Expanding a square

Average Error: 39.1 → 0.0

Time: 1.8s

Precision: binary64

Cost: 448

Math

\[\left(x + 1\right) \cdot \left(x + 1\right) - 1 \]

↓

\[x \cdot x - -2 \cdot x \]

FPCore

(FPCore (x) :precision binary64 (- (* (+ x 1.0) (+ x 1.0)) 1.0))

↓

(FPCore (x) :precision binary64 (- (* x x) (* -2.0 x)))

C

double code(double x) {
	return ((x + 1.0) * (x + 1.0)) - 1.0;
}

↓

double code(double x) {
	return (x * x) - (-2.0 * x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x + 1.0d0) * (x + 1.0d0)) - 1.0d0
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * x) - ((-2.0d0) * x)
end function

Java

public static double code(double x) {
	return ((x + 1.0) * (x + 1.0)) - 1.0;
}

↓

public static double code(double x) {
	return (x * x) - (-2.0 * x);
}

Python

def code(x):
	return ((x + 1.0) * (x + 1.0)) - 1.0

↓

def code(x):
	return (x * x) - (-2.0 * x)

Julia

function code(x)
	return Float64(Float64(Float64(x + 1.0) * Float64(x + 1.0)) - 1.0)
end

↓

function code(x)
	return Float64(Float64(x * x) - Float64(-2.0 * x))
end

MATLAB

function tmp = code(x)
	tmp = ((x + 1.0) * (x + 1.0)) - 1.0;
end

↓

function tmp = code(x)
	tmp = (x * x) - (-2.0 * x);
end

Wolfram

code[x_] := N[(N[(N[(x + 1.0), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

↓

code[x_] := N[(N[(x * x), $MachinePrecision] - N[(-2.0 * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.1

   \[\left(x + 1\right) \cdot \left(x + 1\right) - 1 \]

2. Simplified 0.0

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

   [Start] 39.1	\[ \left(x + 1\right) \cdot \left(x + 1\right) - 1 \]
   rational_best_oopsla_all_46_json_45_simplify-45 [=>] 39.1	\[ \color{blue}{\left(x + 1\right) \cdot \left(x + 1\right) + -1} \]
   rational_best_oopsla_all_46_json_45_simplify-35 [=>] 39.1	\[ \color{blue}{-1 + \left(x + 1\right) \cdot \left(x + 1\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-37 [=>] 39.1	\[ -1 + \color{blue}{\left(x \cdot \left(x + 1\right) + \left(x + 1\right) \cdot 1\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-52 [=>] 39.1	\[ -1 + \left(x \cdot \left(x + 1\right) + \color{blue}{\left(x + 1\right)}\right) \]
   rational_best_oopsla_all_46_json_45_simplify-82 [=>] 33.7	\[ \color{blue}{x \cdot \left(x + 1\right) + \left(-1 + \left(x + 1\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 33.7	\[ \color{blue}{\left(x + 1\right) \cdot x} + \left(-1 + \left(x + 1\right)\right) \]
   rational_best_oopsla_all_46_json_45_simplify-82 [=>] 0.0	\[ \left(x + 1\right) \cdot x + \color{blue}{\left(x + \left(-1 + 1\right)\right)} \]
   metadata-eval [=>] 0.0	\[ \left(x + 1\right) \cdot x + \left(x + \color{blue}{0}\right) \]
   rational_best_oopsla_all_46_json_45_simplify-85 [=>] 0.0	\[ \left(x + 1\right) \cdot x + \color{blue}{x} \]
   rational_best_oopsla_all_46_json_45_simplify-52 [<=] 0.0	\[ \left(x + 1\right) \cdot x + \color{blue}{x \cdot 1} \]
   rational_best_oopsla_all_46_json_45_simplify-23 [=>] 0.0	\[ \color{blue}{x \cdot \left(\left(x + 1\right) + 1\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.0	\[ x \cdot \color{blue}{\left(1 + \left(x + 1\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-82 [=>] 0.0	\[ x \cdot \color{blue}{\left(x + \left(1 + 1\right)\right)} \]
   metadata-eval [=>] 0.0	\[ x \cdot \left(x + \color{blue}{2}\right) \]

3. Applied egg-rr 0.0

   \[\leadsto \color{blue}{x \cdot x - -2 \cdot x} \]

4. Final simplification 0.0

   \[\leadsto x \cdot x - -2 \cdot x \]

Alternatives

Alternative 1
Error	0.0
Cost	320

\[x \cdot \left(x + 2\right) \]

Alternative 2
Error	21.4
Cost	192

\[x + x \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x)
  :name "Expanding a square"
  :precision binary64
  (- (* (+ x 1.0) (+ x 1.0)) 1.0))