Average Error: 37.2 → 0.6
Time: 12.5s
Precision: binary64
\[\tan \left(x + \varepsilon\right) - \tan x \]
↓
\[\begin{array}{l} t_0 := \tan x + \tan \varepsilon\\ \mathbf{if}\;\varepsilon \leq -2.9674248494807262 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.6741431082075642 \cdot 10^{-20}:\\ \;\;\;\;\begin{array}{l} t_1 := {\sin x}^{2}\\ t_2 := {\cos x}^{2}\\ \left(\frac{{\varepsilon}^{3} \cdot {\sin x}^{4}}{{\cos x}^{4}} + \left(\mathsf{fma}\left(\varepsilon, \frac{t_1}{t_2}, \varepsilon\right) + \mathsf{fma}\left(1.3333333333333333, \frac{{\varepsilon}^{3} \cdot t_1}{t_2}, {\varepsilon}^{3} \cdot 0.3333333333333333\right)\right)\right) + \frac{\varepsilon \cdot \varepsilon}{\cos x} \cdot \left(\sin x + \frac{{\sin x}^{3}}{t_2}\right) \end{array}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 - \frac{\tan \varepsilon \cdot \sin x}{\cos x}} - \tan x\\ \end{array} \]
\tan \left(x + \varepsilon\right) - \tan x
↓
\begin{array}{l}
t_0 := \tan x + \tan \varepsilon\\
\mathbf{if}\;\varepsilon \leq -2.9674248494807262 \cdot 10^{-5}:\\
\;\;\;\;\frac{t_0}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\

\mathbf{elif}\;\varepsilon \leq 2.6741431082075642 \cdot 10^{-20}:\\
\;\;\;\;\begin{array}{l}
t_1 := {\sin x}^{2}\\
t_2 := {\cos x}^{2}\\
\left(\frac{{\varepsilon}^{3} \cdot {\sin x}^{4}}{{\cos x}^{4}} + \left(\mathsf{fma}\left(\varepsilon, \frac{t_1}{t_2}, \varepsilon\right) + \mathsf{fma}\left(1.3333333333333333, \frac{{\varepsilon}^{3} \cdot t_1}{t_2}, {\varepsilon}^{3} \cdot 0.3333333333333333\right)\right)\right) + \frac{\varepsilon \cdot \varepsilon}{\cos x} \cdot \left(\sin x + \frac{{\sin x}^{3}}{t_2}\right)
\end{array}\\

\mathbf{else}:\\
\;\;\;\;\frac{t_0}{1 - \frac{\tan \varepsilon \cdot \sin x}{\cos x}} - \tan x\\


\end{array}

(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))
↓
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (tan x) (tan eps))))
   (if (<= eps -2.9674248494807262e-5)
     (- (/ t_0 (- 1.0 (/ (* (tan x) (sin eps)) (cos eps)))) (tan x))
     (if (<= eps 2.6741431082075642e-20)
       (let* ((t_1 (pow (sin x) 2.0)) (t_2 (pow (cos x) 2.0)))
         (+
          (+
           (/ (* (pow eps 3.0) (pow (sin x) 4.0)) (pow (cos x) 4.0))
           (+
            (fma eps (/ t_1 t_2) eps)
            (fma
             1.3333333333333333
             (/ (* (pow eps 3.0) t_1) t_2)
             (* (pow eps 3.0) 0.3333333333333333))))
          (* (/ (* eps eps) (cos x)) (+ (sin x) (/ (pow (sin x) 3.0) t_2)))))
       (- (/ t_0 (- 1.0 (/ (* (tan eps) (sin x)) (cos x)))) (tan x))))))
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 tmp;
	if (eps <= -2.9674248494807262e-5) {
		tmp = (t_0 / (1.0 - ((tan(x) * sin(eps)) / cos(eps)))) - tan(x);
	} else if (eps <= 2.6741431082075642e-20) {
		double t_1 = pow(sin(x), 2.0);
		double t_2 = pow(cos(x), 2.0);
		tmp = (((pow(eps, 3.0) * pow(sin(x), 4.0)) / pow(cos(x), 4.0)) + (fma(eps, (t_1 / t_2), eps) + fma(1.3333333333333333, ((pow(eps, 3.0) * t_1) / t_2), (pow(eps, 3.0) * 0.3333333333333333)))) + (((eps * eps) / cos(x)) * (sin(x) + (pow(sin(x), 3.0) / t_2)));
	} else {
		tmp = (t_0 / (1.0 - ((tan(eps) * sin(x)) / cos(x)))) - tan(x);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus eps

Target

Original37.2
Target15.3
Herbie0.6
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]

Derivation

  1. Split input into 3 regimes

  2. if eps < -2.96742484948072617e-5

    1. Initial program 30.3

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

    2. Applied tan-sum_binary640.4

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

    3. Applied tan-quot_binary640.4

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

    4. Applied associate-*r/_binary640.4

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

    5. Simplified0.4

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

    if -2.96742484948072617e-5 < eps < 2.67414310820756422e-20

    1. Initial program 44.5

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

    2. Applied tan-sum_binary6444.3

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

    3. Taylor expanded in eps around 0 0.2

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

    4. Simplified0.2

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

    if 2.67414310820756422e-20 < eps

    1. Initial program 30.4

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

    2. Applied tan-sum_binary641.4

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

    3. Applied *-commutative_binary641.4

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

    4. Applied tan-quot_binary641.4

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

    5. Applied associate-*r/_binary641.4

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\tan \varepsilon \cdot \sin x}{\cos x}}} - \tan x \]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.9674248494807262 \cdot 10^{-5}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \mathbf{elif}\;\varepsilon \leq 2.6741431082075642 \cdot 10^{-20}:\\ \;\;\;\;\left(\frac{{\varepsilon}^{3} \cdot {\sin x}^{4}}{{\cos x}^{4}} + \left(\mathsf{fma}\left(\varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \varepsilon\right) + \mathsf{fma}\left(1.3333333333333333, \frac{{\varepsilon}^{3} \cdot {\sin x}^{2}}{{\cos x}^{2}}, {\varepsilon}^{3} \cdot 0.3333333333333333\right)\right)\right) + \frac{\varepsilon \cdot \varepsilon}{\cos x} \cdot \left(\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan \varepsilon \cdot \sin x}{\cos x}} - \tan x\\ \end{array} \]

Reproduce

herbie shell --seed 2022082 
(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)))