Average Error: 36.8 → 16.0
Time: 7.6s
Precision: binary64
\[\]
↓
\[\]
↓
double code(double x, double eps) {
	return ((double) (((double) tan(((double) (x + eps)))) - ((double) tan(x))));
}

↓
double code(double x, double eps) {
	double VAR;
	if ((eps <= -4.683864053063239e-144)) {
		VAR = ((double) (((double) (((double) (1.0 - ((double) (((double) tan(x)) * ((double) (((double) tan(x)) * ((double) (((double) tan(eps)) * ((double) tan(eps)))))))))) * ((double) (((double) (((double) (((double) tan(x)) + ((double) tan(eps)))) * ((double) cos(x)))) + ((double) (((double) sin(x)) * ((double) (((double) (((double) tan(x)) * ((double) tan(eps)))) + -1.0)))))))) / ((double) (((double) (1.0 - ((double) (((double) tan(x)) * ((double) (((double) tan(x)) * ((double) (((double) tan(eps)) * ((double) tan(eps)))))))))) * ((double) (((double) cos(x)) * ((double) (1.0 - ((double) (((double) tan(x)) * ((double) tan(eps))))))))))));
	} else {
		double VAR_1;
		if ((eps <= 1.9461005157827348e-72)) {
			VAR_1 = ((double) (eps + ((double) (x * ((double) (eps * ((double) (eps + x))))))));
		} else {
			VAR_1 = ((double) (((double) (((double) (((double) pow(((double) tan(x)), 3.0)) + ((double) pow(((double) tan(eps)), 3.0)))) / ((double) (((double) (1.0 - ((double) (((double) tan(x)) * ((double) tan(eps)))))) * ((double) (((double) (((double) tan(x)) * ((double) tan(x)))) + ((double) (((double) tan(eps)) * ((double) (((double) tan(eps)) - ((double) tan(x)))))))))))) - ((double) tan(x))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

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
Target14.6
Herbie16.0
\[\]
Derivation
  1. Split input into 3 regimes
  2. if  eps  < -4.6838640530632392e-144
    1. Initial program 31.1
      \[\]
    2. Using strategy rm
    3. Applied tan-sum11.1
      \[\leadsto \]
    4. Using strategy rm
    5. Applied flip--11.1
      \[\leadsto \]
    6. Applied associate-/r/11.1
      \[\leadsto \]
    7. Simplified11.1
      \[\leadsto \]
    8. Using strategy rm
    9. Applied tan-quot11.2
      \[\leadsto \]
    10. Applied flip-+11.2
      \[\leadsto \]
    11. Applied frac-times11.2
      \[\leadsto \]
    12. Applied frac-sub11.2
      \[\leadsto \]
    13. Simplified11.2
      \[\leadsto \]
    14. Simplified11.2
      \[\leadsto \]
    if -4.6838640530632392e-144 <  eps  < 1.9461005157827348e-72
    1. Initial program 48.8
      \[\]
    2. Taylor expanded around 0 30.4
      \[\leadsto \]
    3. Simplified30.2
      \[\leadsto \]
    if 1.9461005157827348e-72 <  eps 
    1. Initial program 30.5
      \[\]
    2. Using strategy rm
    3. Applied tan-sum6.5
      \[\leadsto \]
    4. Using strategy rm
    5. Applied flip3-+6.7
      \[\leadsto \]
    6. Applied associate-/l/6.7
      \[\leadsto \]
    7. Simplified6.7
      \[\leadsto \]
  3. Recombined 3 regimes into one program.
  4. Final simplification16.0
    \[\leadsto \]
Reproduce
herbie shell --seed 2020190 
(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)))