2isqrt (example 3.6)

Average Error: 30.65% → 0.41%

Time: 12.3s

Precision: binary64

Cost: 26240

Math

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

↓

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

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))) (+ (sqrt x) (sqrt (+ 1.0 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))) / (sqrt(x) + sqrt((1.0 + 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))) / (Math.sqrt(x) + Math.sqrt((1.0 + 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))) / (math.sqrt(x) + math.sqrt((1.0 + 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(sqrt(x) + sqrt(Float64(1.0 + 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))) / (sqrt(x) + sqrt((1.0 + 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[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	30.65%
Target	1.02%
Herbie	0.41%

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

Derivation

1. Initial program 30.65

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

2. Applied egg-rr 8.65

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

3. Simplified 8.65

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

   [Start] 8.65	\[ \frac{1 + \left(x - x\right)}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
   +-commutative [=>] 8.65	\[ \frac{\color{blue}{\left(x - x\right) + 1}}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
   +-inverses [=>] 8.65	\[ \frac{\color{blue}{0} + 1}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
   metadata-eval [=>] 8.65	\[ \frac{\color{blue}{1}}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
   +-commutative [=>] 8.65	\[ \frac{1}{\sqrt{x + x \cdot x} \cdot \color{blue}{\left(\sqrt{1 + x} + \sqrt{x}\right)}} \]

4. Applied egg-rr 1.07

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

5. Applied egg-rr 34.6

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(x, \sqrt{x}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)}\right)} - 1} \]

6. Simplified 0.41

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

   [Start] 34.6	\[ e^{\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(x, \sqrt{x}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)}\right)} - 1 \]
   expm1-def [=>] 4.47	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(x, \sqrt{x}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)}\right)\right)} \]
   expm1-log1p [=>] 1.03	\[ \color{blue}{\frac{1}{\mathsf{hypot}\left(x, \sqrt{x}\right) \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)}} \]
   associate-/r* [=>] 0.41	\[ \color{blue}{\frac{\frac{1}{\mathsf{hypot}\left(x, \sqrt{x}\right)}}{\sqrt{x} + \sqrt{1 + x}}} \]

7. Final simplification 0.41

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

Alternatives

Alternative 1
Error	1.08%
Cost	26948

\[\begin{array}{l} t_0 := \sqrt{1 + x}\\ t_1 := \frac{-1}{t_0}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + t_1 \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{\left(\sqrt{x} + t_0\right) \cdot \left(x + 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} + t_1\\ \end{array} \]

Alternative 2
Error	0.68%
Cost	26820

\[\begin{array}{l} t_0 := \sqrt{1 + x}\\ t_1 := \frac{-1}{t_0}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + t_1 \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{1}{x}}{\sqrt{x} + t_0}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} + t_1\\ \end{array} \]

Alternative 3
Error	0.7%
Cost	26756

\[\begin{array}{l} t_0 := \frac{-1}{\sqrt{1 + x}}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + t_0 \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\sqrt{x} \cdot 0.5}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} + t_0\\ \end{array} \]

Alternative 4
Error	0.61%
Cost	13760

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

Alternative 5
Error	0.69%
Cost	13380

\[\begin{array}{l} \mathbf{if}\;x \leq 60000000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\sqrt{x} \cdot 0.5}{1 + x}\\ \end{array} \]

Alternative 6
Error	1.55%
Cost	7236

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

Alternative 7
Error	31.24%
Cost	7172

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

Alternative 8
Error	32.11%
Cost	6916

\[\begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{x + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 - {x}^{-0.5}\right)\\ \end{array} \]

Alternative 9
Error	47.35%
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.58:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + 0.5}\\ \end{array} \]

Alternative 10
Error	47.91%
Cost	6720

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

Alternative 11
Error	92.57%
Cost	320

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

Alternative 12
Error	98.06%
Cost	64

\[-1 \]

Alternative 13
Error	94.23%
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2023103 
(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)))))