Average Error: 36.7 → 0.5
Time: 1.2m
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\left(\frac{\sin x}{\frac{\cos \varepsilon \cdot \cos x}{\sin x}} + \frac{\cos x}{\cos \varepsilon}\right) \cdot \sin \varepsilon}{\cos x - \left(\cos x \cdot \sin \varepsilon\right) \cdot \frac{\sin x}{\cos \varepsilon \cdot \cos x}}\]
double f(double x, double eps) {
        double r9749579 = x;
        double r9749580 = eps;
        double r9749581 = r9749579 + r9749580;
        double r9749582 = tan(r9749581);
        double r9749583 = tan(r9749579);
        double r9749584 = r9749582 - r9749583;
        return r9749584;
}
↓
double f(double x, double eps) {
        double r9749585 = x;
        double r9749586 = sin(r9749585);
        double r9749587 = eps;
        double r9749588 = cos(r9749587);
        double r9749589 = cos(r9749585);
        double r9749590 = r9749588 * r9749589;
        double r9749591 = r9749590 / r9749586;
        double r9749592 = r9749586 / r9749591;
        double r9749593 = r9749589 / r9749588;
        double r9749594 = r9749592 + r9749593;
        double r9749595 = sin(r9749587);
        double r9749596 = r9749594 * r9749595;
        double r9749597 = r9749589 * r9749595;
        double r9749598 = r9749586 / r9749590;
        double r9749599 = r9749597 * r9749598;
        double r9749600 = r9749589 - r9749599;
        double r9749601 = r9749596 / r9749600;
        return r9749601;
}


\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\left(\frac{\sin x}{\frac{\cos \varepsilon \cdot \cos x}{\sin x}} + \frac{\cos x}{\cos \varepsilon}\right) \cdot \sin \varepsilon}{\cos x - \left(\cos x \cdot \sin \varepsilon\right) \cdot \frac{\sin x}{\cos \varepsilon \cdot \cos x}}

Error

Bits error versus x

Bits error versus eps

Target

Original36.7
Target15.1
Herbie0.5
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.7

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

  5. Applied tan-quot21.6

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

  6. Applied frac-sub21.6

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

  7. Taylor expanded around inf 0.4

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

  8. Simplified0.5

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

  9. Final simplification0.5

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

Reproduce

herbie shell --seed 2019102 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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