Main:bigenough3 from C

Average Accuracy: 53.6% → 99.7%

Time: 7.6s

Precision: binary64

Cost: 13248

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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 = 1.0d0 / (sqrt((1.0d0 + x)) + sqrt(x))
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 1.0 / (Math.sqrt((1.0 + x)) + Math.sqrt(x));
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Target

Original	53.6%
Target	99.7%
Herbie	99.7%

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

Derivation

1. Initial program 53.6%

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

2. Applied egg-rr 54.6%

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

   [Start] 53.6	\[ \sqrt{x + 1} - \sqrt{x} \]
   flip-- [=>] 53.9	\[ \color{blue}{\frac{\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}} \]
   div-inv [=>] 53.9	\[ \color{blue}{\left(\sqrt{x + 1} \cdot \sqrt{x + 1} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}} \]
   add-sqr-sqrt [<=] 54.0	\[ \left(\color{blue}{\left(x + 1\right)} - \sqrt{x} \cdot \sqrt{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} \]
   add-sqr-sqrt [<=] 54.6	\[ \left(\left(x + 1\right) - \color{blue}{x}\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} \]

3. Simplified 99.7%

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

   [Start] 54.6	\[ \left(\left(x + 1\right) - x\right) \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}} \]
   associate-*r/ [=>] 54.6	\[ \color{blue}{\frac{\left(\left(x + 1\right) - x\right) \cdot 1}{\sqrt{x + 1} + \sqrt{x}}} \]
   *-rgt-identity [=>] 54.6	\[ \frac{\color{blue}{\left(x + 1\right) - x}}{\sqrt{x + 1} + \sqrt{x}} \]
   remove-double-neg [<=] 54.6	\[ \frac{\left(x + 1\right) - x}{\sqrt{x + 1} + \color{blue}{\left(-\left(-\sqrt{x}\right)\right)}} \]
   sub-neg [<=] 54.6	\[ \frac{\left(x + 1\right) - x}{\color{blue}{\sqrt{x + 1} - \left(-\sqrt{x}\right)}} \]
   div-sub [=>] 53.6	\[ \color{blue}{\frac{x + 1}{\sqrt{x + 1} - \left(-\sqrt{x}\right)} - \frac{x}{\sqrt{x + 1} - \left(-\sqrt{x}\right)}} \]
   rem-square-sqrt [<=] 53.6	\[ \frac{x + 1}{\sqrt{x + 1} - \left(-\sqrt{x}\right)} - \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} - \left(-\sqrt{x}\right)} \]
   sqr-neg [<=] 53.6	\[ \frac{x + 1}{\sqrt{x + 1} - \left(-\sqrt{x}\right)} - \frac{\color{blue}{\left(-\sqrt{x}\right) \cdot \left(-\sqrt{x}\right)}}{\sqrt{x + 1} - \left(-\sqrt{x}\right)} \]
   div-sub [<=] 54.0	\[ \color{blue}{\frac{\left(x + 1\right) - \left(-\sqrt{x}\right) \cdot \left(-\sqrt{x}\right)}{\sqrt{x + 1} - \left(-\sqrt{x}\right)}} \]
   sqr-neg [=>] 54.0	\[ \frac{\left(x + 1\right) - \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{x + 1} - \left(-\sqrt{x}\right)} \]
   +-commutative [=>] 54.0	\[ \frac{\color{blue}{\left(1 + x\right)} - \sqrt{x} \cdot \sqrt{x}}{\sqrt{x + 1} - \left(-\sqrt{x}\right)} \]
   rem-square-sqrt [=>] 54.6	\[ \frac{\left(1 + x\right) - \color{blue}{x}}{\sqrt{x + 1} - \left(-\sqrt{x}\right)} \]
   associate--l+ [=>] 99.7	\[ \frac{\color{blue}{1 + \left(x - x\right)}}{\sqrt{x + 1} - \left(-\sqrt{x}\right)} \]
   +-inverses [=>] 99.7	\[ \frac{1 + \color{blue}{0}}{\sqrt{x + 1} - \left(-\sqrt{x}\right)} \]
   metadata-eval [=>] 99.7	\[ \frac{\color{blue}{1}}{\sqrt{x + 1} - \left(-\sqrt{x}\right)} \]
   sub-neg [=>] 99.7	\[ \frac{1}{\color{blue}{\sqrt{x + 1} + \left(-\left(-\sqrt{x}\right)\right)}} \]

4. Final simplification 99.7%

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

Alternatives

Alternative 1
Accuracy	60.6%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \sqrt{x}}\\ \end{array} \]

Alternative 2
Accuracy	99.3%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;x \leq 58000000:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x} + \sqrt{x}}\\ \end{array} \]

Alternative 3
Accuracy	58.6%
Cost	6720

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

Alternative 4
Accuracy	51.6%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023146 
(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)))