2sqrt (example 3.1)

Average Accuracy: 53.6% → 99.1%

Time: 8.0s

Precision: binary64

Cost: 26308

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 0.0) (* 0.5 (pow x -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 <= 0.0) {
		tmp = 0.5 * pow(x, -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 <= 0.0d0) then
        tmp = 0.5d0 * (x ** (-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 <= 0.0) {
		tmp = 0.5 * Math.pow(x, -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 <= 0.0:
		tmp = 0.5 * math.pow(x, -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 <= 0.0)
		tmp = Float64(0.5 * (x ^ -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 <= 0.0)
		tmp = 0.5 * (x ^ -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, 0.0], N[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], t$95$0]]

Target

Original	53.6%
Target	99.8%
Herbie	99.1%

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 3.9%

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

   2. Applied egg-rr 2.3%

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

      [Start] 3.9	\[ \sqrt{x + 1} - \sqrt{x} \]
      flip3-- [=>] 2.3	\[ \color{blue}{\frac{{\left(\sqrt{x + 1}\right)}^{3} - {\left(\sqrt{x}\right)}^{3}}{\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} + \sqrt{x + 1} \cdot \sqrt{x}\right)}} \]
      div-inv [=>] 2.3	\[ \color{blue}{\left({\left(\sqrt{x + 1}\right)}^{3} - {\left(\sqrt{x}\right)}^{3}\right) \cdot \frac{1}{\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} + \sqrt{x + 1} \cdot \sqrt{x}\right)}} \]
      sqrt-pow2 [=>] 2.5	\[ \left(\color{blue}{{\left(x + 1\right)}^{\left(\frac{3}{2}\right)}} - {\left(\sqrt{x}\right)}^{3}\right) \cdot \frac{1}{\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} + \sqrt{x + 1} \cdot \sqrt{x}\right)} \]
      metadata-eval [=>] 2.5	\[ \left({\left(x + 1\right)}^{\color{blue}{1.5}} - {\left(\sqrt{x}\right)}^{3}\right) \cdot \frac{1}{\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} + \sqrt{x + 1} \cdot \sqrt{x}\right)} \]
      sqrt-pow2 [=>] 2.3	\[ \left({\left(x + 1\right)}^{1.5} - \color{blue}{{x}^{\left(\frac{3}{2}\right)}}\right) \cdot \frac{1}{\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} + \sqrt{x + 1} \cdot \sqrt{x}\right)} \]
      metadata-eval [=>] 2.3	\[ \left({\left(x + 1\right)}^{1.5} - {x}^{\color{blue}{1.5}}\right) \cdot \frac{1}{\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} + \sqrt{x + 1} \cdot \sqrt{x}\right)} \]
      add-sqr-sqrt [<=] 2.3	\[ \left({\left(x + 1\right)}^{1.5} - {x}^{1.5}\right) \cdot \frac{1}{\color{blue}{\left(x + 1\right)} + \left(\sqrt{x} \cdot \sqrt{x} + \sqrt{x + 1} \cdot \sqrt{x}\right)} \]
      add-sqr-sqrt [<=] 2.3	\[ \left({\left(x + 1\right)}^{1.5} - {x}^{1.5}\right) \cdot \frac{1}{\left(x + 1\right) + \left(\color{blue}{x} + \sqrt{x + 1} \cdot \sqrt{x}\right)} \]
      sqrt-unprod [=>] 2.3	\[ \left({\left(x + 1\right)}^{1.5} - {x}^{1.5}\right) \cdot \frac{1}{\left(x + 1\right) + \left(x + \color{blue}{\sqrt{\left(x + 1\right) \cdot x}}\right)} \]

   3. Taylor expanded in x around inf 99.8%

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

   4. Applied egg-rr 6.2%

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

      [Start] 99.8	\[ 0.5 \cdot \sqrt{\frac{1}{x}} \]
      expm1-log1p-u [=>] 99.8	\[ 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{x}}\right)\right)} \]
      expm1-udef [=>] 6.2	\[ 0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{x}}\right)} - 1\right)} \]
      log1p-udef [=>] 6.2	\[ 0.5 \cdot \left(e^{\color{blue}{\log \left(1 + \sqrt{\frac{1}{x}}\right)}} - 1\right) \]
      add-exp-log [<=] 6.2	\[ 0.5 \cdot \left(\color{blue}{\left(1 + \sqrt{\frac{1}{x}}\right)} - 1\right) \]
      inv-pow [=>] 6.2	\[ 0.5 \cdot \left(\left(1 + \sqrt{\color{blue}{{x}^{-1}}}\right) - 1\right) \]
      metadata-eval [<=] 6.2	\[ 0.5 \cdot \left(\left(1 + \sqrt{{x}^{\color{blue}{\left(-1\right)}}}\right) - 1\right) \]
      sqrt-pow1 [=>] 6.2	\[ 0.5 \cdot \left(\left(1 + \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right) - 1\right) \]
      metadata-eval [=>] 6.2	\[ 0.5 \cdot \left(\left(1 + {x}^{\left(\frac{\color{blue}{-1}}{2}\right)}\right) - 1\right) \]
      metadata-eval [=>] 6.2	\[ 0.5 \cdot \left(\left(1 + {x}^{\color{blue}{-0.5}}\right) - 1\right) \]

   5. Simplified 100.0%

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

      [Start] 6.2	\[ 0.5 \cdot \left(\left(1 + {x}^{-0.5}\right) - 1\right) \]
      +-commutative [=>] 6.2	\[ 0.5 \cdot \left(\color{blue}{\left({x}^{-0.5} + 1\right)} - 1\right) \]
      associate--l+ [=>] 100.0	\[ 0.5 \cdot \color{blue}{\left({x}^{-0.5} + \left(1 - 1\right)\right)} \]
      metadata-eval [=>] 100.0	\[ 0.5 \cdot \left({x}^{-0.5} + \color{blue}{0}\right) \]
      +-rgt-identity [=>] 100.0	\[ 0.5 \cdot \color{blue}{{x}^{-0.5}} \]

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

   1. Initial program 98.3%

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

4. Final simplification 99.1%

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

Alternatives

Alternative 1
Accuracy	99.8%
Cost	13248

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

Alternative 2
Accuracy	97.1%
Cost	6916

\[\begin{array}{l} \mathbf{if}\;x \leq 1.8:\\ \;\;\;\;\frac{1}{1 + {x}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \]

Alternative 3
Accuracy	97.1%
Cost	6788

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

Alternative 4
Accuracy	97.1%
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.74:\\ \;\;\;\;1 - {x}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \]

Alternative 5
Accuracy	51.7%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023138 
(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)))