Average Error: 37.4 → 0.4
Time: 32.7s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\mathsf{fma}\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}, \frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin \varepsilon \cdot \sin x}{\left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\cos \varepsilon \cdot \cos x\right)} \cdot \frac{\left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)}{\cos \varepsilon \cdot \cos x}\right)}, \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin \varepsilon \cdot \sin x}{\left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\cos \varepsilon \cdot \cos x\right)} \cdot \frac{\left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)}{\cos \varepsilon \cdot \cos x}} - \frac{\sin x}{\cos x}\right) + \mathsf{fma}\left(\frac{\frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{1 - \frac{\sin \varepsilon \cdot \sin x}{\left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\cos \varepsilon \cdot \cos x\right)} \cdot \frac{\left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)}{\cos \varepsilon \cdot \cos x}}, \mathsf{fma}\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}, \frac{\sin x}{\cos x}, \frac{\sin x}{\cos x}\right), \frac{\frac{\left(\left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)\right) \cdot \sin \varepsilon}{\left(\left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\cos \varepsilon \cdot \cos x\right)\right) \cdot \cos \varepsilon}}{1 - \frac{\sin \varepsilon \cdot \sin x}{\left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\cos \varepsilon \cdot \cos x\right)} \cdot \frac{\left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)}{\cos \varepsilon \cdot \cos x}} + \frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin \varepsilon \cdot \sin x}{\left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\cos \varepsilon \cdot \cos x\right)} \cdot \frac{\left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)}{\cos \varepsilon \cdot \cos x}\right)}\right)\right)\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\mathsf{fma}\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}, \frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin \varepsilon \cdot \sin x}{\left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\cos \varepsilon \cdot \cos x\right)} \cdot \frac{\left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)}{\cos \varepsilon \cdot \cos x}\right)}, \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin \varepsilon \cdot \sin x}{\left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\cos \varepsilon \cdot \cos x\right)} \cdot \frac{\left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)}{\cos \varepsilon \cdot \cos x}} - \frac{\sin x}{\cos x}\right) + \mathsf{fma}\left(\frac{\frac{\sin \varepsilon \cdot \sin \varepsilon}{\cos \varepsilon \cdot \cos \varepsilon}}{1 - \frac{\sin \varepsilon \cdot \sin x}{\left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\cos \varepsilon \cdot \cos x\right)} \cdot \frac{\left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)}{\cos \varepsilon \cdot \cos x}}, \mathsf{fma}\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}, \frac{\sin x}{\cos x}, \frac{\sin x}{\cos x}\right), \frac{\frac{\left(\left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)\right) \cdot \sin \varepsilon}{\left(\left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\cos \varepsilon \cdot \cos x\right)\right) \cdot \cos \varepsilon}}{1 - \frac{\sin \varepsilon \cdot \sin x}{\left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\cos \varepsilon \cdot \cos x\right)} \cdot \frac{\left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)}{\cos \varepsilon \cdot \cos x}} + \frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin \varepsilon \cdot \sin x}{\left(\cos \varepsilon \cdot \cos x\right) \cdot \left(\cos \varepsilon \cdot \cos x\right)} \cdot \frac{\left(\sin \varepsilon \cdot \sin x\right) \cdot \left(\sin \varepsilon \cdot \sin x\right)}{\cos \varepsilon \cdot \cos x}\right)}\right)\right)
double f(double x, double eps) {
        double r3988998 = x;
        double r3988999 = eps;
        double r3989000 = r3988998 + r3988999;
        double r3989001 = tan(r3989000);
        double r3989002 = tan(r3988998);
        double r3989003 = r3989001 - r3989002;
        return r3989003;
}
↓
double f(double x, double eps) {
        double r3989004 = x;
        double r3989005 = sin(r3989004);
        double r3989006 = cos(r3989004);
        double r3989007 = r3989005 / r3989006;
        double r3989008 = r3989007 * r3989007;
        double r3989009 = eps;
        double r3989010 = sin(r3989009);
        double r3989011 = cos(r3989009);
        double r3989012 = 1.0;
        double r3989013 = r3989010 * r3989005;
        double r3989014 = r3989011 * r3989006;
        double r3989015 = r3989014 * r3989014;
        double r3989016 = r3989013 / r3989015;
        double r3989017 = r3989013 * r3989013;
        double r3989018 = r3989017 / r3989014;
        double r3989019 = r3989016 * r3989018;
        double r3989020 = r3989012 - r3989019;
        double r3989021 = r3989011 * r3989020;
        double r3989022 = r3989010 / r3989021;
        double r3989023 = r3989007 / r3989020;
        double r3989024 = r3989023 - r3989007;
        double r3989025 = r3989010 * r3989010;
        double r3989026 = r3989011 * r3989011;
        double r3989027 = r3989025 / r3989026;
        double r3989028 = r3989027 / r3989020;
        double r3989029 = fma(r3989008, r3989007, r3989007);
        double r3989030 = r3989017 * r3989010;
        double r3989031 = r3989015 * r3989011;
        double r3989032 = r3989030 / r3989031;
        double r3989033 = r3989032 / r3989020;
        double r3989034 = r3989033 + r3989022;
        double r3989035 = fma(r3989028, r3989029, r3989034);
        double r3989036 = r3989024 + r3989035;
        double r3989037 = fma(r3989008, r3989022, r3989036);
        return r3989037;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.4
Target15.6
Herbie0.4
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.4

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

  3. Applied tan-sum21.8

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

  5. Applied flip3--21.8

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

  6. Applied associate-/r/21.8

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

  7. Applied fma-neg21.8

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

  8. Taylor expanded around inf 21.9

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

  9. Simplified19.6

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

  11. Applied associate--l+0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019162 +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)))