Average Error: 13.3 → 0.2
Time: 1.3m
Precision: 64
\[\left(x = 0.0 \lor 0.5884142 \le x \le 505.5909\right) \land \left(-1.796658 \cdot 10^{+308} \le y \le -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le y \le 1.751224 \cdot 10^{+308}\right) \land \left(-1.776707 \cdot 10^{+308} \le z \le -8.599796 \cdot 10^{-310} \lor 3.293145 \cdot 10^{-311} \le z \le 1.725154 \cdot 10^{+308}\right) \land \left(-1.796658 \cdot 10^{+308} \le a \le -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le a \le 1.751224 \cdot 10^{+308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[\mathsf{fma}\left(\frac{\tan y + \tan z}{1 - \frac{\left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \sin y\right)}{\cos y}}, \tan z \cdot \tan y + 1, -\tan a\right) + x\]
x + \left(\tan \left(y + z\right) - \tan a\right)
↓
\mathsf{fma}\left(\frac{\tan y + \tan z}{1 - \frac{\left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \sin y\right)}{\cos y}}, \tan z \cdot \tan y + 1, -\tan a\right) + x
double f(double x, double y, double z, double a) {
        double r6168111 = x;
        double r6168112 = y;
        double r6168113 = z;
        double r6168114 = r6168112 + r6168113;
        double r6168115 = tan(r6168114);
        double r6168116 = a;
        double r6168117 = tan(r6168116);
        double r6168118 = r6168115 - r6168117;
        double r6168119 = r6168111 + r6168118;
        return r6168119;
}
↓
double f(double x, double y, double z, double a) {
        double r6168120 = y;
        double r6168121 = tan(r6168120);
        double r6168122 = z;
        double r6168123 = tan(r6168122);
        double r6168124 = r6168121 + r6168123;
        double r6168125 = 1.0;
        double r6168126 = r6168123 * r6168121;
        double r6168127 = sin(r6168120);
        double r6168128 = r6168123 * r6168127;
        double r6168129 = r6168126 * r6168128;
        double r6168130 = cos(r6168120);
        double r6168131 = r6168129 / r6168130;
        double r6168132 = r6168125 - r6168131;
        double r6168133 = r6168124 / r6168132;
        double r6168134 = r6168126 + r6168125;
        double r6168135 = a;
        double r6168136 = tan(r6168135);
        double r6168137 = -r6168136;
        double r6168138 = fma(r6168133, r6168134, r6168137);
        double r6168139 = x;
        double r6168140 = r6168138 + r6168139;
        return r6168140;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

  1. Initial program 13.3

    \[x + \left(\tan \left(y + z\right) - \tan a\right)\]
  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. Using strategy rm

  9. Applied tan-quot0.2

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

  10. Applied associate-*l/0.2

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

  11. Applied associate-*r/0.2

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

  12. Final simplification0.2

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

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (x y z a)
  :name "(+ x (- (tan (+ y z)) (tan a)))"
  :pre (and (or (== x 0.0) (<= 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))))