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

Average Error: 0.2 → 0.2

Time: 2.5s

Precision: binary64

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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
    code = x * (1.0d0 / (1.0d0 + sqrt((x + 1.0d0))))
end function

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 0.2

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

2. Applied egg-rr 0.2

   \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({\left(x + 1\right)}^{0.25}, {\left(x + 1\right)}^{0.25}, 1\right)}} \]

3. Applied egg-rr 0.2

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

4. Final simplification 0.2

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

Reproduce

herbie shell --seed 2022210 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))