Average Error: 13.1 → 0.2
Time: 16.1s
Precision: 64
\[\left(x = 0.0 \lor 0.588414199999999998 \le x \le 505.590899999999976\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le y \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le y \le 1.7512240000000001 \cdot 10^{308}\right) \land \left(-1.7767070000000002 \cdot 10^{308} \le z \le -8.59979600000002 \cdot 10^{-310} \lor 3.29314499999998 \cdot 10^{-311} \le z \le 1.72515400000000009 \cdot 10^{308}\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le a \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le a \le 1.7512240000000001 \cdot 10^{308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[\mathsf{fma}\left(\frac{\tan y + \tan z}{1 - \left(\tan y \cdot \tan y\right) \cdot \left(\tan z \cdot \tan z\right)}, \mathsf{fma}\left(\tan y, \tan z, 1\right), \left(-\tan a\right) + x\right) + \mathsf{fma}\left(-\tan a, 1, \tan a \cdot 1\right)\]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
\mathsf{fma}\left(\frac{\tan y + \tan z}{1 - \left(\tan y \cdot \tan y\right) \cdot \left(\tan z \cdot \tan z\right)}, \mathsf{fma}\left(\tan y, \tan z, 1\right), \left(-\tan a\right) + x\right) + \mathsf{fma}\left(-\tan a, 1, \tan a \cdot 1\right)
double f(double x, double y, double z, double a) {
        double r210146 = x;
        double r210147 = y;
        double r210148 = z;
        double r210149 = r210147 + r210148;
        double r210150 = tan(r210149);
        double r210151 = a;
        double r210152 = tan(r210151);
        double r210153 = r210150 - r210152;
        double r210154 = r210146 + r210153;
        return r210154;
}
↓
double f(double x, double y, double z, double a) {
        double r210155 = y;
        double r210156 = tan(r210155);
        double r210157 = z;
        double r210158 = tan(r210157);
        double r210159 = r210156 + r210158;
        double r210160 = 1.0;
        double r210161 = r210156 * r210156;
        double r210162 = r210158 * r210158;
        double r210163 = r210161 * r210162;
        double r210164 = r210160 - r210163;
        double r210165 = r210159 / r210164;
        double r210166 = fma(r210156, r210158, r210160);
        double r210167 = a;
        double r210168 = tan(r210167);
        double r210169 = -r210168;
        double r210170 = x;
        double r210171 = r210169 + r210170;
        double r210172 = fma(r210165, r210166, r210171);
        double r210173 = r210168 * r210160;
        double r210174 = fma(r210169, r210160, r210173);
        double r210175 = r210172 + r210174;
        return r210175;
}

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 *-un-lft-identity0.2

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

  6. Applied flip--0.2

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

  7. Applied associate-/r/0.2

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

  8. Applied prod-diff0.2

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

  9. Applied associate-+r+0.2

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

  10. Simplified0.2

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

  12. Applied swap-sqr0.2

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

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2020035 +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))))