Average Error: 13.1 → 0.2
Time: 1.3m
Precision: 64
\[\left(x = 0 \lor 0.5884142 \le x \le 505.5909\right) \land \left(-1.796658 \cdot 10^{+308} \le y \le -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le y \le 1.751224 \cdot 10^{+308}\right) \land \left(-1.776707 \cdot 10^{+308} \le z \le -8.599796 \cdot 10^{-310} \lor 3.293145 \cdot 10^{-311} \le z \le 1.725154 \cdot 10^{+308}\right) \land \left(-1.796658 \cdot 10^{+308} \le a \le -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le a \le 1.751224 \cdot 10^{+308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[\mathsf{fma}\left(\left(\frac{\tan y + \tan z}{1 - \tan y \cdot \left(\tan z \cdot \left(\tan z \cdot \tan y\right)\right)}\right), \left(\tan z \cdot \tan y + 1\right), \left(-\tan a\right)\right) + x\]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
\mathsf{fma}\left(\left(\frac{\tan y + \tan z}{1 - \tan y \cdot \left(\tan z \cdot \left(\tan z \cdot \tan y\right)\right)}\right), \left(\tan z \cdot \tan y + 1\right), \left(-\tan a\right)\right) + x
double f(double x, double y, double z, double a) {
        double r32346998 = x;
        double r32346999 = y;
        double r32347000 = z;
        double r32347001 = r32346999 + r32347000;
        double r32347002 = tan(r32347001);
        double r32347003 = a;
        double r32347004 = tan(r32347003);
        double r32347005 = r32347002 - r32347004;
        double r32347006 = r32346998 + r32347005;
        return r32347006;
}
↓
double f(double x, double y, double z, double a) {
        double r32347007 = y;
        double r32347008 = tan(r32347007);
        double r32347009 = z;
        double r32347010 = tan(r32347009);
        double r32347011 = r32347008 + r32347010;
        double r32347012 = 1.0;
        double r32347013 = r32347010 * r32347008;
        double r32347014 = r32347010 * r32347013;
        double r32347015 = r32347008 * r32347014;
        double r32347016 = r32347012 - r32347015;
        double r32347017 = r32347011 / r32347016;
        double r32347018 = r32347013 + r32347012;
        double r32347019 = a;
        double r32347020 = tan(r32347019);
        double r32347021 = -r32347020;
        double r32347022 = fma(r32347017, r32347018, r32347021);
        double r32347023 = x;
        double r32347024 = r32347022 + r32347023;
        return r32347024;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

  1. Initial program 13.1

    \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
  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 flip--0.2

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

  6. Applied associate-/r/0.2

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

  7. Applied fma-neg0.2

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

  9. Applied associate-*l*0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x y z a)
  :name "(+ x (- (tan (+ y z)) (tan a)))"
  :pre (and (or (== x 0) (<= 0.5884142 x 505.5909)) (or (<= -1.796658e+308 y -9.425585e-310) (<= 1.284938e-309 y 1.751224e+308)) (or (<= -1.776707e+308 z -8.599796e-310) (<= 3.293145e-311 z 1.725154e+308)) (or (<= -1.796658e+308 a -9.425585e-310) (<= 1.284938e-309 a 1.751224e+308)))
  (+ x (- (tan (+ y z)) (tan a))))