2isqrt (example 3.6)

Average Error: 19.5 → 0.0

Time: 12.0s

Precision: binary64

Cost: 27140

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{1}{x + \left(1 + \sqrt{x + x \cdot x}\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))) 0.0)
   (* 0.5 (pow x -1.5))
   (* (pow x -0.5) (/ 1.0 (+ x (+ 1.0 (sqrt (+ x (* x x)))))))))

C

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

↓

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)))) <= 0.0) {
		tmp = 0.5 * pow(x, -1.5);
	} else {
		tmp = pow(x, -0.5) * (1.0 / (x + (1.0 + sqrt((x + (x * x))))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) - (1.0d0 / sqrt((x + 1.0d0)))) <= 0.0d0) then
        tmp = 0.5d0 * (x ** (-1.5d0))
    else
        tmp = (x ** (-0.5d0)) * (1.0d0 / (x + (1.0d0 + sqrt((x + (x * x))))))
    end if
    code = tmp
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) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) - (1.0 / Math.sqrt((x + 1.0)))) <= 0.0) {
		tmp = 0.5 * Math.pow(x, -1.5);
	} else {
		tmp = Math.pow(x, -0.5) * (1.0 / (x + (1.0 + Math.sqrt((x + (x * x))))));
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) - (1.0 / math.sqrt((x + 1.0)))) <= 0.0:
		tmp = 0.5 * math.pow(x, -1.5)
	else:
		tmp = math.pow(x, -0.5) * (1.0 / (x + (1.0 + math.sqrt((x + (x * 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)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / sqrt(Float64(x + 1.0)))) <= 0.0)
		tmp = Float64(0.5 * (x ^ -1.5));
	else
		tmp = Float64((x ^ -0.5) * Float64(1.0 / Float64(x + Float64(1.0 + sqrt(Float64(x + Float64(x * x)))))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) - (1.0 / sqrt((x + 1.0)))) <= 0.0)
		tmp = 0.5 * (x ^ -1.5);
	else
		tmp = (x ^ -0.5) * (1.0 / (x + (1.0 + sqrt((x + (x * x))))));
	end
	tmp_2 = 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[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * N[(1.0 / N[(x + N[(1.0 + N[Sqrt[N[(x + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	19.5
Target	0.6
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 (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 0.0

   1. Initial program 39.7

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

   2. Applied egg-rr 39.7

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

   3. Taylor expanded in x around inf 21.5

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

   4. Applied egg-rr 39.7

      \[\leadsto 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-1.5}\right)} - 1\right)} \]

   5. Simplified 0.0

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

      [Start] 39.7	\[ 0.5 \cdot \left(e^{\mathsf{log1p}\left({x}^{-1.5}\right)} - 1\right) \]
      expm1-def [=>] 0.0	\[ 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-1.5}\right)\right)} \]
      expm1-log1p [=>] 0.0	\[ 0.5 \cdot \color{blue}{{x}^{-1.5}} \]

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

   1. Initial program 1.4

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

   2. Applied egg-rr 1.5

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

   3. Applied egg-rr 0.1

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

   4. Simplified 0.1

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

      [Start] 0.1	\[ \frac{\left(1 + \left(x - x\right)\right) \cdot \frac{{x}^{-0.5}}{\sqrt{1 + x}}}{\sqrt{x} + \sqrt{1 + x}} \]
      associate-*r/ [=>] 0.1	\[ \frac{\color{blue}{\frac{\left(1 + \left(x - x\right)\right) \cdot {x}^{-0.5}}{\sqrt{1 + x}}}}{\sqrt{x} + \sqrt{1 + x}} \]
      +-commutative [=>] 0.1	\[ \frac{\frac{\color{blue}{\left(\left(x - x\right) + 1\right)} \cdot {x}^{-0.5}}{\sqrt{1 + x}}}{\sqrt{x} + \sqrt{1 + x}} \]
      +-inverses [=>] 0.1	\[ \frac{\frac{\left(\color{blue}{0} + 1\right) \cdot {x}^{-0.5}}{\sqrt{1 + x}}}{\sqrt{x} + \sqrt{1 + x}} \]
      metadata-eval [=>] 0.1	\[ \frac{\frac{\color{blue}{1} \cdot {x}^{-0.5}}{\sqrt{1 + x}}}{\sqrt{x} + \sqrt{1 + x}} \]
      associate-/l/ [=>] 0.1	\[ \color{blue}{\frac{1 \cdot {x}^{-0.5}}{\left(\sqrt{x} + \sqrt{1 + x}\right) \cdot \sqrt{1 + x}}} \]
      *-lft-identity [=>] 0.1	\[ \frac{\color{blue}{{x}^{-0.5}}}{\left(\sqrt{x} + \sqrt{1 + x}\right) \cdot \sqrt{1 + x}} \]

   5. Applied egg-rr 0.1

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{1}{x + \left(1 + \sqrt{x + x \cdot x}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.0
Cost	27012

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{1 + \left(x + \sqrt{x + x \cdot x}\right)}\\ \end{array} \]

Alternative 2
Error	0.2
Cost	26756

\[\begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{t_0} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{{x}^{-0.5}}{1 + \left(x + \left(x + \left(0.5 - \frac{0.125}{x}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{t_0}\\ \end{array} \]

Alternative 3
Error	0.2
Cost	26368

\[\begin{array}{l} t_0 := \sqrt{x + 1}\\ \frac{{x}^{-0.5}}{t_0 \cdot \left(\sqrt{x} + t_0\right)} \end{array} \]

Alternative 4
Error	0.4
Cost	13760

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

Alternative 5
Error	0.1
Cost	13380

\[\begin{array}{l} \mathbf{if}\;x \leq 5000:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{1 + \left(x + \left(x + \left(0.5 - \frac{0.125}{x}\right)\right)\right)}\\ \end{array} \]

Alternative 6
Error	0.5
Cost	7812

\[\begin{array}{l} \mathbf{if}\;x \leq 0.43:\\ \;\;\;\;{x}^{-0.5} - \left(1 + x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{1 + \left(x + \left(\left(0.5 - \frac{0.125}{x}\right) + \left(x + \frac{0.0625}{x \cdot x}\right)\right)\right)}\\ \end{array} \]

Alternative 7
Error	0.5
Cost	7428

\[\begin{array}{l} \mathbf{if}\;x \leq 0.42:\\ \;\;\;\;{x}^{-0.5} - \left(1 + x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{1 + \left(x + \left(x + \left(0.5 - \frac{0.125}{x}\right)\right)\right)}\\ \end{array} \]

Alternative 8
Error	0.6
Cost	7172

\[\begin{array}{l} \mathbf{if}\;x \leq 0.41:\\ \;\;\;\;{x}^{-0.5} - \left(1 + x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5}}{1 + \left(0.5 + x \cdot 2\right)}\\ \end{array} \]

Alternative 9
Error	1.0
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 10
Error	2.2
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 11
Error	1.2
Cost	6788

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

Alternative 12
Error	31.6
Cost	6528

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

Alternative 13
Error	61.5
Cost	192

\[x \cdot 0.5 \]

Alternative 14
Error	62.8
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023010 
(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)))))