Average Error: 37.3 → 0.7
Time: 14.4s
Precision: binary64
\[\tan \left(x + \varepsilon\right) - \tan x \]
↓
\[\begin{array}{l} t_0 := \tan x \cdot \tan \varepsilon\\ t_1 := {\sin x}^{2}\\ t_2 := -\tan x\\ t_3 := {\cos x}^{2}\\ \mathbf{if}\;\varepsilon \leq -0.00022498455180266905:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \tan \varepsilon \cdot \left(\tan x \cdot t_0\right)}, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), t_2\right)\\ \mathbf{elif}\;\varepsilon \leq 2.941971583774042 \cdot 10^{-26}:\\ \;\;\;\;\left(\varepsilon + \mathsf{fma}\left(\frac{{\varepsilon}^{3}}{{\cos x}^{4}}, {\sin x}^{4}, \mathsf{fma}\left(1.6666666666666667, \frac{{\varepsilon}^{4}}{{\left(\frac{\cos x}{\sin x}\right)}^{3}}, \mathsf{fma}\left(\frac{{\varepsilon}^{4}}{{\cos x}^{5}}, {\sin x}^{5}, \mathsf{fma}\left(1.3333333333333333, \frac{{\varepsilon}^{3} \cdot t_1}{t_3}, \mathsf{fma}\left(\varepsilon, \frac{t_1}{t_3}, \mathsf{fma}\left(0.3333333333333333, {\varepsilon}^{3}, 0.6666666666666666 \cdot \left({\varepsilon}^{4} \cdot \frac{\sin x}{\cos x}\right)\right)\right)\right)\right)\right)\right)\right) + \frac{\varepsilon \cdot \varepsilon}{\cos x} \cdot \left(\sin x + \frac{{\sin x}^{3}}{t_3}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{\tan x - \tan \varepsilon}{{\tan x}^{2} - {\tan \varepsilon}^{2}}}, \frac{1}{1 - t_0}, t_2\right)\\ \end{array} \]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
t_0 := \tan x \cdot \tan \varepsilon\\
t_1 := {\sin x}^{2}\\
t_2 := -\tan x\\
t_3 := {\cos x}^{2}\\
\mathbf{if}\;\varepsilon \leq -0.00022498455180266905:\\
\;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \tan \varepsilon \cdot \left(\tan x \cdot t_0\right)}, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), t_2\right)\\

\mathbf{elif}\;\varepsilon \leq 2.941971583774042 \cdot 10^{-26}:\\
\;\;\;\;\left(\varepsilon + \mathsf{fma}\left(\frac{{\varepsilon}^{3}}{{\cos x}^{4}}, {\sin x}^{4}, \mathsf{fma}\left(1.6666666666666667, \frac{{\varepsilon}^{4}}{{\left(\frac{\cos x}{\sin x}\right)}^{3}}, \mathsf{fma}\left(\frac{{\varepsilon}^{4}}{{\cos x}^{5}}, {\sin x}^{5}, \mathsf{fma}\left(1.3333333333333333, \frac{{\varepsilon}^{3} \cdot t_1}{t_3}, \mathsf{fma}\left(\varepsilon, \frac{t_1}{t_3}, \mathsf{fma}\left(0.3333333333333333, {\varepsilon}^{3}, 0.6666666666666666 \cdot \left({\varepsilon}^{4} \cdot \frac{\sin x}{\cos x}\right)\right)\right)\right)\right)\right)\right)\right) + \frac{\varepsilon \cdot \varepsilon}{\cos x} \cdot \left(\sin x + \frac{{\sin x}^{3}}{t_3}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{\tan x - \tan \varepsilon}{{\tan x}^{2} - {\tan \varepsilon}^{2}}}, \frac{1}{1 - t_0}, t_2\right)\\


\end{array}

(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))
↓
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan eps)))
        (t_1 (pow (sin x) 2.0))
        (t_2 (- (tan x)))
        (t_3 (pow (cos x) 2.0)))
   (if (<= eps -0.00022498455180266905)
     (fma
      (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan eps) (* (tan x) t_0))))
      (fma (tan x) (tan eps) 1.0)
      t_2)
     (if (<= eps 2.941971583774042e-26)
       (+
        (+
         eps
         (fma
          (/ (pow eps 3.0) (pow (cos x) 4.0))
          (pow (sin x) 4.0)
          (fma
           1.6666666666666667
           (/ (pow eps 4.0) (pow (/ (cos x) (sin x)) 3.0))
           (fma
            (/ (pow eps 4.0) (pow (cos x) 5.0))
            (pow (sin x) 5.0)
            (fma
             1.3333333333333333
             (/ (* (pow eps 3.0) t_1) t_3)
             (fma
              eps
              (/ t_1 t_3)
              (fma
               0.3333333333333333
               (pow eps 3.0)
               (*
                0.6666666666666666
                (* (pow eps 4.0) (/ (sin x) (cos x)))))))))))
        (* (/ (* eps eps) (cos x)) (+ (sin x) (/ (pow (sin x) 3.0) t_3))))
       (fma
        (/
         1.0
         (/ (- (tan x) (tan eps)) (- (pow (tan x) 2.0) (pow (tan eps) 2.0))))
        (/ 1.0 (- 1.0 t_0))
        t_2)))))
double code(double x, double eps) {
	return tan((x + eps)) - tan(x);
}
↓
double code(double x, double eps) {
	double t_0 = tan(x) * tan(eps);
	double t_1 = pow(sin(x), 2.0);
	double t_2 = -tan(x);
	double t_3 = pow(cos(x), 2.0);
	double tmp;
	if (eps <= -0.00022498455180266905) {
		tmp = fma(((tan(x) + tan(eps)) / (1.0 - (tan(eps) * (tan(x) * t_0)))), fma(tan(x), tan(eps), 1.0), t_2);
	} else if (eps <= 2.941971583774042e-26) {
		tmp = (eps + fma((pow(eps, 3.0) / pow(cos(x), 4.0)), pow(sin(x), 4.0), fma(1.6666666666666667, (pow(eps, 4.0) / pow((cos(x) / sin(x)), 3.0)), fma((pow(eps, 4.0) / pow(cos(x), 5.0)), pow(sin(x), 5.0), fma(1.3333333333333333, ((pow(eps, 3.0) * t_1) / t_3), fma(eps, (t_1 / t_3), fma(0.3333333333333333, pow(eps, 3.0), (0.6666666666666666 * (pow(eps, 4.0) * (sin(x) / cos(x))))))))))) + (((eps * eps) / cos(x)) * (sin(x) + (pow(sin(x), 3.0) / t_3)));
	} else {
		tmp = fma((1.0 / ((tan(x) - tan(eps)) / (pow(tan(x), 2.0) - pow(tan(eps), 2.0)))), (1.0 / (1.0 - t_0)), t_2);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.3
Target15.0
Herbie0.7
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

