Expression 1, p15

Average Error: 0.4 → 0.2

Time: 6.3s

Precision: binary64

Cost: 576

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

Math

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

↓

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

FPCore

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

↓

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

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

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

Python

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

↓

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

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(Float64(b + c) + Float64(a + d)) + e)
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 = ((b + c) + (a + d)) + e;
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[(N[(b + c), $MachinePrecision] + N[(a + d), $MachinePrecision]), $MachinePrecision] + e), $MachinePrecision]

Target

Original	0.4
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 0.4

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

2. Simplified 0.3

   \[\leadsto \color{blue}{\left(e + d\right) + \left(b + \left(c + a\right)\right)} \]
   Proof
   (+.f64 (+.f64 e d) (+.f64 b (+.f64 c a))): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 e d) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 b c) a))): 7 points increase in error, 10 points decrease in error
   (+.f64 (+.f64 e d) (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 c b)) a)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 e d) (+.f64 c b)) a)): 54 points increase in error, 23 points decrease in error
   (+.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 e d) c) b)) a): 39 points increase in error, 32 points decrease in error

3. Taylor expanded in b around 0 0.3

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

4. Applied egg-rr 1.2

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

5. Applied egg-rr 0.2

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

6. Final simplification 0.2

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

Alternatives

Alternative 1
Error	47.6
Cost	448

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

Alternative 2
Error	49.2
Cost	320

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

Alternative 3
Error	50.5
Cost	192

\[d + e \]

Alternative 4
Error	51.9
Cost	64

\[e \]

Reproduce

herbie shell --seed 2022317 
(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))