Average Error: 37.0 → 15.7
Time: 33.9s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.50639832785694013 \cdot 10^{-96}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\sin x, \cos \varepsilon, \cos x \cdot \sin \varepsilon\right)}{\cos x \cdot \cos \varepsilon}, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \le 6.0287019077622141 \cdot 10^{-70}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, x, 1\right), \varepsilon, x \cdot {\varepsilon}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \end{array}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.50639832785694013 \cdot 10^{-96}:\\
\;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\sin x, \cos \varepsilon, \cos x \cdot \sin \varepsilon\right)}{\cos x \cdot \cos \varepsilon}, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\

\mathbf{elif}\;\varepsilon \le 6.0287019077622141 \cdot 10^{-70}:\\
\;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, x, 1\right), \varepsilon, x \cdot {\varepsilon}^{2}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\

\end{array}
double f(double x, double eps) {
        double r91941 = x;
        double r91942 = eps;
        double r91943 = r91941 + r91942;
        double r91944 = tan(r91943);
        double r91945 = tan(r91941);
        double r91946 = r91944 - r91945;
        return r91946;
}
↓
double f(double x, double eps) {
        double r91947 = eps;
        double r91948 = -1.5063983278569401e-96;
        bool r91949 = r91947 <= r91948;
        double r91950 = x;
        double r91951 = sin(r91950);
        double r91952 = cos(r91947);
        double r91953 = cos(r91950);
        double r91954 = sin(r91947);
        double r91955 = r91953 * r91954;
        double r91956 = fma(r91951, r91952, r91955);
        double r91957 = r91953 * r91952;
        double r91958 = r91956 / r91957;
        double r91959 = 1.0;
        double r91960 = tan(r91950);
        double r91961 = tan(r91947);
        double r91962 = r91960 * r91961;
        double r91963 = r91959 - r91962;
        double r91964 = r91959 / r91963;
        double r91965 = -r91960;
        double r91966 = fma(r91958, r91964, r91965);
        double r91967 = 6.028701907762214e-70;
        bool r91968 = r91947 <= r91967;
        double r91969 = fma(r91950, r91950, r91959);
        double r91970 = 2.0;
        double r91971 = pow(r91947, r91970);
        double r91972 = r91950 * r91971;
        double r91973 = fma(r91969, r91947, r91972);
        double r91974 = r91960 + r91961;
        double r91975 = r91974 * r91953;
        double r91976 = r91963 * r91951;
        double r91977 = r91975 - r91976;
        double r91978 = r91963 * r91953;
        double r91979 = r91977 / r91978;
        double r91980 = r91968 ? r91973 : r91979;
        double r91981 = r91949 ? r91966 : r91980;
        return r91981;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.0
Target15.3
Herbie15.7
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -1.5063983278569401e-96

    1. Initial program 31.4

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

    3. Applied tan-sum7.7

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

    5. Applied div-inv7.7

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

    6. Applied fma-neg7.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(\tan x + \tan \varepsilon, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)}\]
    7. Using strategy rm

    8. Applied tan-quot7.8

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

    9. Applied tan-quot7.9

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

    10. Applied frac-add7.9

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

    11. Simplified7.9

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\mathsf{fma}\left(\sin x, \cos \varepsilon, \cos x \cdot \sin \varepsilon\right)}}{\cos x \cdot \cos \varepsilon}, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\]

    if -1.5063983278569401e-96 < eps < 6.028701907762214e-70

    1. Initial program 47.9

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

    3. Applied tan-sum47.9

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

    4. Taylor expanded around 0 31.3

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

    5. Simplified31.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 1\right), \varepsilon, x \cdot {\varepsilon}^{2}\right)}\]

    if 6.028701907762214e-70 < eps

    1. Initial program 30.1

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

    3. Applied tan-quot29.9

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

    4. Applied tan-sum5.5

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

    5. Applied frac-sub5.5

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

  4. Final simplification15.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.50639832785694013 \cdot 10^{-96}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\sin x, \cos \varepsilon, \cos x \cdot \sin \varepsilon\right)}{\cos x \cdot \cos \varepsilon}, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \mathbf{elif}\;\varepsilon \le 6.0287019077622141 \cdot 10^{-70}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, x, 1\right), \varepsilon, x \cdot {\varepsilon}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

  (- (tan (+ x eps)) (tan x)))