Expression, p6

Average Error: 3.6 → 2.8

Time: 6.6s

Precision: binary64

\[-14 \le a \le -13 \land -3 \le b \le -2 \land 3 \le c \le 3.5 \land 12.5 \le d \le 13.5\]

Math

\[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2\]

↓

\[\left(\sqrt[3]{\left(a + \left(\left(b + c\right) + d\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)} \cdot \sqrt[3]{\left(a + \left(b + c\right)\right) + d}\right) \cdot 2\]

C

double code(double a, double b, double c, double d) {
	return ((double) (((double) (a + ((double) (b + ((double) (c + d)))))) * 2.0));
}

↓

double code(double a, double b, double c, double d) {
	return ((double) (((double) (((double) cbrt(((double) (((double) (a + ((double) (((double) (b + c)) + d)))) * ((double) (a + ((double) (((double) (b + c)) + d)))))))) * ((double) cbrt(((double) (((double) (a + ((double) (b + c)))) + d)))))) * 2.0));
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	3.6
Target	3.8
Herbie	2.8

\[\left(a + b\right) \cdot 2 + \left(c + d\right) \cdot 2\]

Derivation

1. Initial program 3.6

   \[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2\]
2. Using strategy rm

3. Applied associate-+r+ 2.8

   \[\leadsto \left(a + \color{blue}{\left(\left(b + c\right) + d\right)}\right) \cdot 2\]
4. Using strategy rm

5. Applied add-cbrt-cube 2.9

   \[\leadsto \color{blue}{\sqrt[3]{\left(\left(a + \left(\left(b + c\right) + d\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)}} \cdot 2\]

6. Simplified 2.9

   \[\leadsto \sqrt[3]{\color{blue}{{\left(a + \left(\left(b + c\right) + d\right)\right)}^{3}}} \cdot 2\]
7. Using strategy rm

8. Applied add-cube-cbrt 3.1

   \[\leadsto \sqrt[3]{{\color{blue}{\left(\left(\sqrt[3]{a + \left(\left(b + c\right) + d\right)} \cdot \sqrt[3]{a + \left(\left(b + c\right) + d\right)}\right) \cdot \sqrt[3]{a + \left(\left(b + c\right) + d\right)}\right)}}^{3}} \cdot 2\]

9. Applied unpow-prod-down 3.1

   \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt[3]{a + \left(\left(b + c\right) + d\right)} \cdot \sqrt[3]{a + \left(\left(b + c\right) + d\right)}\right)}^{3} \cdot {\left(\sqrt[3]{a + \left(\left(b + c\right) + d\right)}\right)}^{3}}} \cdot 2\]

10. Applied cbrt-prod 3.1

    \[\leadsto \color{blue}{\left(\sqrt[3]{{\left(\sqrt[3]{a + \left(\left(b + c\right) + d\right)} \cdot \sqrt[3]{a + \left(\left(b + c\right) + d\right)}\right)}^{3}} \cdot \sqrt[3]{{\left(\sqrt[3]{a + \left(\left(b + c\right) + d\right)}\right)}^{3}}\right)} \cdot 2\]

11. Simplified 3.0

    \[\leadsto \left(\color{blue}{\sqrt[3]{\left(a + \left(\left(b + c\right) + d\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)}} \cdot \sqrt[3]{{\left(\sqrt[3]{a + \left(\left(b + c\right) + d\right)}\right)}^{3}}\right) \cdot 2\]

12. Simplified 3.0

    \[\leadsto \left(\sqrt[3]{\left(a + \left(\left(b + c\right) + d\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)} \cdot \color{blue}{\sqrt[3]{a + \left(\left(b + c\right) + d\right)}}\right) \cdot 2\]
13. Using strategy rm

14. Applied associate-+r+ 2.8

    \[\leadsto \left(\sqrt[3]{\left(a + \left(\left(b + c\right) + d\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)} \cdot \sqrt[3]{\color{blue}{\left(a + \left(b + c\right)\right) + d}}\right) \cdot 2\]

15. Final simplification 2.8

    \[\leadsto \left(\sqrt[3]{\left(a + \left(\left(b + c\right) + d\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)} \cdot \sqrt[3]{\left(a + \left(b + c\right)\right) + d}\right) \cdot 2\]

Reproduce

herbie shell --seed 2020153 
(FPCore (a b c d)
  :name "Expression, p6"
  :precision binary64
  :pre (and (<= -14.0 a -13.0) (<= -3.0 b -2.0) (<= 3.0 c 3.5) (<= 12.5 d 13.5))

  :herbie-target
  (+ (* (+ a b) 2.0) (* (+ c d) 2.0))

  (* (+ a (+ b (+ c d))) 2.0))