Expression 4, p15

Percentage Accurate: 100.0% → 100.0%

Time: 4.2s

Precision: binary64

Cost: 704

• Unsound rule application detected in e-graph. Results may not be sound.

\[\left(5 \leq a \land a \leq 10\right) \land \left(0 \leq b \land b \leq 0.001\right)\]

\[ \begin{array}{c}[a, b] = \mathsf{sort}([a, b])\\ \end{array} \]

Math

\[\left(a + b\right) \cdot \left(a + b\right) \]

↓

\[a \cdot \left(b + \left(b + a\right)\right) + b \cdot b \]

FPCore

(FPCore (a b) :precision binary64 (* (+ a b) (+ a b)))

↓

(FPCore (a b) :precision binary64 (+ (* a (+ b (+ b a))) (* b b)))

C

double code(double a, double b) {
	return (a + b) * (a + b);
}

↓

double code(double a, double b) {
	return (a * (b + (b + a))) + (b * b);
}

Fortran

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

↓

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

Java

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

↓

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

Python

def code(a, b):
	return (a + b) * (a + b)

↓

def code(a, b):
	return (a * (b + (b + a))) + (b * b)

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

code[a_, b_] := N[(N[(a + b), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_] := N[(N[(a * N[(b + N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]

Target

Original	100.0%
Target	100.0%
Herbie	100.0%

\[\left(\left(b \cdot a + b \cdot b\right) + b \cdot a\right) + a \cdot a \]

Derivation

1. Initial program 100.0%

   \[\left(a + b\right) \cdot \left(a + b\right) \]

2. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\left(a \cdot \left(a + b\right)\right) \cdot \left(a \cdot \left(a + b\right)\right) - \left(b \cdot \left(a + b\right)\right) \cdot \left(b \cdot \left(a + b\right)\right)}{a \cdot \left(a + b\right) - b \cdot \left(a + b\right)}} \]
   Step-by-step derivation

   [Start] 100.0	\[ \left(a + b\right) \cdot \left(a + b\right) \]
   distribute-rgt-in [=>] 100.0	\[ \color{blue}{a \cdot \left(a + b\right) + b \cdot \left(a + b\right)} \]
   flip-+ [=>] 99.8	\[ \color{blue}{\frac{\left(a \cdot \left(a + b\right)\right) \cdot \left(a \cdot \left(a + b\right)\right) - \left(b \cdot \left(a + b\right)\right) \cdot \left(b \cdot \left(a + b\right)\right)}{a \cdot \left(a + b\right) - b \cdot \left(a + b\right)}} \]

3. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left(a \cdot \left(a + b\right) + a \cdot b\right) + b \cdot b} \]
   Step-by-step derivation

   [Start] 99.8	\[ \frac{\left(a \cdot \left(a + b\right)\right) \cdot \left(a \cdot \left(a + b\right)\right) - \left(b \cdot \left(a + b\right)\right) \cdot \left(b \cdot \left(a + b\right)\right)}{a \cdot \left(a + b\right) - b \cdot \left(a + b\right)} \]
   flip-+ [<=] 100.0	\[ \color{blue}{a \cdot \left(a + b\right) + b \cdot \left(a + b\right)} \]
   distribute-rgt-in [=>] 100.0	\[ a \cdot \left(a + b\right) + \color{blue}{\left(a \cdot b + b \cdot b\right)} \]
   associate-+r+ [=>] 100.0	\[ \color{blue}{\left(a \cdot \left(a + b\right) + a \cdot b\right) + b \cdot b} \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left(b + \left(a + b\right)\right) \cdot a} + b \cdot b \]
   Step-by-step derivation

   [Start] 100.0	\[ \left(a \cdot \left(a + b\right) + a \cdot b\right) + b \cdot b \]
   distribute-lft-out [=>] 100.0	\[ \color{blue}{a \cdot \left(\left(a + b\right) + b\right)} + b \cdot b \]
   *-commutative [=>] 100.0	\[ \color{blue}{\left(\left(a + b\right) + b\right) \cdot a} + b \cdot b \]
   +-commutative [=>] 100.0	\[ \color{blue}{\left(b + \left(a + b\right)\right)} \cdot a + b \cdot b \]

5. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	99.7%
Cost	448

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

Alternative 2
Accuracy	100.0%
Cost	448

\[\left(b + a\right) \cdot \left(b + a\right) \]

Alternative 3
Accuracy	98.7%
Cost	192

\[b \cdot b \]

Reproduce

herbie shell --seed 2023161 
(FPCore (a b)
  :name "Expression 4, p15"
  :precision binary64
  :pre (and (and (<= 5.0 a) (<= a 10.0)) (and (<= 0.0 b) (<= b 0.001)))

  :herbie-target
  (+ (+ (+ (* b a) (* b b)) (* b a)) (* a a))

  (* (+ a b) (+ a b)))