Average Error: 0.4 → 0.0
Time: 38.8s
Precision: 64
\[\left(\left(\left(e + d\right) + c\right) + b\right) + a\]
↓
\[\log \left(e^{e} \cdot \left(e^{a} \cdot \left(e^{b} \cdot \left(e^{d} \cdot e^{c}\right)\right)\right)\right)\]
double f(double a, double b, double c, double d, double e) {
        double r14646940 = e;
        double r14646941 = d;
        double r14646942 = r14646940 + r14646941;
        double r14646943 = c;
        double r14646944 = r14646942 + r14646943;
        double r14646945 = b;
        double r14646946 = r14646944 + r14646945;
        double r14646947 = a;
        double r14646948 = r14646946 + r14646947;
        return r14646948;
}
↓
double f(double a, double b, double c, double d, double e) {
        double r14646949 = e;
        double r14646950 = exp(r14646949);
        double r14646951 = a;
        double r14646952 = exp(r14646951);
        double r14646953 = b;
        double r14646954 = exp(r14646953);
        double r14646955 = d;
        double r14646956 = exp(r14646955);
        double r14646957 = c;
        double r14646958 = exp(r14646957);
        double r14646959 = r14646956 * r14646958;
        double r14646960 = r14646954 * r14646959;
        double r14646961 = r14646952 * r14646960;
        double r14646962 = r14646950 * r14646961;
        double r14646963 = log(r14646962);
        return r14646963;
}


\left(\left(\left(e + d\right) + c\right) + b\right) + a
↓
\log \left(e^{e} \cdot \left(e^{a} \cdot \left(e^{b} \cdot \left(e^{d} \cdot e^{c}\right)\right)\right)\right)

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Bits error versus e

Target

Original0.4
Target0.2
Herbie0.0
\[\left(d + \left(c + \left(a + b\right)\right)\right) + e\]

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

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

  4. Applied add-log-exp0.4

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

  5. Applied add-log-exp0.4

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

  6. Applied add-log-exp0.4

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

  7. Applied sum-log0.4

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

  8. Applied sum-log0.3

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

  9. Applied sum-log0.2

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

  10. Simplified0.3

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

  12. Applied add-log-exp0.3

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

  13. Applied add-log-exp0.3

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

  14. Applied add-log-exp0.3

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

  15. Applied sum-log0.3

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

  16. Applied sum-log0.1

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

  17. Simplified0.3

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

  18. Taylor expanded around inf 0.3

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

  20. Applied add-log-exp0.3

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

  21. Applied add-log-exp0.3

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

  22. Applied sum-log0.3

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

  23. Applied add-log-exp0.3

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

  24. Applied sum-log0.2

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

  25. Applied add-log-exp0.2

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

  26. Applied sum-log0.2

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

  27. Applied add-log-exp0.2

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

  28. Applied sum-log0.0

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

  29. Applied rem-exp-log0.0

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

  30. Final simplification0.0

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

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(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))