Expression 1, p15

Average Error: 0.61% → 0.38%

Time: 11.0s

Precision: binary64

Cost: 576

\[\left(\left(\left(\left(\left(\left(\left(\left(1 \leq a \land a \leq 2\right) \land 2 \leq b\right) \land b \leq 4\right) \land 4 \leq c\right) \land c \leq 8\right) \land 8 \leq d\right) \land d \leq 16\right) \land 16 \leq e\right) \land e \leq 32\]

Math

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

↓

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

FPCore

(FPCore (a b c d e) :precision binary64 (+ (+ (+ (+ e d) c) b) a))

↓

(FPCore (a b c d e) :precision binary64 (+ (+ d b) (+ e (+ c a))))

C

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) {
	return (d + b) + (e + (c + a));
}

Fortran

real(8) function code(a, b, c, d, e)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8), intent (in) :: e
    code = (((e + d) + c) + b) + a
end function

↓

real(8) function code(a, b, c, d, e)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8), intent (in) :: e
    code = (d + b) + (e + (c + a))
end function

Java

public static double code(double a, double b, double c, double d, double e) {
	return (((e + d) + c) + b) + a;
}

↓

public static double code(double a, double b, double c, double d, double e) {
	return (d + b) + (e + (c + a));
}

Python

def code(a, b, c, d, e):
	return (((e + d) + c) + b) + a

↓

def code(a, b, c, d, e):
	return (d + b) + (e + (c + a))

Julia

function code(a, b, c, d, e)
	return Float64(Float64(Float64(Float64(e + d) + c) + b) + a)
end

↓

function code(a, b, c, d, e)
	return Float64(Float64(d + b) + Float64(e + Float64(c + a)))
end

MATLAB

function tmp = code(a, b, c, d, e)
	tmp = (((e + d) + c) + b) + a;
end

↓

function tmp = code(a, b, c, d, e)
	tmp = (d + b) + (e + (c + a));
end

Wolfram

code[a_, b_, c_, d_, e_] := N[(N[(N[(N[(e + d), $MachinePrecision] + c), $MachinePrecision] + b), $MachinePrecision] + a), $MachinePrecision]

↓

code[a_, b_, c_, d_, e_] := N[(N[(d + b), $MachinePrecision] + N[(e + N[(c + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	0.61%
Target	0.37%
Herbie	0.38%

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

Derivation

1. Initial program 0.61

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

2. Taylor expanded in e around 0 0.57

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

3. Simplified 0.38

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

   [Start] 0.57	\[ c + \left(a + \left(d + \left(b + e\right)\right)\right) \]
   associate-+r+ [=>] 0.46	\[ \color{blue}{\left(c + a\right) + \left(d + \left(b + e\right)\right)} \]
   +-commutative [=>] 0.46	\[ \color{blue}{\left(d + \left(b + e\right)\right) + \left(c + a\right)} \]
   associate-+l+ [<=] 0.58	\[ \color{blue}{\left(\left(d + \left(b + e\right)\right) + c\right) + a} \]
   associate-+r+ [=>] 0.57	\[ \left(\color{blue}{\left(\left(d + b\right) + e\right)} + c\right) + a \]
   associate-+r+ [<=] 0.52	\[ \color{blue}{\left(\left(d + b\right) + \left(e + c\right)\right)} + a \]
   associate-+l+ [=>] 0.44	\[ \color{blue}{\left(d + b\right) + \left(\left(e + c\right) + a\right)} \]
   associate-+l+ [=>] 0.38	\[ \left(d + b\right) + \color{blue}{\left(e + \left(c + a\right)\right)} \]

4. Final simplification 0.38

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

Alternatives

Alternative 1
Error	0.52%
Cost	576

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

Alternative 2
Error	0.47%
Cost	576

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

Alternative 3
Error	0.38%
Cost	576

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

Alternative 4
Error	74.32%
Cost	448

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

Alternative 5
Error	76.81%
Cost	320

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

Alternative 6
Error	78.84%
Cost	192

\[d + e \]

Alternative 7
Error	81.1%
Cost	64

\[e \]

Reproduce

herbie shell --seed 2023125 
(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))