Average Error: 13.2 → 0.3
Time: 18.4s
Precision: 64
\[\left(x = 0.0 \lor 0.588414199999999998 \le x \le 505.590899999999976\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le y \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le y \le 1.7512240000000001 \cdot 10^{308}\right) \land \left(-1.7767070000000002 \cdot 10^{308} \le z \le -8.59979600000002 \cdot 10^{-310} \lor 3.29314499999998 \cdot 10^{-311} \le z \le 1.72515400000000009 \cdot 10^{308}\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le a \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le a \le 1.7512240000000001 \cdot 10^{308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[\mathsf{fma}\left(\frac{\tan y + \tan z}{1 - \sqrt[3]{{\left({\left(\tan y \cdot \tan z\right)}^{3}\right)}^{3}}}, \mathsf{fma}\left(\mathsf{fma}\left(\tan y, \tan z, 1\right), \tan y \cdot \tan z, 1\right), \left(-\tan a\right) + x\right) + \mathsf{fma}\left(-\tan a, 1, \tan a \cdot 1\right)\]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
\mathsf{fma}\left(\frac{\tan y + \tan z}{1 - \sqrt[3]{{\left({\left(\tan y \cdot \tan z\right)}^{3}\right)}^{3}}}, \mathsf{fma}\left(\mathsf{fma}\left(\tan y, \tan z, 1\right), \tan y \cdot \tan z, 1\right), \left(-\tan a\right) + x\right) + \mathsf{fma}\left(-\tan a, 1, \tan a \cdot 1\right)
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 (fma(((tan(y) + tan(z)) / (1.0 - cbrt(pow(pow((tan(y) * tan(z)), 3.0), 3.0)))), fma(fma(tan(y), tan(z), 1.0), (tan(y) * tan(z)), 1.0), (-tan(a) + x)) + fma(-tan(a), 1.0, (tan(a) * 1.0)));
}

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 13.2

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

  3. Applied tan-sum0.2

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

  5. Applied *-un-lft-identity0.2

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

  6. Applied flip3--0.2

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

  7. Applied associate-/r/0.2

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

  8. Applied prod-diff0.2

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

  9. Applied associate-+r+0.2

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

  10. Simplified0.3

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

  12. Applied add-cbrt-cube0.3

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

  13. Simplified0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2020057 +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))))