Expression, p6

Average Accuracy: 94.3% → 100.0%

Time: 17.6s

Precision: binary64

Cost: 576

\[\left(\left(\left(-14 \leq a \land a \leq -13\right) \land \left(-3 \leq b \land b \leq -2\right)\right) \land \left(3 \leq c \land c \leq 3.5\right)\right) \land \left(12.5 \leq d \land d \leq 13.5\right)\]

Math

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

↓

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

FPCore

(FPCore (a b c d) :precision binary64 (* (+ a (+ b (+ c d))) 2.0))

↓

(FPCore (a b c d) :precision binary64 (* (+ (+ d a) (+ c b)) 2.0))

C

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 ((d + a) + (c + b)) * 2.0;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = (a + (b + (c + d))) * 2.0d0
end function

↓

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = ((d + a) + (c + b)) * 2.0d0
end function

Java

public static double code(double a, double b, double c, double d) {
	return (a + (b + (c + d))) * 2.0;
}

↓

public static double code(double a, double b, double c, double d) {
	return ((d + a) + (c + b)) * 2.0;
}

Python

def code(a, b, c, d):
	return (a + (b + (c + d))) * 2.0

↓

def code(a, b, c, d):
	return ((d + a) + (c + b)) * 2.0

Julia

function code(a, b, c, d)
	return Float64(Float64(a + Float64(b + Float64(c + d))) * 2.0)
end

↓

function code(a, b, c, d)
	return Float64(Float64(Float64(d + a) + Float64(c + b)) * 2.0)
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = (a + (b + (c + d))) * 2.0;
end

↓

function tmp = code(a, b, c, d)
	tmp = ((d + a) + (c + b)) * 2.0;
end

Wolfram

code[a_, b_, c_, d_] := N[(N[(a + N[(b + N[(c + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]

↓

code[a_, b_, c_, d_] := N[(N[(N[(d + a), $MachinePrecision] + N[(c + b), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]

Target

Original	94.3%
Target	94.1%
Herbie	100.0%

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

Derivation

1. Initial program 94.3%

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

2. Applied egg-rr 93.8%

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

   [Start] 94.3	\[ \left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2 \]
   +-commutative [=>] 94.3	\[ \color{blue}{\left(\left(b + \left(c + d\right)\right) + a\right)} \cdot 2 \]
   flip-+ [=>] 93.8	\[ \color{blue}{\frac{\left(b + \left(c + d\right)\right) \cdot \left(b + \left(c + d\right)\right) - a \cdot a}{\left(b + \left(c + d\right)\right) - a}} \cdot 2 \]
   pow2 [=>] 93.8	\[ \frac{\color{blue}{{\left(b + \left(c + d\right)\right)}^{2}} - a \cdot a}{\left(b + \left(c + d\right)\right) - a} \cdot 2 \]
   associate--l+ [=>] 93.8	\[ \frac{{\left(b + \left(c + d\right)\right)}^{2} - a \cdot a}{\color{blue}{b + \left(\left(c + d\right) - a\right)}} \cdot 2 \]

3. Applied egg-rr 94.4%

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

   [Start] 93.8	\[ \frac{{\left(b + \left(c + d\right)\right)}^{2} - a \cdot a}{b + \left(\left(c + d\right) - a\right)} \cdot 2 \]
   unpow2 [=>] 93.8	\[ \frac{\color{blue}{\left(b + \left(c + d\right)\right) \cdot \left(b + \left(c + d\right)\right)} - a \cdot a}{b + \left(\left(c + d\right) - a\right)} \cdot 2 \]
   difference-of-squares [=>] 94.3	\[ \frac{\color{blue}{\left(\left(b + \left(c + d\right)\right) + a\right) \cdot \left(\left(b + \left(c + d\right)\right) - a\right)}}{b + \left(\left(c + d\right) - a\right)} \cdot 2 \]
   associate-+l+ [=>] 94.4	\[ \frac{\color{blue}{\left(b + \left(\left(c + d\right) + a\right)\right)} \cdot \left(\left(b + \left(c + d\right)\right) - a\right)}{b + \left(\left(c + d\right) - a\right)} \cdot 2 \]
   +-commutative [=>] 94.4	\[ \frac{\left(b + \left(\left(c + d\right) + a\right)\right) \cdot \left(\color{blue}{\left(\left(c + d\right) + b\right)} - a\right)}{b + \left(\left(c + d\right) - a\right)} \cdot 2 \]
   associate--l+ [=>] 94.4	\[ \frac{\left(b + \left(\left(c + d\right) + a\right)\right) \cdot \color{blue}{\left(\left(c + d\right) + \left(b - a\right)\right)}}{b + \left(\left(c + d\right) - a\right)} \cdot 2 \]

4. Simplified 99.3%

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

   [Start] 94.4	\[ \frac{\left(b + \left(\left(c + d\right) + a\right)\right) \cdot \left(\left(c + d\right) + \left(b - a\right)\right)}{b + \left(\left(c + d\right) - a\right)} \cdot 2 \]
   associate-+l+ [=>] 99.5	\[ \frac{\left(b + \color{blue}{\left(c + \left(d + a\right)\right)}\right) \cdot \left(\left(c + d\right) + \left(b - a\right)\right)}{b + \left(\left(c + d\right) - a\right)} \cdot 2 \]
   associate-+l+ [=>] 99.3	\[ \frac{\left(b + \left(c + \left(d + a\right)\right)\right) \cdot \color{blue}{\left(c + \left(d + \left(b - a\right)\right)\right)}}{b + \left(\left(c + d\right) - a\right)} \cdot 2 \]

