x / (x^2 + 1)

Average Error: 14.9 → 0.1

Time: 2.8s

Precision: binary64

Cost: 448

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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 = 1.0d0 / (x + (1.0d0 / x))
end function

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Target

Original	14.9
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 14.9

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

2. Applied egg-rr 15.0

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

3. Taylor expanded in x around 0 0.1

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

4. Applied egg-rr 58.7

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{x + \frac{1}{x}}\right)} - 1} \]

5. Simplified 0.1

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

   [Start] 58.7	\[ e^{\mathsf{log1p}\left(\frac{1}{x + \frac{1}{x}}\right)} - 1 \]
   expm1-def [=>] 0.1	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x + \frac{1}{x}}\right)\right)} \]
   expm1-log1p [=>] 0.1	\[ \color{blue}{\frac{1}{x + \frac{1}{x}}} \]

6. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -0.86:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 0.86:\\ \;\;\;\;x \cdot \left(1 - x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 2
Error	0.6
Cost	456

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

Alternative 3
Error	30.5
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023056 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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