x / (x^2 + 1)

Average Error: 15.0 → 0.1

Time: 1.1s

Precision: binary64

Math

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

↓

\[{\left(x + \frac{1}{x}\right)}^{-1} \]

FPCore

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

↓

(FPCore (x) :precision binary64 (pow (+ x (/ 1.0 x)) -1.0))

C

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

↓

double code(double x) {
	return pow((x + (1.0 / x)), -1.0);
}

Fortran

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

↓

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

Java

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

↓

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

Python

def code(x):
	return x / ((x * x) + 1.0)

↓

def code(x):
	return math.pow((x + (1.0 / x)), -1.0)

Julia

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

↓

function code(x)
	return Float64(x + Float64(1.0 / x)) ^ -1.0
end

MATLAB

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

↓

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

Wolfram

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

↓

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

Error

Bits error versus x

Target

Original	15.0
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 15.0

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

2. Simplified 15.0

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

3. Applied egg-rr 15.1

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

4. Taylor expanded in x around 0 0.1

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

5. Final simplification 0.1

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

Reproduce

herbie shell --seed 2022160 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

  (/ x (+ (* x x) 1.0)))