2isqrt (example 3.6)

Average Accuracy: 68.4% → 99.8%

Time: 15.3s

Precision: binary64

Cost: 26688

Math

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

↓

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

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (*
  (/ (pow x -0.5) (+ (pow x -0.5) (pow (+ x 1.0) -0.5)))
  (/ (pow x -0.5) (+ x 1.0))))

C

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

↓

double code(double x) {
	return (pow(x, -0.5) / (pow(x, -0.5) + pow((x + 1.0), -0.5))) * (pow(x, -0.5) / (x + 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / sqrt(x)) - (1.0d0 / sqrt((x + 1.0d0)))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x ** (-0.5d0)) / ((x ** (-0.5d0)) + ((x + 1.0d0) ** (-0.5d0)))) * ((x ** (-0.5d0)) / (x + 1.0d0))
end function

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 (Math.pow(x, -0.5) / (Math.pow(x, -0.5) + Math.pow((x + 1.0), -0.5))) * (Math.pow(x, -0.5) / (x + 1.0));
}

Python

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

↓

def code(x):
	return (math.pow(x, -0.5) / (math.pow(x, -0.5) + math.pow((x + 1.0), -0.5))) * (math.pow(x, -0.5) / (x + 1.0))

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((x ^ -0.5) / Float64((x ^ -0.5) + (Float64(x + 1.0) ^ -0.5))) * Float64((x ^ -0.5) / Float64(x + 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)));
end

↓

function tmp = code(x)
	tmp = ((x ^ -0.5) / ((x ^ -0.5) + ((x + 1.0) ^ -0.5))) * ((x ^ -0.5) / (x + 1.0));
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[(N[Power[x, -0.5], $MachinePrecision] / N[(N[Power[x, -0.5], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[x, -0.5], $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	68.4%
Target	99.0%
Herbie	99.8%

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

Derivation

1. Initial program 68.4%

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

2. Applied egg-rr 68.7%

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

   [Start] 68.4	\[ \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   *-un-lft-identity [=>] 68.4	\[ \color{blue}{1 \cdot \frac{1}{\sqrt{x}}} - \frac{1}{\sqrt{x + 1}} \]
   clear-num [=>] 68.4	\[ 1 \cdot \frac{1}{\sqrt{x}} - \color{blue}{\frac{1}{\frac{\sqrt{x + 1}}{1}}} \]
   associate-/r/ [=>] 68.4	\[ 1 \cdot \frac{1}{\sqrt{x}} - \color{blue}{\frac{1}{\sqrt{x + 1}} \cdot 1} \]
   prod-diff [=>] 68.4	\[ \color{blue}{\mathsf{fma}\left(1, \frac{1}{\sqrt{x}}, -1 \cdot \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right)} \]
   *-un-lft-identity [<=] 68.4	\[ \mathsf{fma}\left(1, \frac{1}{\sqrt{x}}, -\color{blue}{\frac{1}{\sqrt{x + 1}}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
   fma-neg [<=] 68.4	\[ \color{blue}{\left(1 \cdot \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\right)} + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
   *-un-lft-identity [<=] 68.4	\[ \left(\color{blue}{\frac{1}{\sqrt{x}}} - \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
   inv-pow [=>] 68.4	\[ \left(\color{blue}{{\left(\sqrt{x}\right)}^{-1}} - \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
   sqrt-pow2 [=>] 64.5	\[ \left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}} - \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
   metadata-eval [=>] 64.5	\[ \left({x}^{\color{blue}{-0.5}} - \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
   pow1/2 [=>] 64.5	\[ \left({x}^{-0.5} - \frac{1}{\color{blue}{{\left(x + 1\right)}^{0.5}}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
   pow-flip [=>] 68.7	\[ \left({x}^{-0.5} - \color{blue}{{\left(x + 1\right)}^{\left(-0.5\right)}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
   +-commutative [=>] 68.7	\[ \left({x}^{-0.5} - {\color{blue}{\left(1 + x\right)}}^{\left(-0.5\right)}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]
   metadata-eval [=>] 68.7	\[ \left({x}^{-0.5} - {\left(1 + x\right)}^{\color{blue}{-0.5}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]

