Average Error: 3.6 → 3.0
Time: 2.9s
Precision: 64
\[-14 \le a \le -13 \land -3 \le b \le -2 \land 3 \le c \le 3.5 \land 12.5 \le d \le 13.5\]
\[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2\]
↓
\[\left(e^{\mathsf{log1p}\left(\left(a + \left(b + c\right)\right) + d\right)} - 1\right) \cdot 2\]
\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2
↓
\left(e^{\mathsf{log1p}\left(\left(a + \left(b + c\right)\right) + d\right)} - 1\right) \cdot 2
double f(double a, double b, double c, double d) {
        double r75569 = a;
        double r75570 = b;
        double r75571 = c;
        double r75572 = d;
        double r75573 = r75571 + r75572;
        double r75574 = r75570 + r75573;
        double r75575 = r75569 + r75574;
        double r75576 = 2.0;
        double r75577 = r75575 * r75576;
        return r75577;
}
↓
double f(double a, double b, double c, double d) {
        double r75578 = a;
        double r75579 = b;
        double r75580 = c;
        double r75581 = r75579 + r75580;
        double r75582 = r75578 + r75581;
        double r75583 = d;
        double r75584 = r75582 + r75583;
        double r75585 = log1p(r75584);
        double r75586 = exp(r75585);
        double r75587 = 1.0;
        double r75588 = r75586 - r75587;
        double r75589 = 2.0;
        double r75590 = r75588 * r75589;
        return r75590;
}

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
Herbie3.0
\[\left(a + b\right) \cdot 2 + \left(c + d\right) \cdot 2\]

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 associate-+r+2.8

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

  7. Applied expm1-log1p-u3.0

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

  9. Applied expm1-udef3.0

    \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\left(a + \left(b + c\right)\right) + d\right)} - 1\right)} \cdot 2\]

  10. Final simplification3.0

    \[\leadsto \left(e^{\mathsf{log1p}\left(\left(a + \left(b + c\right)\right) + d\right)} - 1\right) \cdot 2\]

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (a b c d)
  :name "Expression, p6"
  :precision binary64
  :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))