Average Error: 3.7 → 1.4
Time: 6.1s
Precision: binary64
\[-14 \leq a \land a \leq -13 \land -3 \leq b \land b \leq -2 \land 3 \leq c \land c \leq 3.5 \land 12.5 \leq d \land d \leq 13.5\]
\[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2\]
↓
\[\log \left(e^{d} \cdot \left(e^{b + c} \cdot e^{a}\right)\right) \cdot 2\]
\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2
↓
\log \left(e^{d} \cdot \left(e^{b + c} \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
 (* (log (* (exp d) (* (exp (+ b c)) (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 log(exp(d) * (exp(b + c) * 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
Target3.9
Herbie1.4
\[\left(a + b\right) \cdot 2 + \left(c + d\right) \cdot 2\]

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

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

  3. Simplified2.8

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

  5. Applied add-log-exp_binary64_21632.8

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

  6. Applied add-log-exp_binary64_21632.8

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

  7. Applied sum-log_binary64_22152.8

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

  8. Applied add-log-exp_binary64_21632.8

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

  9. Applied sum-log_binary64_22151.4

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

  10. Final simplification1.4

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

Reproduce

herbie shell --seed 2021015 
(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))