Average Error: 13.0 → 0.3
Time: 41.2s
Precision: binary64
\[\left(x = 0 \lor 0.5884142 \leq x \land x \leq 505.5909\right) \land \left(-1.796658 \cdot 10^{+308} \leq y \land y \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq y \land y \leq 1.7512240000000001 \cdot 10^{+308}\right) \land \left(-1.7767070000000002 \cdot 10^{+308} \leq z \land z \leq -8.599796 \cdot 10^{-310} \lor 3.293145 \cdot 10^{-311} \leq z \land z \leq 1.725154 \cdot 10^{+308}\right) \land \left(-1.796658 \cdot 10^{+308} \leq a \land a \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq a \land a \leq 1.7512240000000001 \cdot 10^{+308}\right)\]
\[[y, z]=\mathsf{sort}([y, z])\]
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[\log \left(e^{x} \cdot \frac{e^{\frac{\tan y + \tan z}{1 - \frac{\tan z \cdot \sin y}{\cos y}}}}{e^{\tan a}}\right)\]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
\log \left(e^{x} \cdot \frac{e^{\frac{\tan y + \tan z}{1 - \frac{\tan z \cdot \sin y}{\cos y}}}}{e^{\tan a}}\right)
(FPCore (x y z a) :precision binary64 (+ x (- (tan (+ y z)) (tan a))))
↓
(FPCore (x y z a)
 :precision binary64
 (log
  (*
   (exp x)
   (/
    (exp (/ (+ (tan y) (tan z)) (- 1.0 (/ (* (tan z) (sin y)) (cos y)))))
    (exp (tan a))))))
double code(double x, double y, double z, double a) {
	return x + (tan(y + z) - tan(a));
}
↓
double code(double x, double y, double z, double a) {
	return log(exp(x) * (exp((tan(y) + tan(z)) / (1.0 - ((tan(z) * sin(y)) / cos(y)))) / exp(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 13.0

    \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
  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 tan-quot_binary640.2

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

  6. Applied associate-*l/_binary640.2

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

  8. Applied add-log-exp_binary640.2

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

  9. Applied add-log-exp_binary640.3

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

  10. Applied diff-log_binary640.3

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

  11. Applied add-log-exp_binary640.3

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

  12. Applied sum-log_binary640.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2021045 
(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))))