x / (x^2 + 1)

Average Accuracy: 76.6% → 99.8%

Time: 6.0s

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	76.6%
Target	99.8%
Herbie	99.8%

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

Derivation

1. Initial program 76.6%

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

2. Applied egg-rr 58.6%

   \[\leadsto \color{blue}{\frac{x}{1 + {x}^{6}} \cdot \left({x}^{4} + \left(1 - x \cdot x\right)\right)} \]
   Proof

   [Start] 76.6	\[ \frac{x}{x \cdot x + 1} \]
   flip3-+ [=>] 58.6	\[ \frac{x}{\color{blue}{\frac{{\left(x \cdot x\right)}^{3} + {1}^{3}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 \cdot 1 - \left(x \cdot x\right) \cdot 1\right)}}} \]
   associate-/r/ [=>] 58.6	\[ \color{blue}{\frac{x}{{\left(x \cdot x\right)}^{3} + {1}^{3}} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 \cdot 1 - \left(x \cdot x\right) \cdot 1\right)\right)} \]
   metadata-eval [=>] 58.6	\[ \frac{x}{{\left(x \cdot x\right)}^{3} + \color{blue}{1}} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 \cdot 1 - \left(x \cdot x\right) \cdot 1\right)\right) \]
   +-commutative [=>] 58.6	\[ \frac{x}{\color{blue}{1 + {\left(x \cdot x\right)}^{3}}} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 \cdot 1 - \left(x \cdot x\right) \cdot 1\right)\right) \]
   pow2 [=>] 58.6	\[ \frac{x}{1 + {\color{blue}{\left({x}^{2}\right)}}^{3}} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 \cdot 1 - \left(x \cdot x\right) \cdot 1\right)\right) \]
   pow-pow [=>] 58.6	\[ \frac{x}{1 + \color{blue}{{x}^{\left(2 \cdot 3\right)}}} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 \cdot 1 - \left(x \cdot x\right) \cdot 1\right)\right) \]
   metadata-eval [=>] 58.6	\[ \frac{x}{1 + {x}^{\color{blue}{6}}} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 \cdot 1 - \left(x \cdot x\right) \cdot 1\right)\right) \]
   metadata-eval [=>] 58.6	\[ \frac{x}{1 + {x}^{6}} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\color{blue}{1} - \left(x \cdot x\right) \cdot 1\right)\right) \]
   *-rgt-identity [=>] 58.6	\[ \frac{x}{1 + {x}^{6}} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(1 - \color{blue}{x \cdot x}\right)\right) \]
   pow2 [=>] 58.6	\[ \frac{x}{1 + {x}^{6}} \cdot \left(\color{blue}{{x}^{2}} \cdot \left(x \cdot x\right) + \left(1 - x \cdot x\right)\right) \]
   pow2 [=>] 58.6	\[ \frac{x}{1 + {x}^{6}} \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}} + \left(1 - x \cdot x\right)\right) \]
   pow-sqr [=>] 58.6	\[ \frac{x}{1 + {x}^{6}} \cdot \left(\color{blue}{{x}^{\left(2 \cdot 2\right)}} + \left(1 - x \cdot x\right)\right) \]
   metadata-eval [=>] 58.6	\[ \frac{x}{1 + {x}^{6}} \cdot \left({x}^{\color{blue}{4}} + \left(1 - x \cdot x\right)\right) \]

3. Applied egg-rr 76.6%

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

   [Start] 58.6	\[ \frac{x}{1 + {x}^{6}} \cdot \left({x}^{4} + \left(1 - x \cdot x\right)\right) \]
   associate-*l/ [=>] 58.5	\[ \color{blue}{\frac{x \cdot \left({x}^{4} + \left(1 - x \cdot x\right)\right)}{1 + {x}^{6}}} \]
   clear-num [=>] 58.4	\[ \color{blue}{\frac{1}{\frac{1 + {x}^{6}}{x \cdot \left({x}^{4} + \left(1 - x \cdot x\right)\right)}}} \]
   *-commutative [=>] 58.4	\[ \frac{1}{\frac{1 + {x}^{6}}{\color{blue}{\left({x}^{4} + \left(1 - x \cdot x\right)\right) \cdot x}}} \]
   associate-/r* [=>] 58.5	\[ \frac{1}{\color{blue}{\frac{\frac{1 + {x}^{6}}{{x}^{4} + \left(1 - x \cdot x\right)}}{x}}} \]

4. Taylor expanded in x around 0 99.8%

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

5. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	99.2%
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
Accuracy	98.9%
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
Accuracy	51.6%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023138 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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