2sqrt (example 3.1)

Average Accuracy: 53.9% → 99.5%

Time: 7.0s

Precision: binary64

Cost: 26308

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;{x}^{-0.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ 1.0 x)) (sqrt x))))
   (if (<= t_0 5e-5) (* (pow x -0.5) 0.5) t_0)))

C

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

↓

double code(double x) {
	double t_0 = sqrt((1.0 + x)) - sqrt(x);
	double tmp;
	if (t_0 <= 5e-5) {
		tmp = pow(x, -0.5) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 + x)) - sqrt(x)
    if (t_0 <= 5d-5) then
        tmp = (x ** (-0.5d0)) * 0.5d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	return Math.sqrt((x + 1.0)) - Math.sqrt(x);
}

↓

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	double tmp;
	if (t_0 <= 5e-5) {
		tmp = Math.pow(x, -0.5) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(x):
	t_0 = math.sqrt((1.0 + x)) - math.sqrt(x)
	tmp = 0
	if t_0 <= 5e-5:
		tmp = math.pow(x, -0.5) * 0.5
	else:
		tmp = t_0
	return tmp

Julia

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

↓

function code(x)
	t_0 = Float64(sqrt(Float64(1.0 + x)) - sqrt(x))
	tmp = 0.0
	if (t_0 <= 5e-5)
		tmp = Float64((x ^ -0.5) * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x)) - sqrt(x);
	tmp = 0.0;
	if (t_0 <= 5e-5)
		tmp = (x ^ -0.5) * 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

↓

code[x_] := Block[{t$95$0 = N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-5], N[(N[Power[x, -0.5], $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]

Target

Original	53.9%
Target	99.8%
Herbie	99.5%

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) < 5.00000000000000024e-5

   1. Initial program 4.8%

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

   2. Applied egg-rr 3.4%

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

   3. Taylor expanded in x around inf 99.4%

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

   4. Simplified 99.4%

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

      [Start] 99.4	\[ 0.5 \cdot \sqrt{\frac{1}{x}} \]
      *-commutative [=>] 99.4	\[ \color{blue}{\sqrt{\frac{1}{x}} \cdot 0.5} \]

   5. Applied egg-rr 7.9%

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

   6. Simplified 99.6%

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

      [Start] 7.9	\[ \left(\left(1 + {x}^{-0.5}\right) - 1\right) \cdot 0.5 \]
      +-commutative [=>] 7.9	\[ \left(\color{blue}{\left({x}^{-0.5} + 1\right)} - 1\right) \cdot 0.5 \]
      associate--l+ [=>] 99.6	\[ \color{blue}{\left({x}^{-0.5} + \left(1 - 1\right)\right)} \cdot 0.5 \]
      metadata-eval [=>] 99.6	\[ \left({x}^{-0.5} + \color{blue}{0}\right) \cdot 0.5 \]
      +-rgt-identity [=>] 99.6	\[ \color{blue}{{x}^{-0.5}} \cdot 0.5 \]

   if 5.00000000000000024e-5 < (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))

   1. Initial program 99.5%

      \[\sqrt{x + 1} - \sqrt{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

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

Alternatives

Alternative 1
Accuracy	99.8%
Cost	13248

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

Alternative 2
Accuracy	98.0%
Cost	6980

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

Alternative 3
Accuracy	98.0%
Cost	6980

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

Alternative 4
Accuracy	98.0%
Cost	6852

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

Alternative 5
Accuracy	96.9%
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot 0.5\\ \end{array} \]

Alternative 6
Accuracy	51.9%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023122 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

  (- (sqrt (+ x 1.0)) (sqrt x)))