Average Error: 3.7 → 0
Time: 6.9s
Precision: binary64
\[-14 \leq a \land a \leq -13 \land -3 \leq b \land b \leq -2 \land 3 \leq c \land c \leq 3.5 \land 12.5 \leq d \land d \leq 13.5\]
\[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2 \]
↓
\[\left(\left(c + b\right) + \left(d + a\right)\right) \cdot 2 \]
\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2
↓
\left(\left(c + b\right) + \left(d + a\right)\right) \cdot 2

(FPCore (a b c d) :precision binary64 (* (+ a (+ b (+ c d))) 2.0))
↓
(FPCore (a b c d) :precision binary64 (* (+ (+ c b) (+ d a)) 2.0))
double code(double a, double b, double c, double d) {
	return (a + (b + (c + d))) * 2.0;
}
↓
double code(double a, double b, double c, double d) {
	return ((c + b) + (d + a)) * 2.0;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original3.7
Target3.8
Herbie0
\[\left(a + b\right) \cdot 2 + \left(c + d\right) \cdot 2 \]

Derivation

  1. Initial program 3.7

    \[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2 \]

  2. Applied add-log-exp_binary643.7

    \[\leadsto \left(a + \left(b + \left(c + \color{blue}{\log \left(e^{d}\right)}\right)\right)\right) \cdot 2 \]

  3. Applied add-log-exp_binary643.7

    \[\leadsto \left(a + \left(b + \left(\color{blue}{\log \left(e^{c}\right)} + \log \left(e^{d}\right)\right)\right)\right) \cdot 2 \]

  4. Applied sum-log_binary643.7

    \[\leadsto \left(a + \left(b + \color{blue}{\log \left(e^{c} \cdot e^{d}\right)}\right)\right) \cdot 2 \]

  5. Applied add-log-exp_binary643.7

    \[\leadsto \left(a + \left(\color{blue}{\log \left(e^{b}\right)} + \log \left(e^{c} \cdot e^{d}\right)\right)\right) \cdot 2 \]

  6. Applied sum-log_binary642.8

    \[\leadsto \left(a + \color{blue}{\log \left(e^{b} \cdot \left(e^{c} \cdot e^{d}\right)\right)}\right) \cdot 2 \]

  7. Applied add-log-exp_binary642.8

    \[\leadsto \left(\color{blue}{\log \left(e^{a}\right)} + \log \left(e^{b} \cdot \left(e^{c} \cdot e^{d}\right)\right)\right) \cdot 2 \]

  8. Applied sum-log_binary641.5

    \[\leadsto \color{blue}{\log \left(e^{a} \cdot \left(e^{b} \cdot \left(e^{c} \cdot e^{d}\right)\right)\right)} \cdot 2 \]

  9. Taylor expanded in c around 0 0.1

    \[\leadsto \color{blue}{\left(c + \log \left(e^{b} \cdot \left(e^{d} \cdot e^{a}\right)\right)\right)} \cdot 2 \]

  10. Applied log-prod_binary640.1

    \[\leadsto \left(c + \color{blue}{\left(\log \left(e^{b}\right) + \log \left(e^{d} \cdot e^{a}\right)\right)}\right) \cdot 2 \]

  11. Simplified0.1

    \[\leadsto \left(c + \left(\color{blue}{b} + \log \left(e^{d} \cdot e^{a}\right)\right)\right) \cdot 2 \]

  12. Simplified0.0

    \[\leadsto \left(c + \left(b + \color{blue}{\left(d + a\right)}\right)\right) \cdot 2 \]

  13. Applied associate-+r+_binary640

    \[\leadsto \color{blue}{\left(\left(c + b\right) + \left(d + a\right)\right)} \cdot 2 \]

  14. Final simplification0

    \[\leadsto \left(\left(c + b\right) + \left(d + a\right)\right) \cdot 2 \]

Reproduce

herbie shell --seed 2021224 
(FPCore (a b c d)
  :name "Expression, p6"
  :precision binary64
  :pre (and (<= -14.0 a -13.0) (<= -3.0 b -2.0) (<= 3.0 c 3.5) (<= 12.5 d 13.5))

  :herbie-target
  (+ (* (+ a b) 2.0) (* (+ c d) 2.0))

  (* (+ a (+ b (+ c d))) 2.0))