(+ x (- (tan (+ y z)) (tan a)))

Average Error: 12.9 → 0.2

Time: 13.2s

Precision: 64

\[\left(x = 0.0 \lor 0.588414199999999998 \le x \le 505.590899999999976\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le y \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le y \le 1.7512240000000001 \cdot 10^{308}\right) \land \left(-1.7767070000000002 \cdot 10^{308} \le z \le -8.59979600000002 \cdot 10^{-310} \lor 3.29314499999998 \cdot 10^{-311} \le z \le 1.72515400000000009 \cdot 10^{308}\right) \land \left(-1.79665800000000009 \cdot 10^{308} \le a \le -9.425585000000013 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \le a \le 1.7512240000000001 \cdot 10^{308}\right)\]

Math

\[x + \left(\tan \left(y + z\right) - \tan a\right)\]

↓

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

C

double f(double x, double y, double z, double a) {
        double r140282 = x;
        double r140283 = y;
        double r140284 = z;
        double r140285 = r140283 + r140284;
        double r140286 = tan(r140285);
        double r140287 = a;
        double r140288 = tan(r140287);
        double r140289 = r140286 - r140288;
        double r140290 = r140282 + r140289;
        return r140290;
}

↓

double f(double x, double y, double z, double a) {
        double r140291 = x;
        double r140292 = y;
        double r140293 = sin(r140292);
        double r140294 = z;
        double r140295 = cos(r140294);
        double r140296 = r140293 / r140295;
        double r140297 = sin(r140294);
        double r140298 = a;
        double r140299 = sin(r140298);
        double r140300 = r140297 * r140299;
        double r140301 = cos(r140292);
        double r140302 = r140300 / r140301;
        double r140303 = cos(r140298);
        double r140304 = r140293 * r140303;
        double r140305 = r140304 / r140301;
        double r140306 = r140297 * r140303;
        double r140307 = r140306 / r140295;
        double r140308 = r140305 + r140307;
        double r140309 = r140308 - r140299;
        double r140310 = fma(r140296, r140302, r140309);
        double r140311 = r140310 / r140303;
        double r140312 = 1.0;
        double r140313 = r140293 * r140297;
        double r140314 = r140295 * r140301;
        double r140315 = r140313 / r140314;
        double r140316 = r140312 - r140315;
        double r140317 = r140311 / r140316;
        double r140318 = r140291 + r140317;
        return r140318;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus a

Derivation

1. Initial program 12.9

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

3. Applied tan-quot 12.9

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

4. Applied tan-sum 0.2

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

5. Applied frac-sub 0.2

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

6. Taylor expanded around inf 0.2

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

7. Simplified 0.2

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

8. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020081 +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))))