Average Error: 0.4 → 0.2
Time: 14.5s
Precision: 64
\[1 \le a \le 2 \le b \le 4 \le c \le 8 \le d \le 16 \le e \le 32\]
\[\left(\left(\left(e + d\right) + c\right) + b\right) + a\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\left(e + \left(b + a\right)\right) + \left(c + d\right)\right)\right)\]
\left(\left(\left(e + d\right) + c\right) + b\right) + a
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(\left(e + \left(b + a\right)\right) + \left(c + d\right)\right)\right)
double f(double a, double b, double c, double d, double e) {
        double r63520 = e;
        double r63521 = d;
        double r63522 = r63520 + r63521;
        double r63523 = c;
        double r63524 = r63522 + r63523;
        double r63525 = b;
        double r63526 = r63524 + r63525;
        double r63527 = a;
        double r63528 = r63526 + r63527;
        return r63528;
}
↓
double f(double a, double b, double c, double d, double e) {
        double r63529 = e;
        double r63530 = b;
        double r63531 = a;
        double r63532 = r63530 + r63531;
        double r63533 = r63529 + r63532;
        double r63534 = c;
        double r63535 = d;
        double r63536 = r63534 + r63535;
        double r63537 = r63533 + r63536;
        double r63538 = expm1(r63537);
        double r63539 = log1p(r63538);
        return r63539;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Bits error versus e

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.4
Target0.2
Herbie0.2
\[\left(d + \left(c + \left(a + b\right)\right)\right) + e\]

Derivation

  1. Initial program 0.4

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

  2. Simplified0.3

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

  4. Applied *-un-lft-identity0.3

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

  5. Applied *-un-lft-identity0.3

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

  6. Applied distribute-lft-out0.3

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

  7. Simplified0.2

    \[\leadsto 1 \cdot \color{blue}{\left(\left(\left(b + a\right) + \left(d + c\right)\right) + e\right)}\]
  8. Using strategy rm

  9. Applied log1p-expm1-u0.2

    \[\leadsto 1 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(\left(b + a\right) + \left(d + c\right)\right) + e\right)\right)}\]

  10. Simplified0.2

    \[\leadsto 1 \cdot \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\left(e + \left(a + b\right)\right) + \left(d + c\right)\right)}\right)\]

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (a b c d e)
  :name "Expression 1, p15"
  :pre (<= 1.0 a 2.0 b 4.0 c 8.0 d 16.0 e 32.0)

  :herbie-target
  (+ (+ d (+ c (+ a b))) e)

  (+ (+ (+ (+ e d) c) b) a))