Expression 4, p15

Average Error: 0.0 → 0.0

Time: 11.7s

Precision: binary64

Cost: 13504

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

Math

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

↓

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

FPCore

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

↓

(FPCore (a b)
 :precision binary64
 (+ (* b (* a 2.0)) (+ (pow b 2.0) (pow a 2.0))))

C

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

↓

double code(double a, double b) {
	return (b * (a * 2.0)) + (pow(b, 2.0) + pow(a, 2.0));
}

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 = (b * (a * 2.0d0)) + ((b ** 2.0d0) + (a ** 2.0d0))
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 (b * (a * 2.0)) + (Math.pow(b, 2.0) + Math.pow(a, 2.0));
}

Python

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

↓

def code(a, b):
	return (b * (a * 2.0)) + (math.pow(b, 2.0) + math.pow(a, 2.0))

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

code[a_, b_] := N[(N[(b * N[(a * 2.0), $MachinePrecision]), $MachinePrecision] + N[(N[Power[b, 2.0], $MachinePrecision] + N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	0.0
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 0.0

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

2. Taylor expanded in a around 0 0.0

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

3. Simplified 0.0

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

   [Start] 0.0	\[ 2 \cdot \left(a \cdot b\right) + \left({b}^{2} + {a}^{2}\right) \]
   rational.json-simplify-2 [=>] 0.0	\[ 2 \cdot \color{blue}{\left(b \cdot a\right)} + \left({b}^{2} + {a}^{2}\right) \]
   rational.json-simplify-43 [=>] 0.0	\[ \color{blue}{b \cdot \left(a \cdot 2\right)} + \left({b}^{2} + {a}^{2}\right) \]

4. Final simplification 0.0

   \[\leadsto b \cdot \left(a \cdot 2\right) + \left({b}^{2} + {a}^{2}\right) \]

Alternatives

Alternative 1
Error	0.0
Cost	448

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

Alternative 2
Error	60.6
Cost	320

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

Reproduce

herbie shell --seed 2023077 
(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)))