Average Error: 37.0 → 12.7
Time: 31.6s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)} + \left(\mathsf{fma}\left(\frac{-1}{\cos x}, \sin x, \frac{\sin x}{\cos x}\right) + \mathsf{fma}\left(\frac{\frac{\sin x}{\sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)}}}{\sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)}}, \frac{1}{\sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)} \cdot \cos x}, -\frac{\sin x}{\cos x}\right)\right)\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)} + \left(\mathsf{fma}\left(\frac{-1}{\cos x}, \sin x, \frac{\sin x}{\cos x}\right) + \mathsf{fma}\left(\frac{\frac{\sin x}{\sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)}}}{\sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)}}, \frac{1}{\sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)} \cdot \cos x}, -\frac{\sin x}{\cos x}\right)\right)
double f(double x, double eps) {
        double r122962 = x;
        double r122963 = eps;
        double r122964 = r122962 + r122963;
        double r122965 = tan(r122964);
        double r122966 = tan(r122962);
        double r122967 = r122965 - r122966;
        return r122967;
}
↓
double f(double x, double eps) {
        double r122968 = eps;
        double r122969 = sin(r122968);
        double r122970 = cos(r122968);
        double r122971 = r122969 / r122970;
        double r122972 = x;
        double r122973 = sin(r122972);
        double r122974 = cos(r122972);
        double r122975 = r122973 / r122974;
        double r122976 = -r122971;
        double r122977 = 1.0;
        double r122978 = fma(r122975, r122976, r122977);
        double r122979 = r122971 / r122978;
        double r122980 = -1.0;
        double r122981 = r122980 / r122974;
        double r122982 = fma(r122981, r122973, r122975);
        double r122983 = cbrt(r122978);
        double r122984 = r122973 / r122983;
        double r122985 = r122984 / r122983;
        double r122986 = r122983 * r122974;
        double r122987 = r122977 / r122986;
        double r122988 = -r122975;
        double r122989 = fma(r122985, r122987, r122988);
        double r122990 = r122982 + r122989;
        double r122991 = r122979 + r122990;
        return r122991;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.0
Target15.2
Herbie12.7
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.0

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

  3. Applied tan-sum21.6

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

  4. Simplified21.6

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

  5. Simplified21.6

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

  6. Taylor expanded around inf 21.7

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

  7. Simplified12.5

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

  9. Applied add-cube-cbrt20.9

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

  10. Applied add-cube-cbrt20.9

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

  11. Applied div-inv20.9

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

  12. Applied times-frac20.9

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

  13. Applied prod-diff22.3

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

  14. Simplified22.2

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

  15. Simplified12.7

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

  16. Final simplification12.7

    \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)} + \left(\mathsf{fma}\left(\frac{-1}{\cos x}, \sin x, \frac{\sin x}{\cos x}\right) + \mathsf{fma}\left(\frac{\frac{\sin x}{\sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)}}}{\sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)}}, \frac{1}{\sqrt[3]{\mathsf{fma}\left(\frac{\sin x}{\cos x}, -\frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)} \cdot \cos x}, -\frac{\sin x}{\cos x}\right)\right)\]

Reproduce

herbie shell --seed 2019194 +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)))