Average Error: 36.7 → 0.5
Time: 9.7s
Precision: binary64
\[\tan \left(x + \varepsilon\right) - \tan x \]
↓
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \varepsilon} \cdot \frac{\cos x + \frac{{\sin x}^{2}}{\cos x}}{1 - \sqrt[3]{{\left(\tan x \cdot \tan \varepsilon\right)}^{3}}} \]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\sin \varepsilon}{\cos x \cdot \cos \varepsilon} \cdot \frac{\cos x + \frac{{\sin x}^{2}}{\cos x}}{1 - \sqrt[3]{{\left(\tan x \cdot \tan \varepsilon\right)}^{3}}}

(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))
↓
(FPCore (x eps)
 :precision binary64
 (*
  (/ (sin eps) (* (cos x) (cos eps)))
  (/
   (+ (cos x) (/ (pow (sin x) 2.0) (cos x)))
   (- 1.0 (cbrt (pow (* (tan x) (tan eps)) 3.0))))))
double code(double x, double eps) {
	return tan(x + eps) - tan(x);
}
↓
double code(double x, double eps) {
	return (sin(eps) / (cos(x) * cos(eps))) * ((cos(x) + (pow(sin(x), 2.0) / cos(x))) / (1.0 - cbrt(pow((tan(x) * tan(eps)), 3.0))));
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.7
Target14.9
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-quot_binary6436.7

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

  4. Applied tan-sum_binary6421.9

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

  5. Applied frac-sub_binary6421.9

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

  6. Simplified21.9

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

  7. Simplified21.9

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

  8. Taylor expanded around inf 0.4

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

  9. Simplified0.4

    \[\leadsto \frac{\color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon} \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)}}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)} \]
  10. Using strategy rm

  11. Applied times-frac_binary640.4

    \[\leadsto \color{blue}{\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{\cos x} \cdot \frac{\cos x + \frac{{\sin x}^{2}}{\cos x}}{1 - \tan x \cdot \tan \varepsilon}} \]

  12. Simplified0.4

    \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos x \cdot \cos \varepsilon}} \cdot \frac{\cos x + \frac{{\sin x}^{2}}{\cos x}}{1 - \tan x \cdot \tan \varepsilon} \]
  13. Using strategy rm

  14. Applied add-cbrt-cube_binary640.5

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

  15. Simplified0.5

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

  16. Final simplification0.5

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

Reproduce

herbie shell --seed 2021210 
(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)))