3. Simplified 68.7%

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

   [Start] 68.7	\[ \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right) \]
   fma-udef [=>] 68.7	\[ \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \color{blue}{\left(-1 \cdot {\left(1 + x\right)}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right)} \]
   distribute-lft1-in [=>] 68.7	\[ \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \color{blue}{\left(-1 + 1\right) \cdot {\left(1 + x\right)}^{-0.5}} \]
   metadata-eval [=>] 68.7	\[ \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5} \]
   mul0-lft [=>] 68.7	\[ \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \color{blue}{0} \]
   +-rgt-identity [=>] 68.7	\[ \color{blue}{{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}} \]

4. Applied egg-rr 68.3%

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

   [Start] 68.7	\[ {x}^{-0.5} - {\left(1 + x\right)}^{-0.5} \]
   flip-- [=>] 68.6	\[ \color{blue}{\frac{{x}^{-0.5} \cdot {x}^{-0.5} - {\left(1 + x\right)}^{-0.5} \cdot {\left(1 + x\right)}^{-0.5}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}} \]
   pow-prod-up [=>] 60.7	\[ \frac{\color{blue}{{x}^{\left(-0.5 + -0.5\right)}} - {\left(1 + x\right)}^{-0.5} \cdot {\left(1 + x\right)}^{-0.5}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
   metadata-eval [=>] 60.7	\[ \frac{{x}^{\color{blue}{-1}} - {\left(1 + x\right)}^{-0.5} \cdot {\left(1 + x\right)}^{-0.5}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
   unpow-1 [=>] 60.7	\[ \frac{\color{blue}{\frac{1}{x}} - {\left(1 + x\right)}^{-0.5} \cdot {\left(1 + x\right)}^{-0.5}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
   pow-prod-up [=>] 68.3	\[ \frac{\frac{1}{x} - \color{blue}{{\left(1 + x\right)}^{\left(-0.5 + -0.5\right)}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
   metadata-eval [=>] 68.3	\[ \frac{\frac{1}{x} - {\left(1 + x\right)}^{\color{blue}{-1}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
   unpow-1 [=>] 68.3	\[ \frac{\frac{1}{x} - \color{blue}{\frac{1}{1 + x}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
   +-commutative [=>] 68.3	\[ \frac{\frac{1}{x} - \frac{1}{\color{blue}{x + 1}}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \]
   +-commutative [=>] 68.3	\[ \frac{\frac{1}{x} - \frac{1}{x + 1}}{{x}^{-0.5} + {\color{blue}{\left(x + 1\right)}}^{-0.5}} \]

5. Applied egg-rr 91.5%

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

   [Start] 68.3	\[ \frac{\frac{1}{x} - \frac{1}{x + 1}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}} \]
   frac-sub [=>] 69.3	\[ \frac{\color{blue}{\frac{1 \cdot \left(x + 1\right) - x \cdot 1}{x \cdot \left(x + 1\right)}}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}} \]
   associate-/r* [=>] 69.3	\[ \frac{\color{blue}{\frac{\frac{1 \cdot \left(x + 1\right) - x \cdot 1}{x}}{x + 1}}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}} \]
   *-un-lft-identity [<=] 69.3	\[ \frac{\frac{\frac{\color{blue}{\left(x + 1\right)} - x \cdot 1}{x}}{x + 1}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}} \]
   +-commutative [=>] 69.3	\[ \frac{\frac{\frac{\color{blue}{\left(1 + x\right)} - x \cdot 1}{x}}{x + 1}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}} \]
   *-rgt-identity [=>] 69.3	\[ \frac{\frac{\frac{\left(1 + x\right) - \color{blue}{x}}{x}}{x + 1}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}} \]
   associate--l+ [=>] 91.5	\[ \frac{\frac{\frac{\color{blue}{1 + \left(x - x\right)}}{x}}{x + 1}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}} \]
   +-commutative [=>] 91.5	\[ \frac{\frac{\frac{1 + \left(x - x\right)}{x}}{\color{blue}{1 + x}}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}} \]

