Expression, p6

Average Error: 3.6 → 2.7

Time: 14.5s

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

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

↓

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

C

double f(double a, double b, double c, double d) {
        double r1961258 = a;
        double r1961259 = b;
        double r1961260 = c;
        double r1961261 = d;
        double r1961262 = r1961260 + r1961261;
        double r1961263 = r1961259 + r1961262;
        double r1961264 = r1961258 + r1961263;
        double r1961265 = 2.0;
        double r1961266 = r1961264 * r1961265;
        return r1961266;
}

↓

double f(double a, double b, double c, double d) {
        double r1961267 = 2.0;
        double r1961268 = d;
        double r1961269 = b;
        double r1961270 = c;
        double r1961271 = r1961269 + r1961270;
        double r1961272 = a;
        double r1961273 = r1961271 + r1961272;
        double r1961274 = r1961268 + r1961273;
        double r1961275 = r1961271 + r1961268;
        double r1961276 = r1961275 + r1961272;
        double r1961277 = r1961276 * r1961276;
        double r1961278 = r1961277 * r1961276;
        double r1961279 = cbrt(r1961278);
        double r1961280 = r1961279 * r1961276;
        double r1961281 = r1961274 * r1961280;
        double r1961282 = cbrt(r1961281);
        double r1961283 = r1961267 * r1961282;
        return r1961283;
}

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	2.7

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

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

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

9. Applied add-cbrt-cube 2.7

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

10. Final simplification 2.7

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

Reproduce

herbie shell --seed 2019153 
(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))