Derivation

  1. Split input into 3 regimes

  2. if eps < -2.2498455180266905e-4

    1. Initial program 30.5

      \[\tan \left(x + \varepsilon\right) - \tan x \]

    2. Applied egg-rr0.4

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

    3. Applied egg-rr0.4

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

    4. Applied egg-rr0.4

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

    if -2.2498455180266905e-4 < eps < 2.9419715837740421e-26

    1. Initial program 45.0

      \[\tan \left(x + \varepsilon\right) - \tan x \]

    2. Taylor expanded in eps around 0 0.2

      \[\leadsto \color{blue}{\frac{{\varepsilon}^{2} \cdot {\sin x}^{3}}{{\cos x}^{3}} + \left(\frac{{\varepsilon}^{2} \cdot \sin x}{\cos x} + \left(\varepsilon + \left(\frac{{\varepsilon}^{3} \cdot {\sin x}^{4}}{{\cos x}^{4}} + \left(1.6666666666666667 \cdot \frac{{\varepsilon}^{4} \cdot {\sin x}^{3}}{{\cos x}^{3}} + \left(\frac{{\varepsilon}^{4} \cdot {\sin x}^{5}}{{\cos x}^{5}} + \left(1.3333333333333333 \cdot \frac{{\varepsilon}^{3} \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(\frac{\varepsilon \cdot {\sin x}^{2}}{{\cos x}^{2}} + \left(0.6666666666666666 \cdot \frac{{\varepsilon}^{4} \cdot \sin x}{\cos x} + 0.3333333333333333 \cdot {\varepsilon}^{3}\right)\right)\right)\right)\right)\right)\right)\right)} \]

    3. Simplified0.2

      \[\leadsto \color{blue}{\left(\varepsilon + \mathsf{fma}\left(\frac{{\varepsilon}^{3}}{{\cos x}^{4}}, {\sin x}^{4}, \mathsf{fma}\left(1.6666666666666667, \frac{{\varepsilon}^{4}}{{\left(\frac{\cos x}{\sin x}\right)}^{3}}, \mathsf{fma}\left(\frac{{\varepsilon}^{4}}{{\cos x}^{5}}, {\sin x}^{5}, \mathsf{fma}\left(1.3333333333333333, \frac{{\sin x}^{2} \cdot {\varepsilon}^{3}}{{\cos x}^{2}}, \mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(0.3333333333333333, {\varepsilon}^{3}, 0.6666666666666666 \cdot \left(\frac{\sin x}{\cos x} \cdot {\varepsilon}^{4}\right)\right)\right)\right)\right)\right)\right)\right) + \frac{\varepsilon \cdot \varepsilon}{\cos x} \cdot \left(\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}\right)} \]

    if 2.9419715837740421e-26 < eps

    1. Initial program 29.7

      \[\tan \left(x + \varepsilon\right) - \tan x \]

    2. Applied egg-rr1.9

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

    3. Applied egg-rr2.1

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{\tan x - \tan \varepsilon}{{\tan x}^{2} - {\tan \varepsilon}^{2}}}}, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right) \]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00022498455180266905:\\ \;\;\;\;\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \tan \varepsilon \cdot \left(\tan x \cdot \left(\tan x \cdot \tan \varepsilon\right)\right)}, \mathsf{fma}\left(\tan x, \tan \varepsilon, 1\right), -\tan x\right)\\ \mathbf{elif}\;\varepsilon \leq 2.941971583774042 \cdot 10^{-26}:\\ \;\;\;\;\left(\varepsilon + \mathsf{fma}\left(\frac{{\varepsilon}^{3}}{{\cos x}^{4}}, {\sin x}^{4}, \mathsf{fma}\left(1.6666666666666667, \frac{{\varepsilon}^{4}}{{\left(\frac{\cos x}{\sin x}\right)}^{3}}, \mathsf{fma}\left(\frac{{\varepsilon}^{4}}{{\cos x}^{5}}, {\sin x}^{5}, \mathsf{fma}\left(1.3333333333333333, \frac{{\varepsilon}^{3} \cdot {\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(0.3333333333333333, {\varepsilon}^{3}, 0.6666666666666666 \cdot \left({\varepsilon}^{4} \cdot \frac{\sin x}{\cos x}\right)\right)\right)\right)\right)\right)\right)\right) + \frac{\varepsilon \cdot \varepsilon}{\cos x} \cdot \left(\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{\tan x - \tan \varepsilon}{{\tan x}^{2} - {\tan \varepsilon}^{2}}}, \frac{1}{1 - \tan x \cdot \tan \varepsilon}, -\tan x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022130 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64

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

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