Expression, p6

Average Error: 3.7 → 2.7

Time: 52.2s

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 r10633092 = a;
        double r10633093 = b;
        double r10633094 = c;
        double r10633095 = d;
        double r10633096 = r10633094 + r10633095;
        double r10633097 = r10633093 + r10633096;
        double r10633098 = r10633092 + r10633097;
        double r10633099 = 2.0;
        double r10633100 = r10633098 * r10633099;
        return r10633100;
}

↓

double f(double a, double b, double c, double d) {
        double r10633101 = a;
        double r10633102 = d;
        double r10633103 = c;
        double r10633104 = b;
        double r10633105 = r10633103 + r10633104;
        double r10633106 = r10633102 + r10633105;
        double r10633107 = r10633101 + r10633106;
        double r10633108 = r10633101 + r10633104;
        double r10633109 = r10633108 + r10633102;
        double r10633110 = r10633103 + r10633109;
        double r10633111 = exp(r10633110);
        double r10633112 = log(r10633111);
        double r10633113 = r10633107 * r10633112;
        double r10633114 = r10633105 + r10633101;
        double r10633115 = r10633114 + r10633102;
        double r10633116 = r10633113 * r10633115;
        double r10633117 = cbrt(r10633116);
        double r10633118 = 2.0;
        double r10633119 = r10633117 * r10633118;
        return r10633119;
}

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 +o rules:numerics
(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))