tan-example

Average Error: 12.9 → 0.2

Time: 47.1s

Precision: binary64

\[\left(\left(\left(x = 0 \lor 0.5884142 \leq x \land x \leq 505.5909\right) \land \left(-1.796658 \cdot 10^{+308} \leq y \land y \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq y \land y \leq 1.751224 \cdot 10^{+308}\right)\right) \land \left(-1.776707 \cdot 10^{+308} \leq z \land z \leq -8.599796 \cdot 10^{-310} \lor 3.293145 \cdot 10^{-311} \leq z \land z \leq 1.725154 \cdot 10^{+308}\right)\right) \land \left(-1.796658 \cdot 10^{+308} \leq a \land a \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq a \land a \leq 1.751224 \cdot 10^{+308}\right)\]

Math

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

↓

\[\begin{array}{l} t_0 := \tan y \cdot \tan z\\ x + \left(\mathsf{fma}\left(\frac{\tan y + \tan z}{1 - {t_0}^{3}}, \mathsf{fma}\left(t_0, \mathsf{fma}\left(\tan y, \tan z, 1\right), 1\right), -\tan a\right) + \tan a \cdot 0\right) \end{array} \]

FPCore

(FPCore (x y z a) :precision binary64 (+ x (- (tan (+ y z)) (tan a))))

↓

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (* (tan y) (tan z))))
   (+
    x
    (+
     (fma
      (/ (+ (tan y) (tan z)) (- 1.0 (pow t_0 3.0)))
      (fma t_0 (fma (tan y) (tan z) 1.0) 1.0)
      (- (tan a)))
     (* (tan a) 0.0)))))

C

double code(double x, double y, double z, double a) {
	return x + (tan(y + z) - tan(a));
}

↓

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) * tan(z);
	return x + (fma(((tan(y) + tan(z)) / (1.0 - pow(t_0, 3.0))), fma(t_0, fma(tan(y), tan(z), 1.0), 1.0), -tan(a)) + (tan(a) * 0.0));
}

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. Applied tan-sum_binary64 0.2

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

3. Applied add-cube-cbrt_binary64 0.3

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

4. Applied flip3--_binary64 0.3

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

5. Applied associate-/r/_binary64 0.3

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

6. Applied prod-diff_binary64 0.3

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

7. Simplified 0.2

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

8. Simplified 0.2

   \[\leadsto x + \left(\mathsf{fma}\left(\frac{\tan y + \tan z}{1 - {\left(\tan y \cdot \tan z\right)}^{3}}, \mathsf{fma}\left(\tan y \cdot \tan z, \mathsf{fma}\left(\tan y, \tan z, 1\right), 1\right), -\tan a\right) + \color{blue}{\tan a \cdot 0}\right) \]

9. Final simplification 0.2

   \[\leadsto x + \left(\mathsf{fma}\left(\frac{\tan y + \tan z}{1 - {\left(\tan y \cdot \tan z\right)}^{3}}, \mathsf{fma}\left(\tan y \cdot \tan z, \mathsf{fma}\left(\tan y, \tan z, 1\right), 1\right), -\tan a\right) + \tan a \cdot 0\right) \]

Reproduce

herbie shell --seed 2022024 
(FPCore (x y z a)
  :name "tan-example"
  :precision binary64
  :pre (and (and (and (or (== x 0.0) (and (<= 0.5884142 x) (<= x 505.5909))) (or (and (<= -1.796658e+308 y) (<= y -9.425585e-310)) (and (<= 1.284938e-309 y) (<= y 1.751224e+308)))) (or (and (<= -1.776707e+308 z) (<= z -8.599796e-310)) (and (<= 3.293145e-311 z) (<= z 1.725154e+308)))) (or (and (<= -1.796658e+308 a) (<= a -9.425585e-310)) (and (<= 1.284938e-309 a) (<= a 1.751224e+308))))
  (+ x (- (tan (+ y z)) (tan a))))