Average Error: 13.4 → 0.2
Time: 41.3s
Precision: 64
\[\left(x = 0 \lor 0.5884142 \le x \le 505.5909\right) \land \left(-1.796658 \cdot 10^{+308} \le y \le -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le y \le 1.751224 \cdot 10^{+308}\right) \land \left(-1.776707 \cdot 10^{+308} \le z \le -8.599796 \cdot 10^{-310} \lor 3.293145 \cdot 10^{-311} \le z \le 1.725154 \cdot 10^{+308}\right) \land \left(-1.796658 \cdot 10^{+308} \le a \le -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le a \le 1.751224 \cdot 10^{+308}\right)\]
\[x + \left(\tan \left(y + z\right) - \tan a\right)\]
↓
\[\mathsf{fma}\left(\frac{\tan y + \tan z}{1 - {\left(\tan z \cdot \tan y\right)}^{3}}, \left(\sqrt[3]{\tan z \cdot \tan y} \cdot \left(\sqrt[3]{\tan z \cdot \tan y} \cdot \sqrt[3]{\tan z \cdot \tan y}\right) + \left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)\right) + 1, -\tan a\right) + x\]
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)}^{3}}, \left(\sqrt[3]{\tan z \cdot \tan y} \cdot \left(\sqrt[3]{\tan z \cdot \tan y} \cdot \sqrt[3]{\tan z \cdot \tan y}\right) + \left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)\right) + 1, -\tan a\right) + x
double f(double x, double y, double z, double a) {
        double r3160041 = x;
        double r3160042 = y;
        double r3160043 = z;
        double r3160044 = r3160042 + r3160043;
        double r3160045 = tan(r3160044);
        double r3160046 = a;
        double r3160047 = tan(r3160046);
        double r3160048 = r3160045 - r3160047;
        double r3160049 = r3160041 + r3160048;
        return r3160049;
}
↓
double f(double x, double y, double z, double a) {
        double r3160050 = y;
        double r3160051 = tan(r3160050);
        double r3160052 = z;
        double r3160053 = tan(r3160052);
        double r3160054 = r3160051 + r3160053;
        double r3160055 = 1.0;
        double r3160056 = r3160053 * r3160051;
        double r3160057 = 3.0;
        double r3160058 = pow(r3160056, r3160057);
        double r3160059 = r3160055 - r3160058;
        double r3160060 = r3160054 / r3160059;
        double r3160061 = cbrt(r3160056);
        double r3160062 = r3160061 * r3160061;
        double r3160063 = r3160061 * r3160062;
        double r3160064 = r3160056 * r3160056;
        double r3160065 = r3160063 + r3160064;
        double r3160066 = r3160065 + r3160055;
        double r3160067 = a;
        double r3160068 = tan(r3160067);
        double r3160069 = -r3160068;
        double r3160070 = fma(r3160060, r3160066, r3160069);
        double r3160071 = x;
        double r3160072 = r3160070 + r3160071;
        return r3160072;
}

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 flip3--0.2

    \[\leadsto x + \left(\frac{\tan y + \tan z}{\color{blue}{\frac{{1}^{3} - {\left(\tan y \cdot \tan z\right)}^{3}}{1 \cdot 1 + \left(\left(\tan y \cdot \tan z\right) \cdot \left(\tan y \cdot \tan z\right) + 1 \cdot \left(\tan y \cdot \tan z\right)\right)}}} - \tan a\right)\]

  6. Applied associate-/r/0.2

    \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{{1}^{3} - {\left(\tan y \cdot \tan z\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\left(\tan y \cdot \tan z\right) \cdot \left(\tan y \cdot \tan z\right) + 1 \cdot \left(\tan y \cdot \tan z\right)\right)\right)} - \tan a\right)\]

  7. Applied fma-neg0.2

    \[\leadsto x + \color{blue}{\mathsf{fma}\left(\frac{\tan y + \tan z}{{1}^{3} - {\left(\tan y \cdot \tan z\right)}^{3}}, 1 \cdot 1 + \left(\left(\tan y \cdot \tan z\right) \cdot \left(\tan y \cdot \tan z\right) + 1 \cdot \left(\tan y \cdot \tan z\right)\right), -\tan a\right)}\]
  8. Using strategy rm

  9. Applied add-cube-cbrt0.2

    \[\leadsto x + \mathsf{fma}\left(\frac{\tan y + \tan z}{{1}^{3} - {\left(\tan y \cdot \tan z\right)}^{3}}, 1 \cdot 1 + \left(\left(\tan y \cdot \tan z\right) \cdot \left(\tan y \cdot \tan z\right) + 1 \cdot \color{blue}{\left(\left(\sqrt[3]{\tan y \cdot \tan z} \cdot \sqrt[3]{\tan y \cdot \tan z}\right) \cdot \sqrt[3]{\tan y \cdot \tan z}\right)}\right), -\tan a\right)\]

  10. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(\frac{\tan y + \tan z}{1 - {\left(\tan z \cdot \tan y\right)}^{3}}, \left(\sqrt[3]{\tan z \cdot \tan y} \cdot \left(\sqrt[3]{\tan z \cdot \tan y} \cdot \sqrt[3]{\tan z \cdot \tan y}\right) + \left(\tan z \cdot \tan y\right) \cdot \left(\tan z \cdot \tan y\right)\right) + 1, -\tan a\right) + x\]

Reproduce

herbie shell --seed 2019144 +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))))