Average Error: 13.4 → 1.1
Time: 1.0min
Precision: binary64
\[\left(x = 0 \lor 0.5884142 \leq x \land x \leq 505.5909\right) \land \left(-1.796658 \cdot 10^{+308} \leq y \land y \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq y \land y \leq 1.7512240000000001 \cdot 10^{+308}\right) \land \left(-1.7767070000000002 \cdot 10^{+308} \leq z \land z \leq -8.599796 \cdot 10^{-310} \lor 3.293145 \cdot 10^{-311} \leq z \land z \leq 1.725154 \cdot 10^{+308}\right) \land \left(-1.796658 \cdot 10^{+308} \leq a \land a \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq a \land a \leq 1.7512240000000001 \cdot 10^{+308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[x + \left(\frac{{\left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\right)}^{3}}{{\tan a}^{2} + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} \cdot \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} + \tan a\right)} - \frac{{\tan a}^{3}}{{\tan a}^{2} + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} \cdot \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} + \tan a\right)}\right)\]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
x + \left(\frac{{\left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\right)}^{3}}{{\tan a}^{2} + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} \cdot \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} + \tan a\right)} - \frac{{\tan a}^{3}}{{\tan a}^{2} + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} \cdot \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
  (-
   (/
    (pow (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))) 3.0)
    (+
     (pow (tan a) 2.0)
     (*
      (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z))))
      (+ (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))) (tan a)))))
   (/
    (pow (tan a) 3.0)
    (+
     (pow (tan a) 2.0)
     (*
      (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z))))
      (+ (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))) (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 x + ((pow(((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))), 3.0) / (pow(tan(a), 2.0) + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) * (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) + tan(a))))) - (pow(tan(a), 3.0) / (pow(tan(a), 2.0) + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) * (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) + 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 13.4

    \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
  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 flip3--_binary641.1

    \[\leadsto x + \color{blue}{\frac{{\left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\right)}^{3} - {\tan a}^{3}}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} \cdot \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} + \left(\tan a \cdot \tan a + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} \cdot \tan a\right)}}\]

  6. Simplified1.1

    \[\leadsto x + \frac{{\left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\right)}^{3} - {\tan a}^{3}}{\color{blue}{\tan a \cdot \tan a + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} \cdot \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} + \tan a\right)}}\]
  7. Using strategy rm

  8. Applied div-sub_binary641.1

    \[\leadsto x + \color{blue}{\left(\frac{{\left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\right)}^{3}}{\tan a \cdot \tan a + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} \cdot \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} + \tan a\right)} - \frac{{\tan a}^{3}}{\tan a \cdot \tan a + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} \cdot \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} + \tan a\right)}\right)}\]

  9. Simplified1.1

    \[\leadsto x + \left(\color{blue}{\frac{{\left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\right)}^{3}}{{\tan a}^{2} + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} \cdot \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} + \tan a\right)}} - \frac{{\tan a}^{3}}{\tan a \cdot \tan a + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} \cdot \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} + \tan a\right)}\right)\]

  10. Simplified1.1

    \[\leadsto x + \left(\frac{{\left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\right)}^{3}}{{\tan a}^{2} + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} \cdot \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} + \tan a\right)} - \color{blue}{\frac{{\tan a}^{3}}{{\tan a}^{2} + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} \cdot \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} + \tan a\right)}}\right)\]

  11. Final simplification1.1

    \[\leadsto x + \left(\frac{{\left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\right)}^{3}}{{\tan a}^{2} + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} \cdot \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} + \tan a\right)} - \frac{{\tan a}^{3}}{{\tan a}^{2} + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} \cdot \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} + \tan a\right)}\right)\]

Reproduce

herbie shell --seed 2020274 
(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))))