sqrt times

Average Error: 0.5 → 0.3

Time: 2.1s

Precision: binary64

Cost: 7424

Math

\[\sqrt{x - 1} \cdot \sqrt{x} \]

↓

\[x - \left(\left(0.5 + 0.125 \cdot \frac{1}{x}\right) + 0.0625 \cdot \frac{1}{{x}^{2}}\right) \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (- x (+ (+ 0.5 (* 0.125 (/ 1.0 x))) (* 0.0625 (/ 1.0 (pow x 2.0))))))

C

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

↓

double code(double x) {
	return x - ((0.5 + (0.125 * (1.0 / x))) + (0.0625 * (1.0 / pow(x, 2.0))));
}

Fortran

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

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = x - ((0.5d0 + (0.125d0 * (1.0d0 / x))) + (0.0625d0 * (1.0d0 / (x ** 2.0d0))))
end function

Java

public static double code(double x) {
	return Math.sqrt((x - 1.0)) * Math.sqrt(x);
}

↓

public static double code(double x) {
	return x - ((0.5 + (0.125 * (1.0 / x))) + (0.0625 * (1.0 / Math.pow(x, 2.0))));
}

Python

def code(x):
	return math.sqrt((x - 1.0)) * math.sqrt(x)

↓

def code(x):
	return x - ((0.5 + (0.125 * (1.0 / x))) + (0.0625 * (1.0 / math.pow(x, 2.0))))

Julia

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

↓

function code(x)
	return Float64(x - Float64(Float64(0.5 + Float64(0.125 * Float64(1.0 / x))) + Float64(0.0625 * Float64(1.0 / (x ^ 2.0)))))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = x - ((0.5 + (0.125 * (1.0 / x))) + (0.0625 * (1.0 / (x ^ 2.0))));
end

Wolfram

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

↓

code[x_] := N[(x - N[(N[(0.5 + N[(0.125 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0625 * N[(1.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.5

   \[\sqrt{x - 1} \cdot \sqrt{x} \]

2. Taylor expanded in x around inf 0.3

   \[\leadsto \color{blue}{x - \left(0.5 + \left(0.0625 \cdot \frac{1}{{x}^{2}} + 0.125 \cdot \frac{1}{x}\right)\right)} \]

3. Simplified 0.3

   \[\leadsto \color{blue}{x - \left(\left(0.5 + 0.125 \cdot \frac{1}{x}\right) + 0.0625 \cdot \frac{1}{{x}^{2}}\right)} \]
   Proof

   [Start] 0.3	\[ x - \left(0.5 + \left(0.0625 \cdot \frac{1}{{x}^{2}} + 0.125 \cdot \frac{1}{x}\right)\right) \]
   rational_best_oopsla_all_46_json_45_simplify-82 [=>] 0.3	\[ x - \color{blue}{\left(0.0625 \cdot \frac{1}{{x}^{2}} + \left(0.5 + 0.125 \cdot \frac{1}{x}\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.3	\[ x - \color{blue}{\left(\left(0.5 + 0.125 \cdot \frac{1}{x}\right) + 0.0625 \cdot \frac{1}{{x}^{2}}\right)} \]

4. Final simplification 0.3

   \[\leadsto x - \left(\left(0.5 + 0.125 \cdot \frac{1}{x}\right) + 0.0625 \cdot \frac{1}{{x}^{2}}\right) \]

Alternatives

Alternative 1
Error	0.4
Cost	576

\[x - \left(0.5 + 0.125 \cdot \frac{1}{x}\right) \]

Alternative 2
Error	0.6
Cost	192

\[x - 0.5 \]

Alternative 3
Error	1.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1.0)) (sqrt x)))