Average Error: 13.9 → 0.2
Time: 12.6s
Precision: 64
\[\left(x = 0.0 \lor 0.588414199999999998 \le x \le 505.590899999999976\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le y \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le y \le 1.7512240000000001 \cdot 10^{308}\right) \land \left(-1.7767070000000002 \cdot 10^{308} \le z \le -8.59979600000002 \cdot 10^{-310} \lor 3.29314499999998 \cdot 10^{-311} \le z \le 1.72515400000000009 \cdot 10^{308}\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le a \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le a \le 1.7512240000000001 \cdot 10^{308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[\left(x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\tan a}, \sqrt[3]{\tan a} \cdot \sqrt[3]{\tan a}, \sqrt[3]{\tan a} \cdot \left(\sqrt[3]{\tan a} \cdot \sqrt[3]{\tan a}\right)\right)\]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
\left(x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \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)
double f(double x, double y, double z, double a) {
        double r166994 = x;
        double r166995 = y;
        double r166996 = z;
        double r166997 = r166995 + r166996;
        double r166998 = tan(r166997);
        double r166999 = a;
        double r167000 = tan(r166999);
        double r167001 = r166998 - r167000;
        double r167002 = r166994 + r167001;
        return r167002;
}
↓
double f(double x, double y, double z, double a) {
        double r167003 = x;
        double r167004 = y;
        double r167005 = tan(r167004);
        double r167006 = z;
        double r167007 = tan(r167006);
        double r167008 = r167005 + r167007;
        double r167009 = 1.0;
        double r167010 = r167005 * r167007;
        double r167011 = r167009 - r167010;
        double r167012 = r167008 / r167011;
        double r167013 = a;
        double r167014 = tan(r167013);
        double r167015 = r167012 - r167014;
        double r167016 = r167003 + r167015;
        double r167017 = cbrt(r167014);
        double r167018 = -r167017;
        double r167019 = r167017 * r167017;
        double r167020 = r167017 * r167019;
        double r167021 = fma(r167018, r167019, r167020);
        double r167022 = r167016 + r167021;
        return r167022;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

  1. Initial program 13.9

    \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
  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 add-cube-cbrt0.3

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

  6. Applied add-sqr-sqrt32.1

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

  7. Applied prod-diff32.1

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

  8. Applied associate-+r+32.1

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

  9. Simplified0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2020062 +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))))