x / (x^2 + 1)

Average Accuracy: 76.9% → 99.9%

Time: 2.9s

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	76.9%
Target	99.9%
Herbie	99.9%

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

Derivation

1. Initial program 76.9%

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

2. Applied egg-rr 76.8%

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

   [Start] 76.9	\[ \frac{x}{x \cdot x + 1} \]
   clear-num [=>] 76.8	\[ \color{blue}{\frac{1}{\frac{x \cdot x + 1}{x}}} \]
   inv-pow [=>] 76.8	\[ \color{blue}{{\left(\frac{x \cdot x + 1}{x}\right)}^{-1}} \]
   fma-def [=>] 76.8	\[ {\left(\frac{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}}{x}\right)}^{-1} \]

3. Taylor expanded in x around 0 99.9%

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

4. Applied egg-rr 99.9%

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

   [Start] 99.9	\[ {\left(\frac{1}{x} + x\right)}^{-1} \]
   unpow-1 [=>] 99.9	\[ \color{blue}{\frac{1}{\frac{1}{x} + x}} \]
   +-commutative [=>] 99.9	\[ \frac{1}{\color{blue}{x + \frac{1}{x}}} \]

5. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.1%
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 2
Accuracy	51.4%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023151 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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