Average Error: 37.2 → 12.8
Time: 27.3s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{\mathsf{fma}\left(-\frac{\sin x}{\cos x}, \frac{\sin \varepsilon}{\cos \varepsilon}, 1\right)} + \left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} - \frac{\sin x}{\cos x}\right)\]
\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(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin \varepsilon}{\cos \varepsilon} \cdot \frac{\sin x}{\cos x}} - \frac{\sin x}{\cos x}\right)
double f(double x, double eps) {
        double r5501002 = x;
        double r5501003 = eps;
        double r5501004 = r5501002 + r5501003;
        double r5501005 = tan(r5501004);
        double r5501006 = tan(r5501002);
        double r5501007 = r5501005 - r5501006;
        return r5501007;
}
↓
double f(double x, double eps) {
        double r5501008 = eps;
        double r5501009 = sin(r5501008);
        double r5501010 = cos(r5501008);
        double r5501011 = r5501009 / r5501010;
        double r5501012 = x;
        double r5501013 = sin(r5501012);
        double r5501014 = cos(r5501012);
        double r5501015 = r5501013 / r5501014;
        double r5501016 = -r5501015;
        double r5501017 = 1.0;
        double r5501018 = fma(r5501016, r5501011, r5501017);
        double r5501019 = r5501011 / r5501018;
        double r5501020 = r5501011 * r5501015;
        double r5501021 = r5501017 - r5501020;
        double r5501022 = r5501015 / r5501021;
        double r5501023 = r5501022 - r5501015;
        double r5501024 = r5501019 + r5501023;
        return r5501024;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.2
Target15.7
Herbie12.8
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.2

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

  3. Applied tan-sum21.5

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

  4. Taylor expanded around inf 21.7

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

  5. Simplified12.8

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

  7. Applied div-inv12.8

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

  8. Applied fma-def12.8

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

  9. Taylor expanded around inf 12.8

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

  10. Simplified12.8

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

  11. Final simplification12.8

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

Reproduce

herbie shell --seed 2019179 +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)))