Average Error: 13.3 → 0.3
Time: 29.9s
Precision: binary64
\[\left(x = 0 \lor 0.5884142 \leq x \land x \leq 505.5909\right) \land \left(-1.796658 \cdot 10^{+308} \leq y \land y \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq y \land y \leq 1.7512240000000001 \cdot 10^{+308}\right) \land \left(-1.7767070000000002 \cdot 10^{+308} \leq z \land z \leq -8.599796 \cdot 10^{-310} \lor 3.293145 \cdot 10^{-311} \leq z \land z \leq 1.725154 \cdot 10^{+308}\right) \land \left(-1.796658 \cdot 10^{+308} \leq a \land a \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq a \land a \leq 1.7512240000000001 \cdot 10^{+308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[\frac{x \cdot x - \frac{\left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right) \cdot \left(\left(\tan y + \tan z\right) \cdot \cos a - \left(1 - \tan y \cdot \tan z\right) \cdot \sin a\right)}{\left(1 - \tan y \cdot \tan z\right) \cdot \cos a}}{x \cdot x - \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right) \cdot \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right)} \cdot \left(x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right)\right)\]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
\frac{x \cdot x - \frac{\left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right) \cdot \left(\left(\tan y + \tan z\right) \cdot \cos a - \left(1 - \tan y \cdot \tan z\right) \cdot \sin a\right)}{\left(1 - \tan y \cdot \tan z\right) \cdot \cos a}}{x \cdot x - \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right) \cdot \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right)} \cdot \left(x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right)\right)
(FPCore (x y z a) :precision binary64 (+ x (- (tan (+ y z)) (tan a))))
↓
(FPCore (x y z a)
 :precision binary64
 (*
  (/
   (-
    (* x x)
    (/
     (*
      (- (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))) (tan a))
      (-
       (* (+ (tan y) (tan z)) (cos a))
       (* (- 1.0 (* (tan y) (tan z))) (sin a))))
     (* (- 1.0 (* (tan y) (tan z))) (cos a))))
   (-
    (* x x)
    (*
     (- (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))) (tan a))
     (- (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))) (tan a)))))
  (+ x (- (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))) (tan a)))))
double code(double x, double y, double z, double a) {
	return ((double) (x + ((double) (((double) tan(((double) (y + z)))) - ((double) tan(a))))));
}
↓
double code(double x, double y, double z, double a) {
	return ((double) ((((double) (((double) (x * x)) - (((double) (((double) ((((double) (((double) tan(y)) + ((double) tan(z)))) / ((double) (1.0 - ((double) (((double) tan(y)) * ((double) tan(z))))))) - ((double) tan(a)))) * ((double) (((double) (((double) (((double) tan(y)) + ((double) tan(z)))) * ((double) cos(a)))) - ((double) (((double) (1.0 - ((double) (((double) tan(y)) * ((double) tan(z)))))) * ((double) sin(a)))))))) / ((double) (((double) (1.0 - ((double) (((double) tan(y)) * ((double) tan(z)))))) * ((double) cos(a))))))) / ((double) (((double) (x * x)) - ((double) (((double) ((((double) (((double) tan(y)) + ((double) tan(z)))) / ((double) (1.0 - ((double) (((double) tan(y)) * ((double) tan(z))))))) - ((double) tan(a)))) * ((double) ((((double) (((double) tan(y)) + ((double) tan(z)))) / ((double) (1.0 - ((double) (((double) tan(y)) * ((double) tan(z))))))) - ((double) tan(a))))))))) * ((double) (x + ((double) ((((double) (((double) tan(y)) + ((double) tan(z)))) / ((double) (1.0 - ((double) (((double) tan(y)) * ((double) tan(z))))))) - ((double) tan(a))))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.3

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

  3. Applied tan-sum_binary640.2

    \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} - \tan a\right)\]
  4. Using strategy rm

  5. Applied flip-+_binary640.5

    \[\leadsto \color{blue}{\frac{x \cdot x - \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right) \cdot \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right)}{x - \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right)}}\]
  6. Using strategy rm

  7. Applied tan-quot_binary640.5

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

  8. Applied frac-sub_binary640.5

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

  9. Applied associate-*r/_binary640.5

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

  11. Applied flip--_binary640.3

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

  12. Applied associate-/r/_binary640.3

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

  13. Final simplification0.3

    \[\leadsto \frac{x \cdot x - \frac{\left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right) \cdot \left(\left(\tan y + \tan z\right) \cdot \cos a - \left(1 - \tan y \cdot \tan z\right) \cdot \sin a\right)}{\left(1 - \tan y \cdot \tan z\right) \cdot \cos a}}{x \cdot x - \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right) \cdot \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right)} \cdot \left(x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right)\right)\]

Reproduce

herbie shell --seed 2020205 
(FPCore (x y z a)
  :name "tan-example"
  :precision binary64
  :pre (and (or (== x 0.0) (<= 0.5884142 x 505.5909)) (or (<= -1.796658e+308 y -9.425585e-310) (<= 1.284938e-309 y 1.7512240000000001e+308)) (or (<= -1.7767070000000002e+308 z -8.599796e-310) (<= 3.293145e-311 z 1.725154e+308)) (or (<= -1.796658e+308 a -9.425585e-310) (<= 1.284938e-309 a 1.7512240000000001e+308)))
  (+ x (- (tan (+ y z)) (tan a))))