Average Error: 36.5 → 15.1
Time: 16.7s
Precision: binary64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.951010323486192 \cdot 10^{-59}:\\ \;\;\;\;\frac{\frac{\tan x \cdot \tan x - \tan \varepsilon \cdot \tan \varepsilon}{\tan x - \tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.0501768016039094 \cdot 10^{-42}:\\ \;\;\;\;\varepsilon + \left(\varepsilon + x\right) \cdot \left(\varepsilon \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) \cdot \left(\tan \varepsilon \cdot \sqrt[3]{\tan x}\right)} - \tan x\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \leq -6.951010323486192 \cdot 10^{-59}:\\
\;\;\;\;\frac{\frac{\tan x \cdot \tan x - \tan \varepsilon \cdot \tan \varepsilon}{\tan x - \tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\

\mathbf{elif}\;\varepsilon \leq 4.0501768016039094 \cdot 10^{-42}:\\
\;\;\;\;\varepsilon + \left(\varepsilon + x\right) \cdot \left(\varepsilon \cdot x\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) \cdot \left(\tan \varepsilon \cdot \sqrt[3]{\tan x}\right)} - \tan x\\

\end{array}
(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))
↓
(FPCore (x eps)
 :precision binary64
 (if (<= eps -6.951010323486192e-59)
   (-
    (/
     (/ (- (* (tan x) (tan x)) (* (tan eps) (tan eps))) (- (tan x) (tan eps)))
     (- 1.0 (* (tan x) (tan eps))))
    (tan x))
   (if (<= eps 4.0501768016039094e-42)
     (+ eps (* (+ eps x) (* eps x)))
     (-
      (/
       (+ (tan x) (tan eps))
       (-
        1.0
        (* (* (cbrt (tan x)) (cbrt (tan x))) (* (tan eps) (cbrt (tan x))))))
      (tan x)))))
double code(double x, double eps) {
	return tan(x + eps) - tan(x);
}
↓
double code(double x, double eps) {
	double tmp;
	if (eps <= -6.951010323486192e-59) {
		tmp = ((((tan(x) * tan(x)) - (tan(eps) * tan(eps))) / (tan(x) - tan(eps))) / (1.0 - (tan(x) * tan(eps)))) - tan(x);
	} else if (eps <= 4.0501768016039094e-42) {
		tmp = eps + ((eps + x) * (eps * x));
	} else {
		tmp = ((tan(x) + tan(eps)) / (1.0 - ((cbrt(tan(x)) * cbrt(tan(x))) * (tan(eps) * cbrt(tan(x)))))) - tan(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

Original36.5
Target15.0
Herbie15.1
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -6.95101032348619227e-59

    1. Initial program 29.9

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

    3. Applied tan-sum_binary64_15774.7

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

    5. Applied flip-+_binary64_14164.7

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

    if -6.95101032348619227e-59 < eps < 4.0501768016039094e-42

    1. Initial program 46.2

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

    2. Taylor expanded around 0 30.9

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

    3. Simplified30.7

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

    if 4.0501768016039094e-42 < eps

    1. Initial program 29.5

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

    3. Applied tan-sum_binary64_15773.4

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

    5. Applied add-cube-cbrt_binary64_14773.5

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

    6. Applied associate-*l*_binary64_13833.5

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

    7. Simplified3.5

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

  4. Final simplification15.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.951010323486192 \cdot 10^{-59}:\\ \;\;\;\;\frac{\frac{\tan x \cdot \tan x - \tan \varepsilon \cdot \tan \varepsilon}{\tan x - \tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 4.0501768016039094 \cdot 10^{-42}:\\ \;\;\;\;\varepsilon + \left(\varepsilon + x\right) \cdot \left(\varepsilon \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) \cdot \left(\tan \varepsilon \cdot \sqrt[3]{\tan x}\right)} - \tan x\\ \end{array}\]

Reproduce

herbie shell --seed 2020354 
(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)))