Average Error: 3.6 → 0
Time: 8.7s
Precision: 64
\[-14 \le a \le -13 \land -3 \le b \le -2 \land 3 \le c \le 3.5 \land 12.5 \le d \le 13.5\]
\[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2\]
↓
\[2 \cdot \left(\left(b + c\right) + \left(a + d\right)\right)\]
\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2
↓
2 \cdot \left(\left(b + c\right) + \left(a + d\right)\right)
double f(double a, double b, double c, double d) {
        double r2613557 = a;
        double r2613558 = b;
        double r2613559 = c;
        double r2613560 = d;
        double r2613561 = r2613559 + r2613560;
        double r2613562 = r2613558 + r2613561;
        double r2613563 = r2613557 + r2613562;
        double r2613564 = 2.0;
        double r2613565 = r2613563 * r2613564;
        return r2613565;
}
↓
double f(double a, double b, double c, double d) {
        double r2613566 = 2.0;
        double r2613567 = b;
        double r2613568 = c;
        double r2613569 = r2613567 + r2613568;
        double r2613570 = a;
        double r2613571 = d;
        double r2613572 = r2613570 + r2613571;
        double r2613573 = r2613569 + r2613572;
        double r2613574 = r2613566 * r2613573;
        return r2613574;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original3.6
Target3.8
Herbie0
\[\left(a + b\right) \cdot 2 + \left(c + d\right) \cdot 2\]

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-log-exp2.8

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

  6. Applied add-log-exp2.8

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

  7. Applied add-log-exp2.8

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

  8. Applied sum-log2.8

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

  9. Applied sum-log2.8

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

  10. Applied add-log-exp2.8

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

  11. Applied sum-log1.5

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

  12. Simplified0.9

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

  14. Applied exp-sum1.2

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

  15. Applied log-prod0.7

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

  16. Simplified0.5

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

  17. Simplified0

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

  18. Final simplification0

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

Reproduce

herbie shell --seed 2019162 +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))