2isqrt (example 3.6)

Average Error: 20.0 → 0.3

Time: 12.1s

Precision: binary64

Cost: 13696

Math

\[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]

↓

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

FPCore

(FPCore (x) :precision binary64 (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))

↓

(FPCore (x)
 :precision binary64
 (/ (/ 1.0 (+ 1.0 x)) (+ (sqrt x) (* x (pow (+ 1.0 x) -0.5)))))

C

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

↓

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

Fortran

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

↓

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

Java

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

↓

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

Python

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

↓

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

Julia

function code(x)
	return Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / sqrt(Float64(x + 1.0))))
end

↓

function code(x)
	return Float64(Float64(1.0 / Float64(1.0 + x)) / Float64(sqrt(x) + Float64(x * (Float64(1.0 + x) ^ -0.5))))
end

MATLAB

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

↓

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

Wolfram

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

↓

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

Target

Original	20.0
Target	0.6
Herbie	0.3

\[\frac{1}{\left(x + 1\right) \cdot \sqrt{x} + x \cdot \sqrt{x + 1}} \]

Derivation

1. Initial program 20.0

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]

2. Applied egg-rr 20.1

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

3. Simplified 20.1

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

   [Start] 20.1	\[ \left(\frac{1}{x} + \frac{-1}{1 + x}\right) \cdot \frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
   associate-*r/ [=>] 20.1	\[ \color{blue}{\frac{\left(\frac{1}{x} + \frac{-1}{1 + x}\right) \cdot 1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}} \]
   *-rgt-identity [=>] 20.1	\[ \frac{\color{blue}{\frac{1}{x} + \frac{-1}{1 + x}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]

4. Applied egg-rr 19.5

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

5. Simplified 5.0

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

   [Start] 19.5	\[ \frac{\frac{x + \left(-1 - x\right)}{x \cdot \left(-1 - x\right)}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
   associate-/r* [=>] 19.5	\[ \frac{\color{blue}{\frac{\frac{x + \left(-1 - x\right)}{x}}{-1 - x}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
   +-commutative [=>] 19.5	\[ \frac{\frac{\frac{\color{blue}{\left(-1 - x\right) + x}}{x}}{-1 - x}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
   associate-+l- [=>] 5.0	\[ \frac{\frac{\frac{\color{blue}{-1 - \left(x - x\right)}}{x}}{-1 - x}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
   +-inverses [=>] 5.0	\[ \frac{\frac{\frac{-1 - \color{blue}{0}}{x}}{-1 - x}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
   metadata-eval [=>] 5.0	\[ \frac{\frac{\frac{\color{blue}{-1}}{x}}{-1 - x}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]

6. Applied egg-rr 0.5

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

7. Simplified 0.7

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

   [Start] 0.5	\[ \frac{-1}{x} \cdot \left(\frac{-1}{x + 1} \cdot \frac{1}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}\right) \]
   *-commutative [=>] 0.5	\[ \frac{-1}{x} \cdot \color{blue}{\left(\frac{1}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}} \cdot \frac{-1}{x + 1}\right)} \]
   associate-*r* [=>] 0.5	\[ \color{blue}{\left(\frac{-1}{x} \cdot \frac{1}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}\right) \cdot \frac{-1}{x + 1}} \]
   associate-*l/ [=>] 0.4	\[ \color{blue}{\frac{-1 \cdot \frac{1}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}}{x}} \cdot \frac{-1}{x + 1} \]
   associate-*r/ [=>] 0.4	\[ \frac{\color{blue}{\frac{-1 \cdot 1}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}}}{x} \cdot \frac{-1}{x + 1} \]
   metadata-eval [=>] 0.4	\[ \frac{\frac{\color{blue}{-1}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}}{x} \cdot \frac{-1}{x + 1} \]
   associate-/r* [<=] 0.4	\[ \color{blue}{\frac{-1}{\left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right) \cdot x}} \cdot \frac{-1}{x + 1} \]
   associate-*l/ [=>] 0.4	\[ \color{blue}{\frac{-1 \cdot \frac{-1}{x + 1}}{\left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right) \cdot x}} \]
   associate-*r/ [=>] 0.4	\[ \frac{\color{blue}{\frac{-1 \cdot -1}{x + 1}}}{\left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right) \cdot x} \]
   metadata-eval [=>] 0.4	\[ \frac{\frac{\color{blue}{1}}{x + 1}}{\left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right) \cdot x} \]
   associate-/r* [<=] 0.7	\[ \color{blue}{\frac{1}{\left(x + 1\right) \cdot \left(\left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right) \cdot x\right)}} \]
   +-commutative [=>] 0.7	\[ \frac{1}{\color{blue}{\left(1 + x\right)} \cdot \left(\left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right) \cdot x\right)} \]
   *-commutative [=>] 0.7	\[ \frac{1}{\left(1 + x\right) \cdot \color{blue}{\left(x \cdot \left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right)\right)}} \]
   +-commutative [=>] 0.7	\[ \frac{1}{\left(1 + x\right) \cdot \left(x \cdot \left({x}^{-0.5} + {\color{blue}{\left(1 + x\right)}}^{-0.5}\right)\right)} \]

8. Applied egg-rr 0.3

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

9. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.6
Cost	26692

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 10^{-20}:\\ \;\;\;\;\frac{\frac{0.5}{x + -1}}{{x}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 2
Error	0.6
Cost	13696

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

Alternative 3
Error	1.3
Cost	7172

\[\begin{array}{l} \mathbf{if}\;x \leq 1.2:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 + x\right) \cdot \left(\sqrt{x} \cdot 2\right)}\\ \end{array} \]

Alternative 4
Error	1.0
Cost	7108

\[\begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;-1 + \left({x}^{-0.5} + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \left(0.5 \cdot \sqrt{\frac{1}{x}}\right)\\ \end{array} \]

Alternative 5
Error	1.0
Cost	7108

\[\begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;-1 + \left({x}^{-0.5} + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sqrt{x} \cdot -2} \cdot \frac{1}{x}\\ \end{array} \]

Alternative 6
Error	1.3
Cost	7108

\[\begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;-1 + \left({x}^{-0.5} + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 + x\right) \cdot \left(\sqrt{x} \cdot 2\right)}\\ \end{array} \]

Alternative 7
Error	20.8
Cost	7044

\[\begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{+76}:\\ \;\;\;\;-1 + \left({x}^{-0.5} + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-1 - {x}^{-0.5}\right)\\ \end{array} \]

Alternative 8
Error	21.3
Cost	6916

\[\begin{array}{l} \mathbf{if}\;x \leq 4:\\ \;\;\;\;-1 + {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-1 - {x}^{-0.5}\right)\\ \end{array} \]

Alternative 9
Error	32.0
Cost	6656

\[-1 + {x}^{-0.5} \]

Alternative 10
Error	62.6
Cost	6592

\[-{x}^{-0.5} \]

Reproduce

herbie shell --seed 2023032 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

  (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))