2isqrt (example 3.6)

Average Error: 20.2 → 0.3

Time: 11.6s

Precision: binary64

Cost: 32704

Math

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

↓

\[\frac{-1}{\mathsf{hypot}\left(x, \sqrt{x}\right)} \cdot \frac{-1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} \]

FPCore

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

↓

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

C

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

↓

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

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 / Math.hypot(x, Math.sqrt(x))) * (-1.0 / (Math.sqrt(x) + Math.hypot(1.0, Math.sqrt(x))));
}

Python

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

↓

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

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 / hypot(x, sqrt(x))) * Float64(-1.0 / Float64(sqrt(x) + hypot(1.0, sqrt(x)))))
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 / hypot(x, sqrt(x))) * (-1.0 / (sqrt(x) + hypot(1.0, sqrt(x))));
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[Sqrt[x ^ 2 + N[Sqrt[x], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[1.0 ^ 2 + N[Sqrt[x], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	20.2
Target	0.7
Herbie	0.3

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

Derivation

1. Initial program 20.2

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

2. Applied egg-rr 5.5

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

3. Simplified 5.5

   \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{x + x \cdot x}}}{\sqrt{1 + x} + \sqrt{x}}} \]
   Proof

   [Start] 5.5	\[ \frac{1 + \left(x - x\right)}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
   associate-/r* [=>] 5.5	\[ \color{blue}{\frac{\frac{1 + \left(x - x\right)}{\sqrt{x + x \cdot x}}}{\sqrt{x} + \sqrt{1 + x}}} \]
   +-commutative [=>] 5.5	\[ \frac{\frac{\color{blue}{\left(x - x\right) + 1}}{\sqrt{x + x \cdot x}}}{\sqrt{x} + \sqrt{1 + x}} \]
   +-inverses [=>] 5.5	\[ \frac{\frac{\color{blue}{0} + 1}{\sqrt{x + x \cdot x}}}{\sqrt{x} + \sqrt{1 + x}} \]
   metadata-eval [=>] 5.5	\[ \frac{\frac{\color{blue}{1}}{\sqrt{x + x \cdot x}}}{\sqrt{x} + \sqrt{1 + x}} \]
   +-commutative [=>] 5.5	\[ \frac{\frac{1}{\sqrt{x + x \cdot x}}}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}} \]

4. Applied egg-rr 0.3

   \[\leadsto \color{blue}{\frac{-1}{\mathsf{hypot}\left(x, \sqrt{x}\right)} \cdot \frac{-1}{\mathsf{hypot}\left(1, \sqrt{x}\right) + \sqrt{x}}} \]

5. Final simplification 0.3

   \[\leadsto \frac{-1}{\mathsf{hypot}\left(x, \sqrt{x}\right)} \cdot \frac{-1}{\sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)} \]

Alternatives

Alternative 1
Error	0.2
Cost	27076

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{1}{x}}{\sqrt{\frac{1}{x}} \cdot 1.5 + \sqrt{x} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \end{array} \]

Alternative 2
Error	0.3
Cost	26948

\[\begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{x + 0.5}}{\sqrt{x} + t_0}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \end{array} \]

Alternative 3
Error	0.4
Cost	26692

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 10^{-13}:\\ \;\;\;\;\frac{{x}^{-0.5}}{x \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \end{array} \]

Alternative 4
Error	0.4
Cost	13696

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

Alternative 5
Error	1.0
Cost	7364

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

Alternative 6
Error	1.0
Cost	7044

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

Alternative 7
Error	1.2
Cost	6916

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

Alternative 8
Error	30.3
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.6:\\ \;\;\;\;-1 + {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{0.5}{x \cdot x}\\ \end{array} \]

Alternative 9
Error	1.6
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;-1 + {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{{x}^{1.5}}\\ \end{array} \]

Alternative 10
Error	31.9
Cost	6528

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

Alternative 11
Error	59.2
Cost	320

\[\frac{1}{x + 0.5} \]

Alternative 12
Error	59.3
Cost	192

\[\frac{1}{x} \]

Alternative 13
Error	62.8
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023083 
(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)))))