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

Average Error: 13.3 → 0.2

Time: 47.3s

Precision: 64

\[\left(x = 0.0 \lor 0.5884141999999999983472775966220069676638 \le x \le 505.5908999999999764440872240811586380005\right) \land \left(-1.7966580000000000931214523812968299911 \cdot 10^{308} \le y \le -9.425585000000013069597555966781986720373 \cdot 10^{-310} \lor 1.284937999999999548796432976649400331091 \cdot 10^{-309} \le y \le 1.751223999999999928063201074847742204824 \cdot 10^{308}\right) \land \left(-1.776707000000000001259808757982040817204 \cdot 10^{308} \le z \le -8.599796000000016667475923823712126825539 \cdot 10^{-310} \lor 3.293144999999983071955117582595641261776 \cdot 10^{-311} \le z \le 1.725154000000000087891269878141591702413 \cdot 10^{308}\right) \land \left(-1.7966580000000000931214523812968299911 \cdot 10^{308} \le a \le -9.425585000000013069597555966781986720373 \cdot 10^{-310} \lor 1.284937999999999548796432976649400331091 \cdot 10^{-309} \le a \le 1.751223999999999928063201074847742204824 \cdot 10^{308}\right)\]

Math

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

↓

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

C

double f(double x, double y, double z, double a) {
        double r106820 = x;
        double r106821 = y;
        double r106822 = z;
        double r106823 = r106821 + r106822;
        double r106824 = tan(r106823);
        double r106825 = a;
        double r106826 = tan(r106825);
        double r106827 = r106824 - r106826;
        double r106828 = r106820 + r106827;
        return r106828;
}

↓

double f(double x, double y, double z, double a) {
        double r106829 = x;
        double r106830 = y;
        double r106831 = tan(r106830);
        double r106832 = z;
        double r106833 = tan(r106832);
        double r106834 = r106831 + r106833;
        double r106835 = 1.0;
        double r106836 = r106831 * r106833;
        double r106837 = r106836 * r106836;
        double r106838 = r106835 - r106837;
        double r106839 = r106834 / r106838;
        double r106840 = fma(r106831, r106833, r106835);
        double r106841 = a;
        double r106842 = tan(r106841);
        double r106843 = -r106842;
        double r106844 = fma(r106839, r106840, r106843);
        double r106845 = r106829 + r106844;
        double r106846 = cbrt(r106842);
        double r106847 = -r106846;
        double r106848 = r106846 * r106846;
        double r106849 = r106846 * r106848;
        double r106850 = fma(r106847, r106848, r106849);
        double r106851 = r106845 + r106850;
        return r106851;
}

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-sum 0.2

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

5. Applied add-cube-cbrt 0.3

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

6. Applied flip-- 0.3

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

7. Applied associate-/r/ 0.3

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

8. Applied prod-diff 0.3

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

9. Applied associate-+r+ 0.3

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

10. Simplified 0.2

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

11. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019325 +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 1.751224e+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))))