Average Error: 13.3 → 0.3
Time: 11.4s
Precision: 64
\[\left(x = 0.0 \lor 0.588414199999999998 \le x \le 505.590899999999976\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le y \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le y \le 1.7512240000000001 \cdot 10^{308}\right) \land \left(-1.7767070000000002 \cdot 10^{308} \le z \le -8.59979600000002 \cdot 10^{-310} \lor 3.29314499999998 \cdot 10^{-311} \le z \le 1.72515400000000009 \cdot 10^{308}\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le a \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le a \le 1.7512240000000001 \cdot 10^{308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[x + \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\tan z, \tan y, 1\right), \tan z \cdot \tan y, 1\right), \frac{\tan y + \tan z}{1 - {\left(\log \left(e^{\tan z \cdot \tan y}\right)\right)}^{3}}, -\tan a\right) + \mathsf{fma}\left(-\tan a, 1, \tan a\right)\right)\]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
x + \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\tan z, \tan y, 1\right), \tan z \cdot \tan y, 1\right), \frac{\tan y + \tan z}{1 - {\left(\log \left(e^{\tan z \cdot \tan y}\right)\right)}^{3}}, -\tan a\right) + \mathsf{fma}\left(-\tan a, 1, \tan a\right)\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 (x + (fma(fma(fma(tan(z), tan(y), 1.0), (tan(z) * tan(y)), 1.0), ((tan(y) + tan(z)) / (1.0 - pow(log(exp((tan(z) * tan(y)))), 3.0))), -tan(a)) + fma(-tan(a), 1.0, 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-sum0.2

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

  4. Simplified0.2

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

  6. Applied add-log-exp0.2

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

  8. Applied add-cube-cbrt0.4

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

  9. Applied flip3--0.4

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

  10. Applied associate-/r/0.4

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

  11. Applied prod-diff0.4

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

  12. Simplified0.3

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

  13. Simplified0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2020066 +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))))