Expression, p6

Average Error: 3.6 → 2.6

Time: 21.7s

Precision: 64

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

Math

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

↓

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

C

double f(double a, double b, double c, double d) {
        double r4960345 = a;
        double r4960346 = b;
        double r4960347 = c;
        double r4960348 = d;
        double r4960349 = r4960347 + r4960348;
        double r4960350 = r4960346 + r4960349;
        double r4960351 = r4960345 + r4960350;
        double r4960352 = 2.0;
        double r4960353 = r4960351 * r4960352;
        return r4960353;
}

↓

double f(double a, double b, double c, double d) {
        double r4960354 = d;
        double r4960355 = b;
        double r4960356 = a;
        double r4960357 = c;
        double r4960358 = r4960356 + r4960357;
        double r4960359 = r4960355 + r4960358;
        double r4960360 = r4960354 + r4960359;
        double r4960361 = exp(r4960360);
        double r4960362 = log(r4960361);
        double r4960363 = r4960355 + r4960357;
        double r4960364 = r4960363 + r4960356;
        double r4960365 = r4960354 + r4960364;
        double r4960366 = r4960363 + r4960354;
        double r4960367 = r4960356 + r4960366;
        double r4960368 = r4960365 * r4960367;
        double r4960369 = r4960362 * r4960368;
        double r4960370 = cbrt(r4960369);
        double r4960371 = 2.0;
        double r4960372 = r4960370 * r4960371;
        return r4960372;
}

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	2.6

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

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. Using strategy rm

7. Applied associate-+r+ 2.7

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

9. Applied add-log-exp 2.7

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

10. Applied add-log-exp 2.7

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

11. Applied add-log-exp 2.7

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

12. Applied sum-log 2.7

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

13. Applied sum-log 2.6

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

14. Applied add-log-exp 2.6

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

15. Applied sum-log 2.3

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

16. Simplified 2.6

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

17. Final simplification 2.6

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

Reproduce

herbie shell --seed 2019179 
(FPCore (a b c d)
  :name "Expression, p6"
  :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))