Average Error: 13.5 → 0.2
Time: 11.8s
Precision: 64
\[\left(x = 0.0 \lor 0.588414199999999998 \le x \le 505.590899999999976\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le y \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le y \le 1.7512240000000001 \cdot 10^{308}\right) \land \left(-1.7767070000000002 \cdot 10^{308} \le z \le -8.59979600000002 \cdot 10^{-310} \lor 3.29314499999998 \cdot 10^{-311} \le z \le 1.72515400000000009 \cdot 10^{308}\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le a \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le a \le 1.7512240000000001 \cdot 10^{308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[x + \frac{\left(\tan y + \tan z\right) \cdot \cos a - \left(1 - \tan y \cdot \tan z\right) \cdot \sin a}{\left(1 - \sqrt[3]{{\left(\tan y \cdot \tan z\right)}^{3}}\right) \cdot \cos a}\]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
x + \frac{\left(\tan y + \tan z\right) \cdot \cos a - \left(1 - \tan y \cdot \tan z\right) \cdot \sin a}{\left(1 - \sqrt[3]{{\left(\tan y \cdot \tan z\right)}^{3}}\right) \cdot \cos a}
double f(double x, double y, double z, double a) {
        double r168937 = x;
        double r168938 = y;
        double r168939 = z;
        double r168940 = r168938 + r168939;
        double r168941 = tan(r168940);
        double r168942 = a;
        double r168943 = tan(r168942);
        double r168944 = r168941 - r168943;
        double r168945 = r168937 + r168944;
        return r168945;
}
↓
double f(double x, double y, double z, double a) {
        double r168946 = x;
        double r168947 = y;
        double r168948 = tan(r168947);
        double r168949 = z;
        double r168950 = tan(r168949);
        double r168951 = r168948 + r168950;
        double r168952 = a;
        double r168953 = cos(r168952);
        double r168954 = r168951 * r168953;
        double r168955 = 1.0;
        double r168956 = r168948 * r168950;
        double r168957 = r168955 - r168956;
        double r168958 = sin(r168952);
        double r168959 = r168957 * r168958;
        double r168960 = r168954 - r168959;
        double r168961 = 3.0;
        double r168962 = pow(r168956, r168961);
        double r168963 = cbrt(r168962);
        double r168964 = r168955 - r168963;
        double r168965 = r168964 * r168953;
        double r168966 = r168960 / r168965;
        double r168967 = r168946 + r168966;
        return r168967;
}

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 13.5

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

  3. Applied tan-quot13.5

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

  4. Applied tan-sum0.2

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

  5. Applied frac-sub0.2

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

  7. Applied add-cbrt-cube0.2

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

  8. Applied add-cbrt-cube0.2

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

  9. Applied cbrt-unprod0.2

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

  10. Simplified0.2

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2020025 +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))))