Average Error: 37.1 → 13.8
Time: 6.7s
Precision: binary64
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\sqrt[3]{\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}\right)} + \frac{\sin x}{\cos x} \cdot \left(\frac{1}{1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}} + -1\right)} \cdot \left(\sqrt[3]{\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}\right)} + \frac{\sin x}{\cos x} \cdot \left(\frac{1}{1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}} + -1\right)} \cdot \sqrt[3]{\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}\right)} + \frac{\sin x}{\cos x} \cdot \left(\frac{1}{1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}} + -1\right)}\right)\]
\tan \left(x + \varepsilon\right) - \tan x
↓
\sqrt[3]{\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}\right)} + \frac{\sin x}{\cos x} \cdot \left(\frac{1}{1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}} + -1\right)} \cdot \left(\sqrt[3]{\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}\right)} + \frac{\sin x}{\cos x} \cdot \left(\frac{1}{1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}} + -1\right)} \cdot \sqrt[3]{\frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}\right)} + \frac{\sin x}{\cos x} \cdot \left(\frac{1}{1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}} + -1\right)}\right)
(FPCore (x eps) :precision binary64 (- (tan (+ x eps)) (tan x)))
↓
(FPCore (x eps)
 :precision binary64
 (*
  (cbrt
   (+
    (/
     (sin eps)
     (* (cos eps) (- 1.0 (/ (* (sin eps) (sin x)) (* (cos eps) (cos x))))))
    (*
     (/ (sin x) (cos x))
     (+
      (/ 1.0 (- 1.0 (/ (* (sin eps) (sin x)) (* (cos eps) (cos x)))))
      -1.0))))
  (*
   (cbrt
    (+
     (/
      (sin eps)
      (* (cos eps) (- 1.0 (/ (* (sin eps) (sin x)) (* (cos eps) (cos x))))))
     (*
      (/ (sin x) (cos x))
      (+
       (/ 1.0 (- 1.0 (/ (* (sin eps) (sin x)) (* (cos eps) (cos x)))))
       -1.0))))
   (cbrt
    (+
     (/
      (sin eps)
      (* (cos eps) (- 1.0 (/ (* (sin eps) (sin x)) (* (cos eps) (cos x))))))
     (*
      (/ (sin x) (cos x))
      (+
       (/ 1.0 (- 1.0 (/ (* (sin eps) (sin x)) (* (cos eps) (cos x)))))
       -1.0)))))))
double code(double x, double eps) {
	return ((double) (((double) tan(((double) (x + eps)))) - ((double) tan(x))));
}
↓
double code(double x, double eps) {
	return ((double) (((double) cbrt(((double) ((((double) sin(eps)) / ((double) (((double) cos(eps)) * ((double) (1.0 - (((double) (((double) sin(eps)) * ((double) sin(x)))) / ((double) (((double) cos(eps)) * ((double) cos(x)))))))))) + ((double) ((((double) sin(x)) / ((double) cos(x))) * ((double) ((1.0 / ((double) (1.0 - (((double) (((double) sin(eps)) * ((double) sin(x)))) / ((double) (((double) cos(eps)) * ((double) cos(x)))))))) + -1.0)))))))) * ((double) (((double) cbrt(((double) ((((double) sin(eps)) / ((double) (((double) cos(eps)) * ((double) (1.0 - (((double) (((double) sin(eps)) * ((double) sin(x)))) / ((double) (((double) cos(eps)) * ((double) cos(x)))))))))) + ((double) ((((double) sin(x)) / ((double) cos(x))) * ((double) ((1.0 / ((double) (1.0 - (((double) (((double) sin(eps)) * ((double) sin(x)))) / ((double) (((double) cos(eps)) * ((double) cos(x)))))))) + -1.0)))))))) * ((double) cbrt(((double) ((((double) sin(eps)) / ((double) (((double) cos(eps)) * ((double) (1.0 - (((double) (((double) sin(eps)) * ((double) sin(x)))) / ((double) (((double) cos(eps)) * ((double) cos(x)))))))))) + ((double) ((((double) sin(x)) / ((double) cos(x))) * ((double) ((1.0 / ((double) (1.0 - (((double) (((double) sin(eps)) * ((double) sin(x)))) / ((double) (((double) cos(eps)) * ((double) cos(x)))))))) + -1.0))))))))))));
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.1
Target15.5
Herbie13.8
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Initial program 37.1

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

  3. Applied tan-sum_binary6421.5

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

  4. Taylor expanded around inf 21.6

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

  5. Simplified21.6

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

  7. Applied add-cube-cbrt_binary6422.2

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

  8. Simplified22.2

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

  9. Simplified13.8

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

  10. Final simplification13.8

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

Reproduce

herbie shell --seed 2020204 
(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)))