Average Error: 3.7 → 0.1
Time: 4.5s
Precision: binary64
\[-14 \leq a \land a \leq -13 \land -3 \leq b \land b \leq -2 \land 3 \leq c \land c \leq 3.5 \land 12.5 \leq d \land d \leq 13.5\]
\[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2 \]
↓
\[\left(c + \log \left(\left(e^{d} \cdot e^{b}\right) \cdot e^{a}\right)\right) \cdot 2 \]
\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2
↓
\left(c + \log \left(\left(e^{d} \cdot e^{b}\right) \cdot e^{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 (log (* (* (exp d) (exp b)) (exp 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 + log((exp(d) * exp(b)) * exp(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
Herbie0.1
\[\left(a + b\right) \cdot 2 + \left(c + d\right) \cdot 2 \]

Derivation

  1. Initial program 3.7

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

  2. Taylor expanded around 0 2.8

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

  3. Simplified2.8

    \[\leadsto \color{blue}{\left(c + \left(\left(d + b\right) + a\right)\right)} \cdot 2 \]
  4. Using strategy rm

  5. Applied add-log-exp_binary642.8

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

  6. Applied add-log-exp_binary642.8

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

  7. Applied add-log-exp_binary642.8

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

  8. Applied sum-log_binary642.8

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

  9. Applied sum-log_binary640.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2021196 
(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))