Expression, p6

Average Error: 3.7 → 2.9

Time: 16.4s

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

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

↓

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

C

double f(double a, double b, double c, double d) {
        double r2810362 = a;
        double r2810363 = b;
        double r2810364 = c;
        double r2810365 = d;
        double r2810366 = r2810364 + r2810365;
        double r2810367 = r2810363 + r2810366;
        double r2810368 = r2810362 + r2810367;
        double r2810369 = 2.0;
        double r2810370 = r2810368 * r2810369;
        return r2810370;
}

↓

double f(double a, double b, double c, double d) {
        double r2810371 = 2.0;
        double r2810372 = d;
        double r2810373 = b;
        double r2810374 = c;
        double r2810375 = r2810373 + r2810374;
        double r2810376 = a;
        double r2810377 = r2810375 + r2810376;
        double r2810378 = r2810372 + r2810377;
        double r2810379 = r2810375 + r2810372;
        double r2810380 = r2810376 + r2810379;
        double r2810381 = r2810378 * r2810380;
        double r2810382 = r2810381 * r2810380;
        double r2810383 = cbrt(r2810382);
        double r2810384 = cbrt(r2810383);
        double r2810385 = r2810384 * r2810384;
        double r2810386 = r2810385 * r2810384;
        double r2810387 = r2810371 * r2810386;
        return r2810387;
}

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	2.9

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

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+ 2.8

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

9. Applied add-cube-cbrt 2.9

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

10. Final simplification 2.9

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

Reproduce

herbie shell --seed 2019135 +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))