1.0 (pow (* (/ (sin x) (cos x)) (/ (sin eps) (cos eps))) 3.0)))))
     (-
      (+
       (/
        (sin x)
        (*
         (cos x)
         (- 1.0 (pow (* (/ (sin x) (cos x)) (/ (sin eps) (cos eps))) 3.0))))
       (+
        (/
         (sin eps)
         (*
          (cos eps)
          (- 1.0 (pow (* (/ (sin x) (cos x)) (/ (sin eps) (cos eps))) 3.0))))
        (*
         (/
          (pow (sin eps) 2.0)
          (*
           (- 1.0 (pow (* (/ (sin x) (cos x)) (/ (sin eps) (cos eps))) 3.0))
           (pow (cos eps) 2.0)))
         (+ (/ (sin x) (cos x)) (pow (/ (sin x) (cos x)) 3.0)))))
      (/ (sin x) (cos x)))))
   (+ eps (* x (* eps (+ eps x))))))
double code(double x, double eps) {
	return ((double) (((double) tan(((double) (x + eps)))) - ((double) tan(x))));
}
↓
double code(double x, double eps) {
	double tmp;
	if (((eps <= -1.7248288871182305e-62) || !(eps <= 5.817203080719242e-77))) {
		tmp = ((double) (((double) ((((double) pow(((double) sin(x)), 2.0)) / ((double) pow(((double) cos(x)), 2.0))) * (((double) pow(((double) sin(eps)), 3.0)) / ((double) (((double) pow(((double) cos(eps)), 3.0)) * ((double) (1.0 - ((double) pow(((double) ((((double) sin(x)) / ((double) cos(x))) * (((double) sin(eps)) / ((double) cos(eps))))), 3.0))))))))) + ((double) (((double) ((((double) pow(((double) sin(x)), 2.0)) / ((double) pow(((double) cos(x)), 2.0))) * (((double) sin(eps)) / ((double) (((double) cos(eps)) * ((double) (1.0 - ((double) pow(((double) ((((double) sin(x)) / ((double) cos(x))) * (((double) sin(eps)) / ((double) cos(eps))))), 3.0))))))))) + ((double) (((double) ((((double) sin(x)) / ((double) (((double) cos(x)) * ((double) (1.0 - ((double) pow(((double) ((((double) sin(x)) / ((double) cos(x))) * (((double) sin(eps)) / ((double) cos(eps))))), 3.0))))))) + ((double) ((((double) sin(eps)) / ((double) (((double) cos(eps)) * ((double) (1.0 - ((double) pow(((double) ((((double) sin(x)) / ((double) cos(x))) * (((double) sin(eps)) / ((double) cos(eps))))), 3.0))))))) + ((double) ((((double) pow(((double) sin(eps)), 2.0)) / ((double) (((double) (1.0 - ((double) pow(((double) ((((double) sin(x)) / ((double) cos(x))) * (((double) sin(eps)) / ((double) cos(eps))))), 3.0)))) * ((double) pow(((double) cos(eps)), 2.0))))) * ((double) ((((double) sin(x)) / ((double) cos(x))) + ((double) pow((((double) sin(x)) / ((double) cos(x))), 3.0)))))))))) - (((double) sin(x)) / ((double) cos(x)))))))));
	} else {
		tmp = ((double) (eps + ((double) (x * ((double) (eps * ((double) (eps + x))))))));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.3
Target15.5
Herbie15.1
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -1.72482888711823047e-62 or 5.81720308071924208e-77 < eps

    1. Initial program Error: 31.1 bits

      \[\tan \left(x + \varepsilon\right) - \tan x\]
    2. Using strategy rm

    3. Applied tan-sumError: 5.9 bits

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
    4. Using strategy rm

    5. Applied flip3--Error: 5.9 bits

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

    6. Applied associate-/r/Error: 5.9 bits

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

    7. SimplifiedError: 5.9 bits

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

    8. Taylor expanded around inf Error: 6.1 bits

      \[\leadsto \color{blue}{\left(\frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{2} \cdot \left(\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos \varepsilon\right)}^{3}\right)} + \left(\frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{\cos \varepsilon \cdot \left({\left(\cos x\right)}^{2} \cdot \left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right)\right)} + \left(\frac{\sin x}{\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot \cos x} + \left(\frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{3} \cdot \left(\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos \varepsilon\right)}^{2}\right)} + \left(\frac{\sin x \cdot {\left(\sin \varepsilon\right)}^{2}}{\cos x \cdot \left(\left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right) \cdot {\left(\cos \varepsilon\right)}^{2}\right)} + \frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{{\left(\sin x\right)}^{3} \cdot {\left(\sin \varepsilon\right)}^{3}}{{\left(\cos x\right)}^{3} \cdot {\left(\cos \varepsilon\right)}^{3}}\right)}\right)\right)\right)\right)\right) - \frac{\sin x}{\cos x}}\]

    9. SimplifiedError: 5.3 bits

      \[\leadsto \color{blue}{\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} \cdot \frac{{\left(\sin \varepsilon\right)}^{3}}{{\left(\cos \varepsilon\right)}^{3} \cdot \left(1 - {\left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}\right)} + \left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - {\left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}\right)} \cdot \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} + \left(\left(\frac{\sin x}{\cos x \cdot \left(1 - {\left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}\right)} + \left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - {\left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}\right)} + \frac{{\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2} \cdot \left(1 - {\left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}\right)} \cdot \left({\left(\frac{\sin x}{\cos x}\right)}^{3} + \frac{\sin x}{\cos x}\right)\right)\right) - \frac{\sin x}{\cos x}\right)\right)}\]

    if -1.72482888711823047e-62 < eps < 5.81720308071924208e-77

    1. Initial program Error: 47.3 bits

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

    2. Taylor expanded around 0 Error: 31.3 bits

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

    3. SimplifiedError: 31.1 bits

      \[\leadsto \color{blue}{\varepsilon + x \cdot \left(\varepsilon \cdot \left(x + \varepsilon\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplificationError: 15.1 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.7248288871182305 \cdot 10^{-62} \lor \neg \left(\varepsilon \leq 5.817203080719242 \cdot 10^{-77}\right):\\ \;\;\;\;\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} \cdot \frac{{\left(\sin \varepsilon\right)}^{3}}{{\left(\cos \varepsilon\right)}^{3} \cdot \left(1 - {\left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}\right)} + \left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} \cdot \frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - {\left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}\right)} + \left(\left(\frac{\sin x}{\cos x \cdot \left(1 - {\left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}\right)} + \left(\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - {\left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}\right)} + \frac{{\left(\sin \varepsilon\right)}^{2}}{\left(1 - {\left(\frac{\sin x}{\cos x} \cdot \frac{\sin \varepsilon}{\cos \varepsilon}\right)}^{3}\right) \cdot {\left(\cos \varepsilon\right)}^{2}} \cdot \left(\frac{\sin x}{\cos x} + {\left(\frac{\sin x}{\cos x}\right)}^{3}\right)\right)\right) - \frac{\sin x}{\cos x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon + x \cdot \left(\varepsilon \cdot \left(\varepsilon + x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(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)))