Average Error: 0.4 → 0.2
Time: 7.3s
Precision: binary64
Cost: 576
\[\left(\left(\left(\left(\left(\left(\left(\left(1 \leq a \land a \leq 2\right) \land 2 \leq b\right) \land b \leq 4\right) \land 4 \leq c\right) \land c \leq 8\right) \land 8 \leq d\right) \land d \leq 16\right) \land 16 \leq e\right) \land e \leq 32\]
\[\left(\left(\left(e + d\right) + c\right) + b\right) + a \]
↓
\[e + \left(c + \left(d + \left(b + a\right)\right)\right) \]
(FPCore (a b c d e) :precision binary64 (+ (+ (+ (+ e d) c) b) a))
↓
(FPCore (a b c d e) :precision binary64 (+ e (+ c (+ d (+ b 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) {
	return e + (c + (d + (b + a)));
}

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 = e + (c + (d + (b + a)))
end function

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

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

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(e + Float64(c + Float64(d + Float64(b + a))))
end

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

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[(e + N[(c + N[(d + N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\left(\left(\left(e + d\right) + c\right) + b\right) + a
↓
e + \left(c + \left(d + \left(b + a\right)\right)\right)

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.4
Target0.2
Herbie0.2
\[\left(d + \left(c + \left(a + b\right)\right)\right) + e \]

Derivation

  1. Initial program 0.4

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

  2. Taylor expanded in e around 0 0.4

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

  3. Taylor expanded in c around 0 0.4

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

  4. Simplified0.2

    \[\leadsto \color{blue}{e + \left(c + \left(d + \left(b + a\right)\right)\right)} \]
    Proof
    (+.f64 e (+.f64 c (+.f64 d (+.f64 b a)))): 0 points increase in error, 0 points decrease in error
    (+.f64 e (+.f64 c (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 d b) a)))): 13 points increase in error, 7 points decrease in error
    (+.f64 e (+.f64 c (Rewrite<= +-commutative_binary64 (+.f64 a (+.f64 d b))))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 c (+.f64 a (+.f64 d b))) e)): 0 points increase in error, 0 points decrease in error
    (Rewrite=> associate-+l+_binary64 (+.f64 c (+.f64 (+.f64 a (+.f64 d b)) e))): 48 points increase in error, 32 points decrease in error
    (+.f64 c (Rewrite<= associate-+r+_binary64 (+.f64 a (+.f64 (+.f64 d b) e)))): 45 points increase in error, 25 points decrease in error
    (+.f64 c (+.f64 a (Rewrite<= associate-+r+_binary64 (+.f64 d (+.f64 b e))))): 20 points increase in error, 12 points decrease in error

  5. Final simplification0.2

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

Alternatives

Alternative 1
Error0.3
Cost576
\[a + \left(d + \left(c + \left(e + b\right)\right)\right) \]
Alternative 2
Error0.4
Cost576
\[c + \left(b + \left(e + \left(d + a\right)\right)\right) \]
Alternative 3
Error47.6
Cost448
\[b + \left(e + \left(c + d\right)\right) \]
Alternative 4
Error49.2
Cost320
\[d + \left(e + c\right) \]
Alternative 5
Error50.5
Cost192
\[e + d \]
Alternative 6
Error51.9
Cost64
\[e \]

Error

Reproduce

herbie shell --seed 2022308 
(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))