Average Error: 0.4 → 0.3
Time: 11.3s
Precision: 64
\[1 \le a \le 2 \le b \le 4 \le c \le 8 \le d \le 16 \le e \le 32\]
\[\left(\left(\left(e + d\right) + c\right) + b\right) + a\]
↓
\[e + \frac{\left(b + a\right) \cdot \left(\left(\left(d - c\right) \cdot d + c \cdot c\right) \cdot \left(b - a\right)\right) + \left(\left(d \cdot d\right) \cdot d + \left(c \cdot c\right) \cdot c\right) \cdot \left(b - a\right)}{\left(\left(d - c\right) \cdot d + c \cdot c\right) \cdot \left(b - a\right)}\]
\left(\left(\left(e + d\right) + c\right) + b\right) + a
↓
e + \frac{\left(b + a\right) \cdot \left(\left(\left(d - c\right) \cdot d + c \cdot c\right) \cdot \left(b - a\right)\right) + \left(\left(d \cdot d\right) \cdot d + \left(c \cdot c\right) \cdot c\right) \cdot \left(b - a\right)}{\left(\left(d - c\right) \cdot d + c \cdot c\right) \cdot \left(b - a\right)}
double f(double a, double b, double c, double d, double e) {
        double r5887068 = e;
        double r5887069 = d;
        double r5887070 = r5887068 + r5887069;
        double r5887071 = c;
        double r5887072 = r5887070 + r5887071;
        double r5887073 = b;
        double r5887074 = r5887072 + r5887073;
        double r5887075 = a;
        double r5887076 = r5887074 + r5887075;
        return r5887076;
}
↓
double f(double a, double b, double c, double d, double e) {
        double r5887077 = e;
        double r5887078 = b;
        double r5887079 = a;
        double r5887080 = r5887078 + r5887079;
        double r5887081 = d;
        double r5887082 = c;
        double r5887083 = r5887081 - r5887082;
        double r5887084 = r5887083 * r5887081;
        double r5887085 = r5887082 * r5887082;
        double r5887086 = r5887084 + r5887085;
        double r5887087 = r5887078 - r5887079;
        double r5887088 = r5887086 * r5887087;
        double r5887089 = r5887080 * r5887088;
        double r5887090 = r5887081 * r5887081;
        double r5887091 = r5887090 * r5887081;
        double r5887092 = r5887085 * r5887082;
        double r5887093 = r5887091 + r5887092;
        double r5887094 = r5887093 * r5887087;
        double r5887095 = r5887089 + r5887094;
        double r5887096 = r5887095 / r5887088;
        double r5887097 = r5887077 + r5887096;
        return r5887097;
}

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

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

  4. Applied associate-+l+0.3

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

  6. Applied associate-+l+0.2

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

  8. Applied flip-+0.2

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

  9. Applied flip3-+0.3

    \[\leadsto e + \left(\color{blue}{\frac{{c}^{3} + {d}^{3}}{c \cdot c + \left(d \cdot d - c \cdot d\right)}} + \frac{b \cdot b - a \cdot a}{b - a}\right)\]

  10. Applied frac-add0.4

    \[\leadsto e + \color{blue}{\frac{\left({c}^{3} + {d}^{3}\right) \cdot \left(b - a\right) + \left(c \cdot c + \left(d \cdot d - c \cdot d\right)\right) \cdot \left(b \cdot b - a \cdot a\right)}{\left(c \cdot c + \left(d \cdot d - c \cdot d\right)\right) \cdot \left(b - a\right)}}\]

  11. Simplified0.4

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

  12. Simplified0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2019171 
(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))