(+ x (- (tan (+ y z)) (tan a)))

Average Error: 13.3 → 0.2

Time: 9.2s

Precision: 64

\[\left(x = 0.0 \lor 0.588414199999999998 \le x \le 505.590899999999976\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le y \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le y \le 1.7512240000000001 \cdot 10^{308}\right) \land \left(-1.7767070000000002 \cdot 10^{308} \le z \le -8.59979600000002 \cdot 10^{-310} \lor 3.29314499999998 \cdot 10^{-311} \le z \le 1.72515400000000009 \cdot 10^{308}\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le a \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le a \le 1.7512240000000001 \cdot 10^{308}\right)\]

Math

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

↓

\[x + \left(\frac{\mathsf{fma}\left(\frac{\sin z \cdot \tan y}{\cos z}, \frac{\sin z \cdot \tan y}{\cos z} + 1, 1\right) \cdot \left(\tan y + \tan z\right)}{1 - {\left(\frac{\sin z \cdot \tan y}{\cos z}\right)}^{3}} + \left(-\tan a\right)\right)\]

C

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 + (((fma(((sin(z) * tan(y)) / cos(z)), (((sin(z) * tan(y)) / cos(z)) + 1.0), 1.0) * (tan(y) + tan(z))) / (1.0 - pow(((sin(z) * tan(y)) / cos(z)), 3.0))) + -tan(a)));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

1. Initial program 13.3

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

3. Applied tan-sum 0.2

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

4. Simplified 0.2

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

6. Applied tan-quot 0.2

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

7. Applied associate-*l/ 0.2

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

9. Applied flip3-- 0.2

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

10. Applied associate-/r/ 0.2

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

11. Applied fma-neg 0.2

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

13. Applied fma-udef 0.2

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

14. Simplified 0.2

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

15. Final simplification 0.2

    \[\leadsto x + \left(\frac{\mathsf{fma}\left(\frac{\sin z \cdot \tan y}{\cos z}, \frac{\sin z \cdot \tan y}{\cos z} + 1, 1\right) \cdot \left(\tan y + \tan z\right)}{1 - {\left(\frac{\sin z \cdot \tan y}{\cos z}\right)}^{3}} + \left(-\tan a\right)\right)\]

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(FPCore (x y z a)
  :name "(+ x (- (tan (+ y z)) (tan a)))"
  :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))))