Average Error: 0.4 → 0.5
Time: 3.1s
Precision: 64
\[1 \le a \le 2 \le b \le 4 \le c \le 8 \le d \le 16 \le e \le 32\]
\[\left(\left(\left(e + d\right) + c\right) + b\right) + a\]
↓
\[\left(b + a\right) + \frac{d \cdot d - \left(e + c\right) \cdot \left(e + c\right)}{d - \left(e + c\right)}\]
\left(\left(\left(e + d\right) + c\right) + b\right) + a
↓
\left(b + a\right) + \frac{d \cdot d - \left(e + c\right) \cdot \left(e + c\right)}{d - \left(e + c\right)}
double code(double a, double b, double c, double d, double e) {
	return ((((e + d) + c) + b) + a);
}
↓
double code(double a, double b, double c, double d, double e) {
	return ((b + a) + (((d * d) - ((e + c) * (e + c))) / (d - (e + c))));
}

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
Herbie0.5
\[\left(d + \left(c + \left(a + b\right)\right)\right) + e\]

Derivation

  1. Initial program 0.4

    \[\left(\left(\left(e + d\right) + c\right) + b\right) + a\]
  2. Using strategy rm

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

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

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

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

  5. Applied distribute-lft-out0.4

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

  6. Simplified0.4

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

  8. Applied associate-+l+0.3

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

  10. Applied flip-+0.5

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

  11. Final simplification0.5

    \[\leadsto \left(b + a\right) + \frac{d \cdot d - \left(e + c\right) \cdot \left(e + c\right)}{d - \left(e + c\right)}\]

Reproduce

herbie shell --seed 2020058 +o rules:numerics
(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))