sqrt times

Average Error: 0.5 → 0.3

Time: 4.9s

Precision: binary64

Cost: 832

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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 = ((-0.5d0) + ((-0.125d0) / x)) + (x + ((-0.0625d0) / (x * x)))
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 (-0.5 + (-0.125 / x)) + (x + (-0.0625 / (x * x)));
}

Python

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

↓

def code(x):
	return (-0.5 + (-0.125 / x)) + (x + (-0.0625 / (x * x)))

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

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

   [Start] 0.3	\[ x - \left(0.5 + \left(0.0625 \cdot \frac{1}{{x}^{2}} + 0.125 \cdot \frac{1}{x}\right)\right) \]
   associate--r+ [=>] 0.3	\[ \color{blue}{\left(x - 0.5\right) - \left(0.0625 \cdot \frac{1}{{x}^{2}} + 0.125 \cdot \frac{1}{x}\right)} \]
   sub-neg [=>] 0.3	\[ \color{blue}{\left(x + \left(-0.5\right)\right)} - \left(0.0625 \cdot \frac{1}{{x}^{2}} + 0.125 \cdot \frac{1}{x}\right) \]
   metadata-eval [=>] 0.3	\[ \left(x + \color{blue}{-0.5}\right) - \left(0.0625 \cdot \frac{1}{{x}^{2}} + 0.125 \cdot \frac{1}{x}\right) \]
   +-commutative [=>] 0.3	\[ \left(x + -0.5\right) - \color{blue}{\left(0.125 \cdot \frac{1}{x} + 0.0625 \cdot \frac{1}{{x}^{2}}\right)} \]
   associate-*r/ [=>] 0.3	\[ \left(x + -0.5\right) - \left(\color{blue}{\frac{0.125 \cdot 1}{x}} + 0.0625 \cdot \frac{1}{{x}^{2}}\right) \]
   metadata-eval [=>] 0.3	\[ \left(x + -0.5\right) - \left(\frac{\color{blue}{0.125}}{x} + 0.0625 \cdot \frac{1}{{x}^{2}}\right) \]
   associate-*r/ [=>] 0.3	\[ \left(x + -0.5\right) - \left(\frac{0.125}{x} + \color{blue}{\frac{0.0625 \cdot 1}{{x}^{2}}}\right) \]
   metadata-eval [=>] 0.3	\[ \left(x + -0.5\right) - \left(\frac{0.125}{x} + \frac{\color{blue}{0.0625}}{{x}^{2}}\right) \]
   unpow2 [=>] 0.3	\[ \left(x + -0.5\right) - \left(\frac{0.125}{x} + \frac{0.0625}{\color{blue}{x \cdot x}}\right) \]

4. Applied egg-rr 0.3

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

5. Applied egg-rr 0.3

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

6. Simplified 0.3

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

   [Start] 0.3	\[ \left(-0.5 + \frac{-0.125}{x}\right) - \left(\frac{0.0625}{x \cdot x} - x\right) \]
   +-commutative [=>] 0.3	\[ \color{blue}{\left(\frac{-0.125}{x} + -0.5\right)} - \left(\frac{0.0625}{x \cdot x} - x\right) \]

7. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.3
Cost	832

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

Alternative 2
Error	0.4
Cost	448

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

Alternative 3
Error	0.6
Cost	192

\[x + -0.5 \]

Alternative 4
Error	1.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023010 
(FPCore (x)
  :name "sqrt times"
  :precision binary64
  (* (sqrt (- x 1.0)) (sqrt x)))