2isqrt (example 3.6)

Average Error: 19.7 → 0.0

Time: 13.3s

Precision: binary64

Cost: 26372

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(x, x, x\right)\right)}^{-0.5}}{\sqrt{1 + x} + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x 5e+15)
   (/ (pow (fma x x x) -0.5) (+ (sqrt (+ 1.0 x)) (sqrt x)))
   (* 0.5 (pow x -1.5))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= 5e+15) {
		tmp = pow(fma(x, x, x), -0.5) / (sqrt((1.0 + x)) + sqrt(x));
	} else {
		tmp = 0.5 * pow(x, -1.5);
	}
	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)
	tmp = 0.0
	if (x <= 5e+15)
		tmp = Float64((fma(x, x, x) ^ -0.5) / Float64(sqrt(Float64(1.0 + x)) + sqrt(x)));
	else
		tmp = Float64(0.5 * (x ^ -1.5));
	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_] := If[LessEqual[x, 5e+15], N[(N[Power[N[(x * x + x), $MachinePrecision], -0.5], $MachinePrecision] / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision]]

Target

Original	19.7
Target	0.7
Herbie	0.0

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

Derivation

1. Split input into 2 regimes

2. if x < 5e15

   1. Initial program 1.5

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

   2. Applied egg-rr 0.3

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

   3. Simplified 0.3

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

      [Start] 0.3	\[ \frac{1 + \left(x - x\right)}{\sqrt{x + x \cdot x} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]
      associate-/r* [=>] 0.3	\[ \color{blue}{\frac{\frac{1 + \left(x - x\right)}{\sqrt{x + x \cdot x}}}{\sqrt{x} + \sqrt{1 + x}}} \]
      +-commutative [=>] 0.3	\[ \frac{\frac{\color{blue}{\left(x - x\right) + 1}}{\sqrt{x + x \cdot x}}}{\sqrt{x} + \sqrt{1 + x}} \]
      +-inverses [=>] 0.3	\[ \frac{\frac{\color{blue}{0} + 1}{\sqrt{x + x \cdot x}}}{\sqrt{x} + \sqrt{1 + x}} \]
      metadata-eval [=>] 0.3	\[ \frac{\frac{\color{blue}{1}}{\sqrt{x + x \cdot x}}}{\sqrt{x} + \sqrt{1 + x}} \]
      +-commutative [=>] 0.3	\[ \frac{\frac{1}{\sqrt{x + x \cdot x}}}{\color{blue}{\sqrt{1 + x} + \sqrt{x}}} \]

   4. Applied egg-rr 6.3

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{\left(\mathsf{fma}\left(x, x, x\right)\right)}^{-0.5}}{\sqrt{1 + x} + \sqrt{x}}\right)} - 1} \]

   5. Simplified 0.1

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

      [Start] 6.3	\[ e^{\mathsf{log1p}\left(\frac{{\left(\mathsf{fma}\left(x, x, x\right)\right)}^{-0.5}}{\sqrt{1 + x} + \sqrt{x}}\right)} - 1 \]
      expm1-def [=>] 4.6	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\mathsf{fma}\left(x, x, x\right)\right)}^{-0.5}}{\sqrt{1 + x} + \sqrt{x}}\right)\right)} \]
      expm1-log1p [=>] 0.1	\[ \color{blue}{\frac{{\left(\mathsf{fma}\left(x, x, x\right)\right)}^{-0.5}}{\sqrt{1 + x} + \sqrt{x}}} \]

