2isqrt (example 3.6)

Average Error: 31.05% → 0.22%

Time: 10.6s

Precision: binary64

Cost: 39684

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{t_0} \leq 0:\\ \;\;\;\;\frac{\frac{0.5}{\sqrt{x}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(x, x, x\right)\right)}^{-0.5}}{\sqrt{x} + t_0}\\ \end{array} \]

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))))
   (if (<= (- (/ 1.0 (sqrt x)) (/ 1.0 t_0)) 0.0)
     (/ (/ 0.5 (sqrt x)) x)
     (/ (pow (fma x x x) -0.5) (+ (sqrt x) t_0)))))

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));
	double tmp;
	if (((1.0 / sqrt(x)) - (1.0 / t_0)) <= 0.0) {
		tmp = (0.5 / sqrt(x)) / x;
	} else {
		tmp = pow(fma(x, x, x), -0.5) / (sqrt(x) + t_0);
	}
	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 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / t_0)) <= 0.0)
		tmp = Float64(Float64(0.5 / sqrt(x)) / x);
	else
		tmp = Float64((fma(x, x, x) ^ -0.5) / Float64(sqrt(x) + t_0));
	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[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(0.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[Power[N[(x * x + x), $MachinePrecision], -0.5], $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Target

Original	31.05%
Target	1.07%
Herbie	0.22%

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 0.0

   1. Initial program 62.16

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

   2. Applied egg-rr 62.15

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

   3. Taylor expanded in x around inf 33.77

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \sqrt{{x}^{3}}}} \]

   4. Applied egg-rr 1.19

      \[\leadsto \color{blue}{{\left(\frac{\sqrt[3]{0.5}}{\sqrt{x}}\right)}^{3}} \]

   5. Simplified 0.36

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

      [Start] 1.19	\[ {\left(\frac{\sqrt[3]{0.5}}{\sqrt{x}}\right)}^{3} \]
      cube-div [=>] 2.53	\[ \color{blue}{\frac{{\left(\sqrt[3]{0.5}\right)}^{3}}{{\left(\sqrt{x}\right)}^{3}}} \]
      cube-mult [=>] 2.54	\[ \frac{{\left(\sqrt[3]{0.5}\right)}^{3}}{\color{blue}{\sqrt{x} \cdot \left(\sqrt{x} \cdot \sqrt{x}\right)}} \]
      associate-/r* [=>] 1.18	\[ \color{blue}{\frac{\frac{{\left(\sqrt[3]{0.5}\right)}^{3}}{\sqrt{x}}}{\sqrt{x} \cdot \sqrt{x}}} \]
      rem-cube-cbrt [=>] 0.74	\[ \frac{\frac{\color{blue}{0.5}}{\sqrt{x}}}{\sqrt{x} \cdot \sqrt{x}} \]
      rem-square-sqrt [=>] 0.36	\[ \frac{\frac{0.5}{\sqrt{x}}}{\color{blue}{x}} \]

   if 0.0 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 2.37

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

   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 0.47

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

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

   4. Applied egg-rr 9.93

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

   5. Simplified 0.1

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

      [Start] 9.93	\[ e^{\mathsf{log1p}\left(\frac{{\left(\mathsf{fma}\left(x, x, x\right)\right)}^{-0.5}}{\sqrt{x + 1} + \sqrt{x}}\right)} - 1 \]
      expm1-def [=>] 7.04	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\mathsf{fma}\left(x, x, x\right)\right)}^{-0.5}}{\sqrt{x + 1} + \sqrt{x}}\right)\right)} \]
      expm1-log1p [=>] 0.1	\[ \color{blue}{\frac{{\left(\mathsf{fma}\left(x, x, x\right)\right)}^{-0.5}}{\sqrt{x + 1} + \sqrt{x}}} \]
      +-commutative [=>] 0.1	\[ \frac{{\left(\mathsf{fma}\left(x, x, x\right)\right)}^{-0.5}}{\color{blue}{\sqrt{x} + \sqrt{x + 1}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.22

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 0:\\ \;\;\;\;\frac{\frac{0.5}{\sqrt{x}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(x, x, x\right)\right)}^{-0.5}}{\sqrt{x} + \sqrt{1 + x}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.31%
Cost	27588

\[\begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{t_0} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{0.5 + \left(x + \frac{-0.125}{x}\right)}}{\sqrt{x} + t_0}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 2
Error	1.19%
Cost	26692

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 0:\\ \;\;\;\;\frac{\frac{0.5}{\sqrt{x}}}{x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 3
Error	1.11%
Cost	26368

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

Alternative 4
Error	0.41%
Cost	26304

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

Alternative 5
Error	1.73%
Cost	7236

\[\begin{array}{l} \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;\frac{1}{\sqrt{x}} - \frac{1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\sqrt{x}}}{x}\\ \end{array} \]

Alternative 6
Error	1.88%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq 0.66:\\ \;\;\;\;-1 + {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{\sqrt{x}}}{x}\\ \end{array} \]

Alternative 7
Error	47.23%
Cost	6788

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

Alternative 8
Error	2.48%
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.66:\\ \;\;\;\;-1 + {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{{x}^{1.5}}\\ \end{array} \]

Alternative 9
Error	92.59%
Cost	320

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

Alternative 10
Error	92.61%
Cost	192

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

Reproduce

herbie shell --seed 2023125 
(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)))))