Main:bigenough3 from C

Average Accuracy: 54.1% → 99.6%

Time: 9.2s

Precision: binary64

Cost: 26048

Math

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

↓

\[{\left({\left(\sqrt{1 + x} + \sqrt{x}\right)}^{2}\right)}^{-0.5} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (pow (pow (+ (sqrt (+ 1.0 x)) (sqrt x)) 2.0) -0.5))

C

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

↓

double code(double x) {
	return pow(pow((sqrt((1.0 + x)) + sqrt(x)), 2.0), -0.5);
}

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
    code = ((sqrt((1.0d0 + x)) + sqrt(x)) ** 2.0d0) ** (-0.5d0)
end function

Java

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

↓

public static double code(double x) {
	return Math.pow(Math.pow((Math.sqrt((1.0 + x)) + Math.sqrt(x)), 2.0), -0.5);
}

Python

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

↓

def code(x):
	return math.pow(math.pow((math.sqrt((1.0 + x)) + math.sqrt(x)), 2.0), -0.5)

Julia

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

↓

function code(x)
	return (Float64(sqrt(Float64(1.0 + x)) + sqrt(x)) ^ 2.0) ^ -0.5
end

MATLAB

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

↓

function tmp = code(x)
	tmp = ((sqrt((1.0 + x)) + sqrt(x)) ^ 2.0) ^ -0.5;
end

Wolfram

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

↓

code[x_] := N[Power[N[Power[N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], -0.5], $MachinePrecision]

Target

Original	54.1%
Target	99.7%
Herbie	99.6%

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

Derivation

1. Initial program 56.1%

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

2. Applied egg-rr 57.8%

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

   [Start] 56.1	\[ \sqrt{x + 1} - \sqrt{x} \]
   flip-- [=>] 57.1	\[ \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} \]
   div-inv [=>] 57.1	\[ \color{blue}{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}} \]
   add-sqr-sqrt [<=] 57.0	\[ \left(\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} \]
   add-sqr-sqrt [<=] 57.8	\[ \left(\left(x + 1\right) - \color{blue}{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} \]

3. Simplified 99.8%

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

   [Start] 57.8	\[ \left(\left(x + 1\right) - x\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} \]
   *-commutative [=>] 57.8	\[ \color{blue}{\frac{1}{\sqrt{x + 1} + \sqrt{x}} \cdot \left(\left(x + 1\right) - x\right)} \]
   associate-/r/ [<=] 57.8	\[ \color{blue}{\frac{1}{\frac{\sqrt{x + 1} + \sqrt{x}}{\left(x + 1\right) - x}}} \]
   +-commutative [=>] 57.8	\[ \frac{1}{\frac{\sqrt{x + 1} + \sqrt{x}}{\color{blue}{\left(1 + x\right)} - x}} \]
   associate--l+ [=>] 99.8	\[ \frac{1}{\frac{\sqrt{x + 1} + \sqrt{x}}{\color{blue}{1 + \left(x - x\right)}}} \]
   +-inverses [=>] 99.8	\[ \frac{1}{\frac{\sqrt{x + 1} + \sqrt{x}}{1 + \color{blue}{0}}} \]
   metadata-eval [=>] 99.8	\[ \frac{1}{\frac{\sqrt{x + 1} + \sqrt{x}}{\color{blue}{1}}} \]
   /-rgt-identity [=>] 99.8	\[ \frac{1}{\color{blue}{\sqrt{x + 1} + \sqrt{x}}} \]
   +-commutative [=>] 99.8	\[ \frac{1}{\sqrt{\color{blue}{1 + x}} + \sqrt{x}} \]

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{{\left({\left(\sqrt{1 + x} + \sqrt{x}\right)}^{2}\right)}^{-0.5}} \]
   Proof

   [Start] 99.8	\[ \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]
   inv-pow [=>] 99.8	\[ \color{blue}{{\left(\sqrt{1 + x} + \sqrt{x}\right)}^{-1}} \]
   sqr-pow [=>] 99.6	\[ \color{blue}{{\left(\sqrt{1 + x} + \sqrt{x}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\sqrt{1 + x} + \sqrt{x}\right)}^{\left(\frac{-1}{2}\right)}} \]
   pow-prod-down [=>] 99.8	\[ \color{blue}{{\left(\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)\right)}^{\left(\frac{-1}{2}\right)}} \]
   pow2 [=>] 99.8	\[ {\color{blue}{\left({\left(\sqrt{1 + x} + \sqrt{x}\right)}^{2}\right)}}^{\left(\frac{-1}{2}\right)} \]
   metadata-eval [=>] 99.8	\[ {\left({\left(\sqrt{1 + x} + \sqrt{x}\right)}^{2}\right)}^{\color{blue}{-0.5}} \]

5. Final simplification 99.8%

   \[\leadsto {\left({\left(\sqrt{1 + x} + \sqrt{x}\right)}^{2}\right)}^{-0.5} \]

Alternatives

Alternative 1
Accuracy	99.5%
Cost	26308

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

Alternative 2
Accuracy	99.7%
Cost	13248

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

Alternative 3
Accuracy	96.8%
Cost	7172

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

Alternative 4
Accuracy	96.8%
Cost	6916

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

Alternative 5
Accuracy	96.8%
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 2023160 
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

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

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