6. Applied egg-rr 99.8%

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

   [Start] 91.5	\[ \frac{\frac{\frac{1 + \left(x - x\right)}{x}}{1 + x}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}} \]
   associate-/l/ [=>] 99.4	\[ \color{blue}{\frac{\frac{1 + \left(x - x\right)}{x}}{\left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right) \cdot \left(1 + x\right)}} \]
   add-sqr-sqrt [=>] 99.4	\[ \frac{\color{blue}{\sqrt{\frac{1 + \left(x - x\right)}{x}} \cdot \sqrt{\frac{1 + \left(x - x\right)}{x}}}}{\left({x}^{-0.5} + {\left(x + 1\right)}^{-0.5}\right) \cdot \left(1 + x\right)} \]
   times-frac [=>] 99.6	\[ \color{blue}{\frac{\sqrt{\frac{1 + \left(x - x\right)}{x}}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}} \cdot \frac{\sqrt{\frac{1 + \left(x - x\right)}{x}}}{1 + x}} \]
   +-inverses [=>] 99.6	\[ \frac{\sqrt{\frac{1 + \color{blue}{0}}{x}}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}} \cdot \frac{\sqrt{\frac{1 + \left(x - x\right)}{x}}}{1 + x} \]
   metadata-eval [=>] 99.6	\[ \frac{\sqrt{\frac{\color{blue}{1}}{x}}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}} \cdot \frac{\sqrt{\frac{1 + \left(x - x\right)}{x}}}{1 + x} \]
   inv-pow [=>] 99.6	\[ \frac{\sqrt{\color{blue}{{x}^{-1}}}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}} \cdot \frac{\sqrt{\frac{1 + \left(x - x\right)}{x}}}{1 + x} \]
   metadata-eval [<=] 99.6	\[ \frac{\sqrt{{x}^{\color{blue}{\left(-0.5 + -0.5\right)}}}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}} \cdot \frac{\sqrt{\frac{1 + \left(x - x\right)}{x}}}{1 + x} \]
   pow-prod-up [<=] 99.7	\[ \frac{\sqrt{\color{blue}{{x}^{-0.5} \cdot {x}^{-0.5}}}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}} \cdot \frac{\sqrt{\frac{1 + \left(x - x\right)}{x}}}{1 + x} \]
   sqrt-unprod [<=] 99.1	\[ \frac{\color{blue}{\sqrt{{x}^{-0.5}} \cdot \sqrt{{x}^{-0.5}}}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}} \cdot \frac{\sqrt{\frac{1 + \left(x - x\right)}{x}}}{1 + x} \]
   add-sqr-sqrt [<=] 99.7	\[ \frac{\color{blue}{{x}^{-0.5}}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}} \cdot \frac{\sqrt{\frac{1 + \left(x - x\right)}{x}}}{1 + x} \]
   +-commutative [<=] 99.7	\[ \frac{{x}^{-0.5}}{{x}^{-0.5} + {\color{blue}{\left(1 + x\right)}}^{-0.5}} \cdot \frac{\sqrt{\frac{1 + \left(x - x\right)}{x}}}{1 + x} \]

7. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	99.3%
Cost	26692

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;0.5 \cdot {\left(\frac{1}{x}\right)}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \end{array} \]

Alternative 2
Accuracy	99.4%
Cost	19968

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

Alternative 3
Accuracy	99.4%
Cost	13892

\[\begin{array}{l} \mathbf{if}\;x \leq 10^{+26}:\\ \;\;\;\;\frac{\frac{1}{x + x \cdot x}}{{x}^{-0.5} + {\left(x + 1\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {\left(\frac{1}{x}\right)}^{1.5}\\ \end{array} \]

Alternative 4
Accuracy	99.3%
Cost	13888

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

Alternative 5
Accuracy	98.3%
Cost	7172

\[\begin{array}{l} \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {\left(\frac{1}{x}\right)}^{1.5}\\ \end{array} \]

Alternative 6
Accuracy	98.3%
Cost	7044

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

Alternative 7
Accuracy	98.0%
Cost	6916

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

Alternative 8
Accuracy	51.5%
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.8:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5}\\ \end{array} \]

Alternative 9
Accuracy	52.0%
Cost	6720

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

Alternative 10
Accuracy	50.0%
Cost	6528

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

Alternative 11
Accuracy	4.0%
Cost	192

\[x \cdot 0.5 \]

Alternative 12
Accuracy	1.9%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023146 
(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)))))