Average Error: 0.4 → 0.3
Time: 4.0s
Precision: 64
\[1 \le a \le 2 \le b \le 4 \le c \le 8 \le d \le 16 \le e \le 32\]
\[\left(\left(\left(e + d\right) + c\right) + b\right) + a\]
↓
\[d + \left(\left(e + \left(b + c\right)\right) + a\right)\]
\left(\left(\left(e + d\right) + c\right) + b\right) + a
↓
d + \left(\left(e + \left(b + c\right)\right) + a\right)
double f(double a, double b, double c, double d, double e) {
        double r100304 = e;
        double r100305 = d;
        double r100306 = r100304 + r100305;
        double r100307 = c;
        double r100308 = r100306 + r100307;
        double r100309 = b;
        double r100310 = r100308 + r100309;
        double r100311 = a;
        double r100312 = r100310 + r100311;
        return r100312;
}
↓
double f(double a, double b, double c, double d, double e) {
        double r100313 = d;
        double r100314 = e;
        double r100315 = b;
        double r100316 = c;
        double r100317 = r100315 + r100316;
        double r100318 = r100314 + r100317;
        double r100319 = a;
        double r100320 = r100318 + r100319;
        double r100321 = r100313 + r100320;
        return r100321;
}

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
Herbie0.3
\[\left(d + \left(c + \left(a + b\right)\right)\right) + e\]

Derivation

  1. Initial program 0.4

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

  2. Taylor expanded around 0 0.4

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

  4. Applied associate-+l+0.4

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

  5. Simplified0.3

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

  7. Applied associate-+l+0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019354 
(FPCore (a b c d e)
  :name "Expression 1, p15"
  :precision binary64
  :pre (<= 1 a 2 b 4 c 8 d 16 e 32)

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

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