Average Error: 3.6 → 0.1
Time: 47.5s
Precision: 64
\[-14 \le a \le -13 \land -3 \le b \le -2 \land 3 \le c \le 3.5 \land 12.5 \le d \le 13.5\]
\[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2\]
↓
\[2 \cdot \mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(\left(b + c\right) + \left(d + a\right)\right)\right)\right)\right)\]
\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2
↓
2 \cdot \mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(\left(b + c\right) + \left(d + a\right)\right)\right)\right)\right)
double f(double a, double b, double c, double d) {
        double r18676574 = a;
        double r18676575 = b;
        double r18676576 = c;
        double r18676577 = d;
        double r18676578 = r18676576 + r18676577;
        double r18676579 = r18676575 + r18676578;
        double r18676580 = r18676574 + r18676579;
        double r18676581 = 2.0;
        double r18676582 = r18676580 * r18676581;
        return r18676582;
}
↓
double f(double a, double b, double c, double d) {
        double r18676583 = 2.0;
        double r18676584 = b;
        double r18676585 = c;
        double r18676586 = r18676584 + r18676585;
        double r18676587 = d;
        double r18676588 = a;
        double r18676589 = r18676587 + r18676588;
        double r18676590 = r18676586 + r18676589;
        double r18676591 = expm1(r18676590);
        double r18676592 = log1p(r18676591);
        double r18676593 = r18676583 * r18676592;
        return r18676593;
}

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
Herbie0.1
\[\left(a + b\right) \cdot 2 + \left(c + d\right) \cdot 2\]

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-cube2.9

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

  7. Applied log1p-expm1-u2.9

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(\sqrt[3]{\left(\left(a + \left(\left(b + c\right) + d\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)\right) \cdot \left(a + \left(\left(b + c\right) + d\right)\right)}\right)\right)\right)\right)} \cdot 2\]

  8. Simplified0.1

    \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(\mathsf{expm1}\left(\left(\left(a + d\right) + \left(b + c\right)\right)\right)\right)}\right) \cdot 2\]

  9. Final simplification0.1

    \[\leadsto 2 \cdot \mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(\left(b + c\right) + \left(d + a\right)\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2019125 +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))