Expression, p6

Average Error: 3.7 → 2.8

Time: 43.9s

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

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

↓

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

C

double f(double a, double b, double c, double d) {
        double r7155939 = a;
        double r7155940 = b;
        double r7155941 = c;
        double r7155942 = d;
        double r7155943 = r7155941 + r7155942;
        double r7155944 = r7155940 + r7155943;
        double r7155945 = r7155939 + r7155944;
        double r7155946 = 2.0;
        double r7155947 = r7155945 * r7155946;
        return r7155947;
}

↓

double f(double a, double b, double c, double d) {
        double r7155948 = 2.0;
        double r7155949 = d;
        double r7155950 = b;
        double r7155951 = c;
        double r7155952 = r7155950 + r7155951;
        double r7155953 = a;
        double r7155954 = r7155952 + r7155953;
        double r7155955 = r7155949 + r7155954;
        double r7155956 = r7155952 + r7155949;
        double r7155957 = r7155956 + r7155953;
        double r7155958 = cbrt(r7155957);
        double r7155959 = r7155958 * r7155958;
        double r7155960 = r7155958 * r7155959;
        double r7155961 = r7155960 * r7155957;
        double r7155962 = r7155955 * r7155961;
        double r7155963 = cbrt(r7155962);
        double r7155964 = r7155948 * r7155963;
        return r7155964;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	3.7
Target	3.8
Herbie	2.8

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

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 3.0

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

7. Applied associate-+r+ 2.8

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

9. Applied add-cube-cbrt 2.8

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

10. Final simplification 2.8

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

Reproduce

herbie shell --seed 2019168 
(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))