x / (x^2 + 1)

Percentage Accurate: 75.8% → 100.0%

Time: 3.7s

Precision: binary64

Cost: 7112

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+24}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 10000:\\ \;\;\;\;x \cdot \frac{1}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - {x}^{-3}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -2e+24)
   (/ 1.0 x)
   (if (<= x 10000.0) (* x (/ 1.0 (fma x x 1.0))) (- (/ 1.0 x) (pow x -3.0)))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -2e+24) {
		tmp = 1.0 / x;
	} else if (x <= 10000.0) {
		tmp = x * (1.0 / fma(x, x, 1.0));
	} else {
		tmp = (1.0 / x) - pow(x, -3.0);
	}
	return tmp;
}

Julia

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

↓

function code(x)
	tmp = 0.0
	if (x <= -2e+24)
		tmp = Float64(1.0 / x);
	elseif (x <= 10000.0)
		tmp = Float64(x * Float64(1.0 / fma(x, x, 1.0)));
	else
		tmp = Float64(Float64(1.0 / x) - (x ^ -3.0));
	end
	return tmp
end

Wolfram

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

↓

code[x_] := If[LessEqual[x, -2e+24], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 10000.0], N[(x * N[(1.0 / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] - N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision]]]

Target

Original	75.8%
Target	99.9%
Herbie	100.0%

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

Derivation

1. Split input into 3 regimes

2. if x < -2e24

   1. Initial program 56.7%

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

   2. Taylor expanded in x around inf 100.0%

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

   if -2e24 < x < 1e4

   1. Initial program 100.0%

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

   2. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(x, x, 1\right)} \cdot x} \]
      Step-by-step derivation

      [Start] 100.0%	\[ \frac{x}{x \cdot x + 1} \]
      clear-num [=>] 99.7%	\[ \color{blue}{\frac{1}{\frac{x \cdot x + 1}{x}}} \]
      associate-/r/ [=>] 100.0%	\[ \color{blue}{\frac{1}{x \cdot x + 1} \cdot x} \]
      fma-def [=>] 100.0%	\[ \frac{1}{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}} \cdot x \]

   if 1e4 < x

   1. Initial program 40.7%

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

   2. Applied egg-rr 40.7%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(x, x, 1\right)} \cdot x} \]
      Step-by-step derivation

      [Start] 40.7%	\[ \frac{x}{x \cdot x + 1} \]
      clear-num [=>] 40.7%	\[ \color{blue}{\frac{1}{\frac{x \cdot x + 1}{x}}} \]
      associate-/r/ [=>] 40.7%	\[ \color{blue}{\frac{1}{x \cdot x + 1} \cdot x} \]
      fma-def [=>] 40.7%	\[ \frac{1}{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}} \cdot x \]

   3. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\frac{1}{x} - \frac{1}{{x}^{3}}} \]

   4. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{x} - {x}^{-3}} \]
      Step-by-step derivation

      [Start] 100.0%	\[ \frac{1}{x} - \frac{1}{{x}^{3}} \]
      unpow-1 [<=] 100.0%	\[ \frac{1}{x} - \color{blue}{{\left({x}^{3}\right)}^{-1}} \]
      exp-to-pow [<=] 100.0%	\[ \frac{1}{x} - \color{blue}{e^{\log \left({x}^{3}\right) \cdot -1}} \]
      *-commutative [=>] 100.0%	\[ \frac{1}{x} - e^{\color{blue}{-1 \cdot \log \left({x}^{3}\right)}} \]
      log-pow [=>] 100.0%	\[ \frac{1}{x} - e^{-1 \cdot \color{blue}{\left(3 \cdot \log x\right)}} \]
      associate-*r* [=>] 100.0%	\[ \frac{1}{x} - e^{\color{blue}{\left(-1 \cdot 3\right) \cdot \log x}} \]
      metadata-eval [=>] 100.0%	\[ \frac{1}{x} - e^{\color{blue}{-3} \cdot \log x} \]
      *-commutative [<=] 100.0%	\[ \frac{1}{x} - e^{\color{blue}{\log x \cdot -3}} \]
      exp-to-pow [=>] 100.0%	\[ \frac{1}{x} - \color{blue}{{x}^{-3}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+24}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 10000:\\ \;\;\;\;x \cdot \frac{1}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - {x}^{-3}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	100.0%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+24}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 10000:\\ \;\;\;\;x \cdot \frac{1}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - {x}^{-3}\\ \end{array} \]

Alternative 2
Accuracy	100.0%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+66}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 20000:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - {x}^{-3}\\ \end{array} \]

Alternative 3
Accuracy	100.0%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+66}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 20000000:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 4
Accuracy	99.0%
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 5
Accuracy	51.1%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023271 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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