Expression, p6

Average Error: 3.7 → 2.7

Time: 14.9s

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

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

↓

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

C

double f(double a, double b, double c, double d) {
        double r2890105 = a;
        double r2890106 = b;
        double r2890107 = c;
        double r2890108 = d;
        double r2890109 = r2890107 + r2890108;
        double r2890110 = r2890106 + r2890109;
        double r2890111 = r2890105 + r2890110;
        double r2890112 = 2.0;
        double r2890113 = r2890111 * r2890112;
        return r2890113;
}

↓

double f(double a, double b, double c, double d) {
        double r2890114 = d;
        double r2890115 = a;
        double r2890116 = c;
        double r2890117 = r2890115 + r2890116;
        double r2890118 = b;
        double r2890119 = r2890117 + r2890118;
        double r2890120 = r2890114 + r2890119;
        double r2890121 = exp(r2890120);
        double r2890122 = log(r2890121);
        double r2890123 = r2890118 + r2890116;
        double r2890124 = r2890123 + r2890115;
        double r2890125 = r2890114 + r2890124;
        double r2890126 = r2890122 * r2890125;
        double r2890127 = r2890123 + r2890114;
        double r2890128 = r2890127 + r2890115;
        double r2890129 = r2890126 * r2890128;
        double r2890130 = cbrt(r2890129);
        double r2890131 = 2.0;
        double r2890132 = r2890130 * r2890131;
        return r2890132;
}

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 add-log-exp 2.9

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

8. Applied add-log-exp 2.9

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

9. Applied add-log-exp 2.9

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

10. Applied sum-log 2.9

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

11. Applied sum-log 2.8

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

12. Applied add-log-exp 2.8

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

13. Applied sum-log 2.6

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

14. Simplified 2.7

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

16. Applied associate-+r+ 2.7

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

17. Final simplification 2.7

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

Reproduce

herbie shell --seed 2019135 
(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))