   if 5e15 < x

   1. Initial program 40.2

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

   2. Applied egg-rr 60.0

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

   3. Simplified 60.0

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

      [Start] 60.0	\[ \frac{\frac{\frac{1}{x}}{x} - \frac{\frac{1}{1 + x}}{1 + x}}{\left({x}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right) \cdot \left(\frac{1}{1 + x} + \frac{1}{x}\right)} \]
      sub-neg [=>] 60.0	\[ \frac{\color{blue}{\frac{\frac{1}{x}}{x} + \left(-\frac{\frac{1}{1 + x}}{1 + x}\right)}}{\left({x}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right) \cdot \left(\frac{1}{1 + x} + \frac{1}{x}\right)} \]
      associate-/l/ [=>] 60.0	\[ \frac{\frac{\frac{1}{x}}{x} + \left(-\color{blue}{\frac{1}{\left(1 + x\right) \cdot \left(1 + x\right)}}\right)}{\left({x}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right) \cdot \left(\frac{1}{1 + x} + \frac{1}{x}\right)} \]
      distribute-neg-frac [=>] 60.0	\[ \frac{\frac{\frac{1}{x}}{x} + \color{blue}{\frac{-1}{\left(1 + x\right) \cdot \left(1 + x\right)}}}{\left({x}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right) \cdot \left(\frac{1}{1 + x} + \frac{1}{x}\right)} \]
      metadata-eval [=>] 60.0	\[ \frac{\frac{\frac{1}{x}}{x} + \frac{\color{blue}{-1}}{\left(1 + x\right) \cdot \left(1 + x\right)}}{\left({x}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right) \cdot \left(\frac{1}{1 + x} + \frac{1}{x}\right)} \]
      +-commutative [=>] 60.0	\[ \frac{\frac{\frac{1}{x}}{x} + \frac{-1}{\left(1 + x\right) \cdot \left(1 + x\right)}}{\left({x}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right) \cdot \color{blue}{\left(\frac{1}{x} + \frac{1}{1 + x}\right)}} \]

   4. Taylor expanded in x around inf 21.8

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}} \]

   5. Applied egg-rr 40.2

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(1 + {x}^{-1.5}\right) - 1\right)} \]

   6. Simplified 0.0

      \[\leadsto 0.5 \cdot \color{blue}{{x}^{-1.5}} \]
      Proof

      [Start] 40.2	\[ 0.5 \cdot \left(\left(1 + {x}^{-1.5}\right) - 1\right) \]
      +-commutative [=>] 40.2	\[ 0.5 \cdot \left(\color{blue}{\left({x}^{-1.5} + 1\right)} - 1\right) \]
      associate--l+ [=>] 0.0	\[ 0.5 \cdot \color{blue}{\left({x}^{-1.5} + \left(1 - 1\right)\right)} \]
      metadata-eval [=>] 0.0	\[ 0.5 \cdot \left({x}^{-1.5} + \color{blue}{0}\right) \]
      +-rgt-identity [=>] 0.0	\[ 0.5 \cdot \color{blue}{{x}^{-1.5}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(x, x, x\right)\right)}^{-0.5}}{\sqrt{1 + x} + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	27588

\[\begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 10^{-5}:\\ \;\;\;\;\frac{\frac{1}{\left(0.5 + \left(x + \frac{0.0625}{x \cdot x}\right)\right) + \frac{-0.125}{x}}}{t_0 + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 2
Error	0.2
Cost	27204

\[\begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{1}{\left(x + 0.5\right) + \frac{-0.125}{x}}}{t_0 + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 3
Error	0.2
Cost	26948

\[\begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{1}{x + 0.5}}{t_0 + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 4
Error	0.2
Cost	26948

\[\begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{-1}{t_0 + \sqrt{x}}}{-0.5 - x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 5
Error	0.3
Cost	26692

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 10^{-15}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 6
Error	0.3
Cost	26304

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

Alternative 7
Error	0.2
Cost	20164

\[\begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x + x \cdot x}}}{\sqrt{1 + x} + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \end{array} \]

Alternative 8
Error	0.9
Cost	7044

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

Alternative 9
Error	2.0
Cost	6788

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

Alternative 10
Error	1.1
Cost	6788

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

Alternative 11
Error	31.3
Cost	6528

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

Alternative 12
Error	59.3
Cost	320

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

Alternative 13
Error	59.3
Cost	192

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

Alternative 14
Error	62.8
Cost	64

\[-1 \]

Alternative 15
Error	60.3
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2023038 
(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)))))