Average Error: 0.4 → 0.3
Time: 17.2s
Precision: 64
\[1 \le a \le 2 \le b \le 4 \le c \le 8 \le d \le 16 \le e \le 32\]
\[\left(\left(\left(e + d\right) + c\right) + b\right) + a\]
↓
\[e + \left(\left(\left(b + d\right) + a\right) + c\right)\]
\left(\left(\left(e + d\right) + c\right) + b\right) + a
↓
e + \left(\left(\left(b + d\right) + a\right) + c\right)
double f(double a, double b, double c, double d, double e) {
        double r3775371 = e;
        double r3775372 = d;
        double r3775373 = r3775371 + r3775372;
        double r3775374 = c;
        double r3775375 = r3775373 + r3775374;
        double r3775376 = b;
        double r3775377 = r3775375 + r3775376;
        double r3775378 = a;
        double r3775379 = r3775377 + r3775378;
        return r3775379;
}
↓
double f(double a, double b, double c, double d, double e) {
        double r3775380 = e;
        double r3775381 = b;
        double r3775382 = d;
        double r3775383 = r3775381 + r3775382;
        double r3775384 = a;
        double r3775385 = r3775383 + r3775384;
        double r3775386 = c;
        double r3775387 = r3775385 + r3775386;
        double r3775388 = r3775380 + r3775387;
        return r3775388;
}

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. Using strategy rm

  3. Applied add-log-exp0.4

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

  4. Applied add-log-exp0.4

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

  5. Applied add-log-exp0.4

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

  6. Applied add-log-exp0.4

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

  7. Applied sum-log0.4

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

  8. Applied sum-log0.3

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

  9. Applied sum-log0.2

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

  10. Simplified0.3

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

  12. Applied add-log-exp0.3

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

  13. Applied add-log-exp0.3

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

  14. Applied sum-log0.3

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

  15. Applied rem-exp-log0.3

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

  16. Taylor expanded around -inf 0.3

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

  17. Simplified0.3

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

  18. Final simplification0.3

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

Reproduce

herbie shell --seed 2019130 
(FPCore (a b c d e)
  :name "Expression 1, p15"
  :pre (<= 1 a 2 b 4 c 8 d 16 e 32)

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

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