Average Error: 36.8 → 0.4
Time: 7.5s
Precision: 64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\frac{\mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{\cos x}, \frac{\sin \varepsilon}{\cos \varepsilon}, \frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon}\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\frac{\mathsf{fma}\left(\frac{{\left(\sin x\right)}^{2}}{\cos x}, \frac{\sin \varepsilon}{\cos \varepsilon}, \frac{\sin \varepsilon \cdot \cos x}{\cos \varepsilon}\right)}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}
double code(double x, double eps) {
	return (tan((x + eps)) - tan(x));
}
↓
double code(double x, double eps) {
	return (fma((pow(sin(x), 2.0) / cos(x)), (sin(eps) / cos(eps)), ((sin(eps) * cos(x)) / cos(eps))) / ((1.0 - (tan(x) * tan(eps))) * cos(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

Original36.8
Target14.9
Herbie0.4
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 36.8

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

  3. Applied tan-quot36.8

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

  4. Applied tan-sum22.0

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

  5. Applied frac-sub22.1

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

  6. Taylor expanded around inf 0.4

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

  7. Simplified0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2020079 +o rules:numerics
(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)))