5. Taylor expanded in b around 0 99.4%

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

6. Simplified 99.8%

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

   [Start] 99.4	\[ \frac{\left(b + \left(c + \left(d + a\right)\right)\right) \cdot \left(c + \left(d + \left(b - a\right)\right)\right)}{\left(c + \left(d + b\right)\right) - a} \cdot 2 \]
   associate-+r+ [=>] 99.4	\[ \frac{\left(b + \left(c + \left(d + a\right)\right)\right) \cdot \left(c + \left(d + \left(b - a\right)\right)\right)}{\color{blue}{\left(\left(c + d\right) + b\right)} - a} \cdot 2 \]
   associate-+r- [<=] 99.4	\[ \frac{\left(b + \left(c + \left(d + a\right)\right)\right) \cdot \left(c + \left(d + \left(b - a\right)\right)\right)}{\color{blue}{\left(c + d\right) + \left(b - a\right)}} \cdot 2 \]
   associate-+r+ [<=] 99.8	\[ \frac{\left(b + \left(c + \left(d + a\right)\right)\right) \cdot \left(c + \left(d + \left(b - a\right)\right)\right)}{\color{blue}{c + \left(d + \left(b - a\right)\right)}} \cdot 2 \]

7. Applied egg-rr 100.0%

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

   [Start] 99.8	\[ \frac{\left(b + \left(c + \left(d + a\right)\right)\right) \cdot \left(c + \left(d + \left(b - a\right)\right)\right)}{c + \left(d + \left(b - a\right)\right)} \cdot 2 \]
   associate-/l* [=>] 100.0	\[ \color{blue}{\frac{b + \left(c + \left(d + a\right)\right)}{\frac{c + \left(d + \left(b - a\right)\right)}{c + \left(d + \left(b - a\right)\right)}}} \cdot 2 \]
   flip3-+ [=>] 98.3	\[ \frac{\color{blue}{\frac{{b}^{3} + {\left(c + \left(d + a\right)\right)}^{3}}{b \cdot b + \left(\left(c + \left(d + a\right)\right) \cdot \left(c + \left(d + a\right)\right) - b \cdot \left(c + \left(d + a\right)\right)\right)}}}{\frac{c + \left(d + \left(b - a\right)\right)}{c + \left(d + \left(b - a\right)\right)}} \cdot 2 \]
   associate-/l/ [=>] 98.3	\[ \color{blue}{\frac{{b}^{3} + {\left(c + \left(d + a\right)\right)}^{3}}{\frac{c + \left(d + \left(b - a\right)\right)}{c + \left(d + \left(b - a\right)\right)} \cdot \left(b \cdot b + \left(\left(c + \left(d + a\right)\right) \cdot \left(c + \left(d + a\right)\right) - b \cdot \left(c + \left(d + a\right)\right)\right)\right)}} \cdot 2 \]
   *-inverses [=>] 98.3	\[ \frac{{b}^{3} + {\left(c + \left(d + a\right)\right)}^{3}}{\color{blue}{1} \cdot \left(b \cdot b + \left(\left(c + \left(d + a\right)\right) \cdot \left(c + \left(d + a\right)\right) - b \cdot \left(c + \left(d + a\right)\right)\right)\right)} \cdot 2 \]
   *-un-lft-identity [<=] 98.3	\[ \frac{{b}^{3} + {\left(c + \left(d + a\right)\right)}^{3}}{\color{blue}{b \cdot b + \left(\left(c + \left(d + a\right)\right) \cdot \left(c + \left(d + a\right)\right) - b \cdot \left(c + \left(d + a\right)\right)\right)}} \cdot 2 \]
   flip3-+ [<=] 100.0	\[ \color{blue}{\left(b + \left(c + \left(d + a\right)\right)\right)} \cdot 2 \]
   associate-+r+ [=>] 100.0	\[ \color{blue}{\left(\left(b + c\right) + \left(d + a\right)\right)} \cdot 2 \]
   +-commutative [=>] 100.0	\[ \color{blue}{\left(\left(d + a\right) + \left(b + c\right)\right)} \cdot 2 \]
   +-commutative [=>] 100.0	\[ \left(\left(d + a\right) + \color{blue}{\left(c + b\right)}\right) \cdot 2 \]

8. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	13.9%
Cost	708

\[\begin{array}{l} \mathbf{if}\;d + c \leq 16:\\ \;\;\;\;2 \cdot \left(\left(c + b\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(c + -1\right)\\ \end{array} \]

Alternative 2
Accuracy	94.3%
Cost	576

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

Alternative 3
Accuracy	95.7%
Cost	576

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

Alternative 4
Accuracy	12.0%
Cost	320

\[2 \cdot \left(c + -1\right) \]

Alternative 5
Accuracy	6.2%
Cost	192

\[b \cdot 2 \]

Alternative 6
Accuracy	11.7%
Cost	192

\[c \cdot 2 \]

Reproduce

herbie shell --seed 2023138 
(FPCore (a b c d)
  :name "Expression, p6"
  :precision binary64
  :pre (and (and (and (and (<= -14.0 a) (<= a -13.0)) (and (<= -3.0 b) (<= b -2.0))) (and (<= 3.0 c) (<= c 3.5))) (and (<= 12.5 d) (<= d 13.5)))

  :herbie-target
  (+ (* (+ a b) 2.0) (* (+ c d) 2.0))

  (* (+ a (+ b (+ c d))) 2.0))