Numeric.Log:$clog1p from log-domain-0.10.2.1, B

Average Error: 0.2 → 0.2

Time: 11.8s

Precision: binary64

Cost: 20544

Math

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

↓

\[\begin{array}{l} t_0 := -1 - \sqrt{x + 1}\\ \frac{\frac{x}{t_0 \cdot t_0}}{\frac{-1}{t_0}} \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- -1.0 (sqrt (+ x 1.0))))) (/ (/ x (* t_0 t_0)) (/ -1.0 t_0))))

C

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

↓

double code(double x) {
	double t_0 = -1.0 - sqrt((x + 1.0));
	return (x / (t_0 * t_0)) / (-1.0 / t_0);
}

Fortran

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

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = (-1.0d0) - sqrt((x + 1.0d0))
    code = (x / (t_0 * t_0)) / ((-1.0d0) / t_0)
end function

Java

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

↓

public static double code(double x) {
	double t_0 = -1.0 - Math.sqrt((x + 1.0));
	return (x / (t_0 * t_0)) / (-1.0 / t_0);
}

Python

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

↓

def code(x):
	t_0 = -1.0 - math.sqrt((x + 1.0))
	return (x / (t_0 * t_0)) / (-1.0 / t_0)

Julia

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

↓

function code(x)
	t_0 = Float64(-1.0 - sqrt(Float64(x + 1.0)))
	return Float64(Float64(x / Float64(t_0 * t_0)) / Float64(-1.0 / t_0))
end

MATLAB

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

↓

function tmp = code(x)
	t_0 = -1.0 - sqrt((x + 1.0));
	tmp = (x / (t_0 * t_0)) / (-1.0 / t_0);
end

Wolfram

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

↓

code[x_] := Block[{t$95$0 = N[(-1.0 - N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[(x / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 0.2

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

2. Applied egg-rr 0.2

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

3. Applied egg-rr 0.2

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

4. Simplified 0.2

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

   [Start] 0.2	\[ \left(\left(1 + \sqrt{1 + x}\right) \cdot \frac{1}{\left(1 + \sqrt{1 + x}\right) \cdot \left(1 + \sqrt{1 + x}\right)}\right) \cdot x \]
   rational.json-simplify-2 [=>] 0.2	\[ \color{blue}{\left(\frac{1}{\left(1 + \sqrt{1 + x}\right) \cdot \left(1 + \sqrt{1 + x}\right)} \cdot \left(1 + \sqrt{1 + x}\right)\right)} \cdot x \]
   rational.json-simplify-7 [<=] 0.2	\[ \left(\frac{1}{\left(1 + \sqrt{1 + x}\right) \cdot \left(1 + \sqrt{1 + x}\right)} \cdot \color{blue}{\frac{1 + \sqrt{1 + x}}{1}}\right) \cdot x \]
   rational.json-simplify-55 [=>] 0.2	\[ \color{blue}{\frac{\frac{1 + \sqrt{1 + x}}{1}}{\frac{\left(1 + \sqrt{1 + x}\right) \cdot \left(1 + \sqrt{1 + x}\right)}{1}}} \cdot x \]
   rational.json-simplify-7 [=>] 0.2	\[ \frac{\color{blue}{1 + \sqrt{1 + x}}}{\frac{\left(1 + \sqrt{1 + x}\right) \cdot \left(1 + \sqrt{1 + x}\right)}{1}} \cdot x \]
   rational.json-simplify-1 [=>] 0.2	\[ \frac{1 + \sqrt{\color{blue}{x + 1}}}{\frac{\left(1 + \sqrt{1 + x}\right) \cdot \left(1 + \sqrt{1 + x}\right)}{1}} \cdot x \]
   rational.json-simplify-7 [=>] 0.2	\[ \frac{1 + \sqrt{x + 1}}{\color{blue}{\left(1 + \sqrt{1 + x}\right) \cdot \left(1 + \sqrt{1 + x}\right)}} \cdot x \]
   rational.json-simplify-21 [=>] 0.2	\[ \frac{1 + \sqrt{x + 1}}{\color{blue}{\left|\left(1 + \sqrt{1 + x}\right) \cdot \left(1 + \sqrt{1 + x}\right)\right|}} \cdot x \]
   rational.json-simplify-39 [=>] 0.2	\[ \frac{1 + \sqrt{x + 1}}{\color{blue}{\left|1 + \sqrt{1 + x}\right| \cdot \left|1 + \sqrt{1 + x}\right|}} \cdot x \]
   rational.json-simplify-17 [=>] 0.2	\[ \frac{1 + \sqrt{x + 1}}{\left|\color{blue}{\sqrt{1 + x} - -1}\right| \cdot \left|1 + \sqrt{1 + x}\right|} \cdot x \]
   rational.json-simplify-58 [=>] 0.2	\[ \frac{1 + \sqrt{x + 1}}{\color{blue}{\left|-1 - \sqrt{1 + x}\right|} \cdot \left|1 + \sqrt{1 + x}\right|} \cdot x \]
   rational.json-simplify-17 [=>] 0.2	\[ \frac{1 + \sqrt{x + 1}}{\left|-1 - \sqrt{1 + x}\right| \cdot \left|\color{blue}{\sqrt{1 + x} - -1}\right|} \cdot x \]
   rational.json-simplify-58 [=>] 0.2	\[ \frac{1 + \sqrt{x + 1}}{\left|-1 - \sqrt{1 + x}\right| \cdot \color{blue}{\left|-1 - \sqrt{1 + x}\right|}} \cdot x \]
   rational.json-simplify-38 [=>] 0.2	\[ \frac{1 + \sqrt{x + 1}}{\color{blue}{\left|\left(-1 - \sqrt{1 + x}\right) \cdot \left(-1 - \sqrt{1 + x}\right)\right|}} \cdot x \]
   rational.json-simplify-21 [<=] 0.2	\[ \frac{1 + \sqrt{x + 1}}{\color{blue}{\left(-1 - \sqrt{1 + x}\right) \cdot \left(-1 - \sqrt{1 + x}\right)}} \cdot x \]

5. Applied egg-rr 0.2

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

6. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.2
Cost	20416

\[\begin{array}{l} t_0 := \sqrt{x + 1}\\ t_1 := -1 - t_0\\ \frac{1 + t_0}{t_1 \cdot t_1} \cdot x \end{array} \]

Alternative 2
Error	0.2
Cost	6848

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

Alternative 3
Error	20.0
Cost	448

\[\frac{2}{1 + \frac{4}{x}} \]

Alternative 4
Error	19.9
Cost	448

\[\frac{x}{0.5 \cdot x + 2} \]

Alternative 5
Error	20.3
Cost	192

\[\frac{x}{2} \]

Alternative 6
Error	60.9
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2023068 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))