2isqrt (example 3.6)

Average Error: 30.19% → 0.31%

Time: 12.8s

Precision: binary64

Cost: 26436

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{1 + x} + \sqrt{x}\\ \mathbf{if}\;x \leq 2 \cdot 10^{+63}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(x, x, x\right)}}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{t_0}}{-0.5 - x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (sqrt (+ 1.0 x)) (sqrt x))))
   (if (<= x 2e+63)
     (/ (sqrt (/ 1.0 (fma x x x))) t_0)
     (/ (/ -1.0 t_0) (- -0.5 x)))))

C

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

↓

double code(double x) {
	double t_0 = sqrt((1.0 + x)) + sqrt(x);
	double tmp;
	if (x <= 2e+63) {
		tmp = sqrt((1.0 / fma(x, x, x))) / t_0;
	} else {
		tmp = (-1.0 / t_0) / (-0.5 - x);
	}
	return tmp;
}

Julia

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

↓

function code(x)
	t_0 = Float64(sqrt(Float64(1.0 + x)) + sqrt(x))
	tmp = 0.0
	if (x <= 2e+63)
		tmp = Float64(sqrt(Float64(1.0 / fma(x, x, x))) / t_0);
	else
		tmp = Float64(Float64(-1.0 / t_0) / Float64(-0.5 - x));
	end
	return tmp
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_] := Block[{t$95$0 = N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2e+63], N[(N[Sqrt[N[(1.0 / N[(x * x + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(-1.0 / t$95$0), $MachinePrecision] / N[(-0.5 - x), $MachinePrecision]), $MachinePrecision]]]

Target

Original	30.19%
Target	1.02%
Herbie	0.31%

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

Derivation

1. Split input into 2 regimes

2. if x < 2.00000000000000012e63

   1. Initial program 14.05

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

   2. Applied egg-rr 0.47

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

   3. Simplified 0.48

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

      [Start] 0.47	\[ \frac{1 + \left(x - x\right)}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
      associate-/r* [=>] 0.48	\[ \color{blue}{\frac{\frac{1 + \left(x - x\right)}{\sqrt{x + x \cdot x}}}{\sqrt{x} + \sqrt{1 + x}}} \]
      +-commutative [=>] 0.48	\[ \frac{\frac{\color{blue}{\left(x - x\right) + 1}}{\sqrt{x + x \cdot x}}}{\sqrt{x} + \sqrt{1 + x}} \]
      +-inverses [=>] 0.48	\[ \frac{\frac{\color{blue}{0} + 1}{\sqrt{x + x \cdot x}}}{\sqrt{x} + \sqrt{1 + x}} \]
      metadata-eval [=>] 0.48	\[ \frac{\frac{\color{blue}{1}}{\sqrt{x + x \cdot x}}}{\sqrt{x} + \sqrt{1 + x}} \]
      +-commutative [=>] 0.48	\[ \frac{\frac{1}{\sqrt{x + x \cdot x}}}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}} \]

   4. Applied egg-rr 0.31

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

   if 2.00000000000000012e63 < x

   1. Initial program 55.17

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

   2. Applied egg-rr 20.32

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

   3. Simplified 20.32

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

      [Start] 20.32	\[ \frac{1 + \left(x - x\right)}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
      associate-/r* [=>] 20.32	\[ \color{blue}{\frac{\frac{1 + \left(x - x\right)}{\sqrt{x + x \cdot x}}}{\sqrt{x} + \sqrt{1 + x}}} \]
      +-commutative [=>] 20.32	\[ \frac{\frac{\color{blue}{\left(x - x\right) + 1}}{\sqrt{x + x \cdot x}}}{\sqrt{x} + \sqrt{1 + x}} \]
      +-inverses [=>] 20.32	\[ \frac{\frac{\color{blue}{0} + 1}{\sqrt{x + x \cdot x}}}{\sqrt{x} + \sqrt{1 + x}} \]
      metadata-eval [=>] 20.32	\[ \frac{\frac{\color{blue}{1}}{\sqrt{x + x \cdot x}}}{\sqrt{x} + \sqrt{1 + x}} \]
      +-commutative [=>] 20.32	\[ \frac{\frac{1}{\sqrt{x + x \cdot x}}}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}} \]

   4. Taylor expanded in x around inf 0.31

      \[\leadsto \frac{\frac{1}{\color{blue}{0.5 + x}}}{\sqrt{1 + x} + \sqrt{x}} \]

   5. Simplified 0.31

      \[\leadsto \frac{\frac{1}{\color{blue}{x + 0.5}}}{\sqrt{1 + x} + \sqrt{x}} \]
      Proof

      [Start] 0.31	\[ \frac{\frac{1}{0.5 + x}}{\sqrt{1 + x} + \sqrt{x}} \]
      +-commutative [=>] 0.31	\[ \frac{\frac{1}{\color{blue}{x + 0.5}}}{\sqrt{1 + x} + \sqrt{x}} \]

   6. Applied egg-rr 0.31

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{-0.5 - x} \cdot -1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.31

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+63}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(x, x, x\right)}}}{\sqrt{1 + x} + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\sqrt{1 + x} + \sqrt{x}}}{-0.5 - x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.34%
Cost	26948

\[\begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{t_0} \leq 10^{-9}:\\ \;\;\;\;\frac{\frac{1}{x + 0.5}}{t_0 + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 2
Error	0.67%
Cost	26820

\[\begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{t_0} \leq 10^{-12}:\\ \;\;\;\;\frac{\frac{1}{x}}{t_0 + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 3
Error	7.75%
Cost	26692

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 10^{-12}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{1 + x}}{2 \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 4
Error	0.41%
Cost	26304

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

Alternative 5
Error	1.03%
Cost	26240

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

Alternative 6
Error	0.41%
Cost	20164

\[\begin{array}{l} t_0 := \sqrt{1 + x} + \sqrt{x}\\ \mathbf{if}\;x \leq 2 \cdot 10^{+63}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x + x \cdot x}}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{t_0}}{-0.5 - x}\\ \end{array} \]

Alternative 7
Error	0.58%
Cost	13892

\[\begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+30}:\\ \;\;\;\;\frac{\frac{1}{x + x \cdot x}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\sqrt{1 + x} + \sqrt{x}}}{-0.5 - x}\\ \end{array} \]

Alternative 8
Error	8.59%
Cost	7364

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

Alternative 9
Error	30.79%
Cost	7236

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

Alternative 10
Error	9.44%
Cost	7236

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

Alternative 11
Error	30.72%
Cost	7044

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

Alternative 12
Error	32.57%
Cost	6848

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

Alternative 13
Error	46.05%
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.65:\\ \;\;\;\;-1 + {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + 0.5}\\ \end{array} \]

Alternative 14
Error	33.04%
Cost	6784

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

Alternative 15
Error	48.58%
Cost	6528

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

Alternative 16
Error	92.59%
Cost	320

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

Alternative 17
Error	98.08%
Cost	64

\[-1 \]

Alternative 18
Error	94.21%
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2023121 
(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)))))