Average Error: 36.7 → 0.5
Time: 12.1s
Precision: binary64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{\cos x \cdot \left(1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)} \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)\right) \cdot \left(1 + \tan \varepsilon \cdot \tan x\right)\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\left(\frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{\cos x \cdot \left(1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)} \cdot \left(\cos x + \frac{{\sin x}^{2}}{\cos x}\right)\right) \cdot \left(1 + \tan \varepsilon \cdot \tan x\right)
(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))
↓
(FPCore (x eps)
 :precision binary64
 (*
  (*
   (/
    (/ (sin eps) (cos eps))
    (* (cos x) (- 1.0 (* (* (tan eps) (tan x)) (* (tan eps) (tan x))))))
   (+ (cos x) (/ (pow (sin x) 2.0) (cos x))))
  (+ 1.0 (* (tan eps) (tan x)))))
double code(double x, double eps) {
	return tan(x + eps) - tan(x);
}
↓
double code(double x, double eps) {
	return (((sin(eps) / cos(eps)) / (cos(x) * (1.0 - ((tan(eps) * tan(x)) * (tan(eps) * tan(x)))))) * (cos(x) + (pow(sin(x), 2.0) / cos(x)))) * (1.0 + (tan(eps) * tan(x)));
}

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
Target14.8
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_binary64_44936.7

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

  4. Applied tan-sum_binary64_42521.9

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

  5. Applied frac-sub_binary64_56521.9

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

  6. Taylor expanded around inf 0.4

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

  7. Simplified0.4

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

  9. Applied unpow2_binary64_5170.4

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

  10. Applied associate-/l*_binary64_6380.4

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

  12. Applied flip--_binary64_6000.4

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

  13. Applied associate-*l/_binary64_6360.5

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

  14. Applied associate-/r/_binary64_6390.4

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

  15. Simplified0.5

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

  16. Final simplification0.5

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

Reproduce

herbie shell --seed 2020231 
(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)))