Expression, p6

Average Error: 3.7 → 2.7

Time: 50.5s

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

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

↓

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

C

double f(double a, double b, double c, double d) {
        double r24038390 = a;
        double r24038391 = b;
        double r24038392 = c;
        double r24038393 = d;
        double r24038394 = r24038392 + r24038393;
        double r24038395 = r24038391 + r24038394;
        double r24038396 = r24038390 + r24038395;
        double r24038397 = 2.0;
        double r24038398 = r24038396 * r24038397;
        return r24038398;
}

↓

double f(double a, double b, double c, double d) {
        double r24038399 = a;
        double r24038400 = d;
        double r24038401 = c;
        double r24038402 = b;
        double r24038403 = r24038401 + r24038402;
        double r24038404 = r24038400 + r24038403;
        double r24038405 = r24038399 + r24038404;
        double r24038406 = r24038399 + r24038402;
        double r24038407 = r24038406 + r24038400;
        double r24038408 = r24038401 + r24038407;
        double r24038409 = exp(r24038408);
        double r24038410 = log(r24038409);
        double r24038411 = r24038405 * r24038410;
        double r24038412 = r24038403 + r24038399;
        double r24038413 = r24038412 + r24038400;
        double r24038414 = r24038411 * r24038413;
        double r24038415 = cbrt(r24038414);
        double r24038416 = 2.0;
        double r24038417 = r24038415 * r24038416;
        return r24038417;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	3.7
Target	3.9
Herbie	2.7

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

Derivation

1. Initial program 3.7

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

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

9. Applied add-log-exp 2.7

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

10. Applied add-log-exp 2.7

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

11. Applied add-log-exp 2.7

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

12. Applied sum-log 2.7

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

13. Applied sum-log 2.6

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

14. Applied add-log-exp 2.6

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

15. Applied sum-log 2.3

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

16. Simplified 2.7

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

17. Final simplification 2.7

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

Reproduce

herbie shell --seed 2019119 
(FPCore (a b c d)
  :name "Expression, p6"
  :pre (and (<= -14 a -13) (<= -3 b -2) (<= 3 c 3.5) (<= 12.5 d 13.5))

  :herbie-target
  (+ (* (+ a b) 2) (* (+ c d) 2))

  (* (+ a (+ b (+ c d))) 2))