Average Error: 0.4 → 0.0
Time: 44.8s
Precision: binary64
\[\left(\left(\left(\left(\left(\left(\left(\left(1 \leq a \land a \leq 2\right) \land 2 \leq b\right) \land b \leq 4\right) \land 4 \leq c\right) \land c \leq 8\right) \land 8 \leq d\right) \land d \leq 16\right) \land 16 \leq e\right) \land e \leq 32\]
\[\left(\left(\left(e + d\right) + c\right) + b\right) + a \]
↓
\[\begin{array}{l} t_0 := \left(e^{e} \cdot e^{d}\right) \cdot e^{c}\\ \log \left(\sqrt[3]{\left(t_0 \cdot \left(t_0 \cdot t_0\right)\right) \cdot \left(e^{b} \cdot \left(e^{b} \cdot e^{b}\right)\right)} \cdot e^{a}\right) \end{array} \]
\left(\left(\left(e + d\right) + c\right) + b\right) + a
↓
\begin{array}{l}
t_0 := \left(e^{e} \cdot e^{d}\right) \cdot e^{c}\\
\log \left(\sqrt[3]{\left(t_0 \cdot \left(t_0 \cdot t_0\right)\right) \cdot \left(e^{b} \cdot \left(e^{b} \cdot e^{b}\right)\right)} \cdot e^{a}\right)
\end{array}

(FPCore (a b c d e) :precision binary64 (+ (+ (+ (+ e d) c) b) a))
↓
(FPCore (a b c d e)
 :precision binary64
 (let* ((t_0 (* (* (exp e) (exp d)) (exp c))))
   (log
    (*
     (cbrt (* (* t_0 (* t_0 t_0)) (* (exp b) (* (exp b) (exp b)))))
     (exp a)))))
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) {
	double t_0 = (exp(e) * exp(d)) * exp(c);
	return log(cbrt((t_0 * (t_0 * t_0)) * (exp(b) * (exp(b) * exp(b)))) * exp(a));
}

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

Derivation

  1. Initial program 0.4

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

  2. Applied add-log-exp_binary640.4

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

  3. Applied add-log-exp_binary640.4

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

  4. Applied add-log-exp_binary640.4

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

  5. Applied add-log-exp_binary640.4

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

  6. Applied add-log-exp_binary640.4

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

  7. Applied sum-log_binary640.4

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

  8. Applied sum-log_binary640.3

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

  9. Applied sum-log_binary640.2

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

  10. Applied sum-log_binary640.0

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

  11. Applied add-cbrt-cube_binary640.0

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

  12. Applied add-cbrt-cube_binary640.0

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

  13. Applied cbrt-unprod_binary640.0

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

  14. Final simplification0.0

    \[\leadsto \log \left(\sqrt[3]{\left(\left(\left(e^{e} \cdot e^{d}\right) \cdot e^{c}\right) \cdot \left(\left(\left(e^{e} \cdot e^{d}\right) \cdot e^{c}\right) \cdot \left(\left(e^{e} \cdot e^{d}\right) \cdot e^{c}\right)\right)\right) \cdot \left(e^{b} \cdot \left(e^{b} \cdot e^{b}\right)\right)} \cdot e^{a}\right) \]

Reproduce

herbie shell --seed 2021313 
(FPCore (a b c d e)
  :name "Expression 1, p15"
  :precision binary64
  :pre (and (and (and (and (and (and (and (and (and (<= 1.0 a) (<= a 2.0)) (<= 2.0 b)) (<= b 4.0)) (<= 4.0 c)) (<= c 8.0)) (<= 8.0 d)) (<= d 16.0)) (<= 16.0 e)) (<= e 32.0))

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

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