Average Error: 13.4 → 0.2
Time: 17.2s
Precision: 64
\[\left(x = 0.0 \lor 0.588414199999999998 \le x \le 505.590899999999976\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le y \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le y \le 1.7512240000000001 \cdot 10^{308}\right) \land \left(-1.7767070000000002 \cdot 10^{308} \le z \le -8.59979600000002 \cdot 10^{-310} \lor 3.29314499999998 \cdot 10^{-311} \le z \le 1.72515400000000009 \cdot 10^{308}\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le a \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le a \le 1.7512240000000001 \cdot 10^{308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[x + \mathsf{fma}\left(\tan y + \tan z, \frac{1}{1 - \tan y \cdot \tan z}, -\tan a\right)\]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
x + \mathsf{fma}\left(\tan y + \tan z, \frac{1}{1 - \tan y \cdot \tan z}, -\tan a\right)
double f(double x, double y, double z, double a) {
        double r143087 = x;
        double r143088 = y;
        double r143089 = z;
        double r143090 = r143088 + r143089;
        double r143091 = tan(r143090);
        double r143092 = a;
        double r143093 = tan(r143092);
        double r143094 = r143091 - r143093;
        double r143095 = r143087 + r143094;
        return r143095;
}
↓
double f(double x, double y, double z, double a) {
        double r143096 = x;
        double r143097 = y;
        double r143098 = tan(r143097);
        double r143099 = z;
        double r143100 = tan(r143099);
        double r143101 = r143098 + r143100;
        double r143102 = 1.0;
        double r143103 = r143098 * r143100;
        double r143104 = r143102 - r143103;
        double r143105 = r143102 / r143104;
        double r143106 = a;
        double r143107 = tan(r143106);
        double r143108 = -r143107;
        double r143109 = fma(r143101, r143105, r143108);
        double r143110 = r143096 + r143109;
        return r143110;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

  1. Initial program 13.4

    \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
  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 div-inv0.2

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

  6. Applied fma-neg0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2020047 +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))))