Average Error: 13.1 → 0.2
Time: 34.5s
Precision: 64
\[\left(x = 0 \lor 0.5884142 \le x \le 505.5909\right) \land \left(-1.796658 \cdot 10^{+308} \le y \le -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le y \le 1.751224 \cdot 10^{+308}\right) \land \left(-1.776707 \cdot 10^{+308} \le z \le -8.599796 \cdot 10^{-310} \lor 3.293145 \cdot 10^{-311} \le z \le 1.725154 \cdot 10^{+308}\right) \land \left(-1.796658 \cdot 10^{+308} \le a \le -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le a \le 1.751224 \cdot 10^{+308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[\mathsf{fma}\left(\frac{\tan y + \tan z}{1 - \left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)}, \tan z \cdot \tan y + 1, -\tan a\right) + x\]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
\mathsf{fma}\left(\frac{\tan y + \tan z}{1 - \left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)}, \tan z \cdot \tan y + 1, -\tan a\right) + x
double f(double x, double y, double z, double a) {
        double r2908143 = x;
        double r2908144 = y;
        double r2908145 = z;
        double r2908146 = r2908144 + r2908145;
        double r2908147 = tan(r2908146);
        double r2908148 = a;
        double r2908149 = tan(r2908148);
        double r2908150 = r2908147 - r2908149;
        double r2908151 = r2908143 + r2908150;
        return r2908151;
}
↓
double f(double x, double y, double z, double a) {
        double r2908152 = y;
        double r2908153 = tan(r2908152);
        double r2908154 = z;
        double r2908155 = tan(r2908154);
        double r2908156 = r2908153 + r2908155;
        double r2908157 = 1.0;
        double r2908158 = r2908155 * r2908153;
        double r2908159 = r2908158 * r2908158;
        double r2908160 = r2908157 - r2908159;
        double r2908161 = r2908156 / r2908160;
        double r2908162 = r2908158 + r2908157;
        double r2908163 = a;
        double r2908164 = tan(r2908163);
        double r2908165 = -r2908164;
        double r2908166 = fma(r2908161, r2908162, r2908165);
        double r2908167 = x;
        double r2908168 = r2908166 + r2908167;
        return r2908168;
}

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

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019153 +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))))