x / (x^2 + 1)

Average Accuracy: 76.7% → 100.0%

Time: 3.6s

Precision: binary64

Cost: 13640

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 2000:\\ \;\;\;\;\frac{{x}^{3} - x}{-1 + {x}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{1}{x \cdot x}}{x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -2e+14)
   (/ 1.0 x)
   (if (<= x 2000.0)
     (/ (- (pow x 3.0) x) (+ -1.0 (pow x 4.0)))
     (/ (- 1.0 (/ 1.0 (* x x))) x))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -2e+14) {
		tmp = 1.0 / x;
	} else if (x <= 2000.0) {
		tmp = (pow(x, 3.0) - x) / (-1.0 + pow(x, 4.0));
	} else {
		tmp = (1.0 - (1.0 / (x * x))) / x;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (x <= (-2d+14)) then
        tmp = 1.0d0 / x
    else if (x <= 2000.0d0) then
        tmp = ((x ** 3.0d0) - x) / ((-1.0d0) + (x ** 4.0d0))
    else
        tmp = (1.0d0 - (1.0d0 / (x * x))) / x
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x) {
	double tmp;
	if (x <= -2e+14) {
		tmp = 1.0 / x;
	} else if (x <= 2000.0) {
		tmp = (Math.pow(x, 3.0) - x) / (-1.0 + Math.pow(x, 4.0));
	} else {
		tmp = (1.0 - (1.0 / (x * x))) / x;
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -2e+14:
		tmp = 1.0 / x
	elif x <= 2000.0:
		tmp = (math.pow(x, 3.0) - x) / (-1.0 + math.pow(x, 4.0))
	else:
		tmp = (1.0 - (1.0 / (x * x))) / x
	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+14)
		tmp = Float64(1.0 / x);
	elseif (x <= 2000.0)
		tmp = Float64(Float64((x ^ 3.0) - x) / Float64(-1.0 + (x ^ 4.0)));
	else
		tmp = Float64(Float64(1.0 - Float64(1.0 / Float64(x * x))) / x);
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2e+14)
		tmp = 1.0 / x;
	elseif (x <= 2000.0)
		tmp = ((x ^ 3.0) - x) / (-1.0 + (x ^ 4.0));
	else
		tmp = (1.0 - (1.0 / (x * x))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_] := If[LessEqual[x, -2e+14], N[(1.0 / x), $MachinePrecision], If[LessEqual[x, 2000.0], N[(N[(N[Power[x, 3.0], $MachinePrecision] - x), $MachinePrecision] / N[(-1.0 + N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Target

Original	76.7%
Target	99.8%
Herbie	100.0%

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

Derivation

1. Split input into 3 regimes

2. if x < -2e14

   1. Initial program 51.8%

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

   2. Taylor expanded in x around inf 100.0%

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

   if -2e14 < x < 2e3

   1. Initial program 100.0%

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

   2. Applied egg-rr 100.0%

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

      [Start] 100.0	\[ \frac{x}{x \cdot x + 1} \]
      flip-+ [=>] 100.0	\[ \frac{x}{\color{blue}{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot 1}{x \cdot x - 1}}} \]
      associate-/r/ [=>] 100.0	\[ \color{blue}{\frac{x}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1 \cdot 1} \cdot \left(x \cdot x - 1\right)} \]
      metadata-eval [=>] 100.0	\[ \frac{x}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - \color{blue}{1}} \cdot \left(x \cdot x - 1\right) \]
      sub-neg [=>] 100.0	\[ \frac{x}{\color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(-1\right)}} \cdot \left(x \cdot x - 1\right) \]
      pow2 [=>] 100.0	\[ \frac{x}{\color{blue}{{x}^{2}} \cdot \left(x \cdot x\right) + \left(-1\right)} \cdot \left(x \cdot x - 1\right) \]
      pow2 [=>] 100.0	\[ \frac{x}{{x}^{2} \cdot \color{blue}{{x}^{2}} + \left(-1\right)} \cdot \left(x \cdot x - 1\right) \]
      pow-sqr [=>] 100.0	\[ \frac{x}{\color{blue}{{x}^{\left(2 \cdot 2\right)}} + \left(-1\right)} \cdot \left(x \cdot x - 1\right) \]
      metadata-eval [=>] 100.0	\[ \frac{x}{{x}^{\color{blue}{4}} + \left(-1\right)} \cdot \left(x \cdot x - 1\right) \]
      metadata-eval [=>] 100.0	\[ \frac{x}{{x}^{4} + \color{blue}{-1}} \cdot \left(x \cdot x - 1\right) \]
      fma-neg [=>] 100.0	\[ \frac{x}{{x}^{4} + -1} \cdot \color{blue}{\mathsf{fma}\left(x, x, -1\right)} \]
      metadata-eval [=>] 100.0	\[ \frac{x}{{x}^{4} + -1} \cdot \mathsf{fma}\left(x, x, \color{blue}{-1}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{{x}^{3} - x}{-1 + {x}^{4}}} \]
      Proof

      [Start] 100.0	\[ \frac{x}{{x}^{4} + -1} \cdot \mathsf{fma}\left(x, x, -1\right) \]
      associate-*l/ [=>] 100.0	\[ \color{blue}{\frac{x \cdot \mathsf{fma}\left(x, x, -1\right)}{{x}^{4} + -1}} \]
      fma-udef [=>] 100.0	\[ \frac{x \cdot \color{blue}{\left(x \cdot x + -1\right)}}{{x}^{4} + -1} \]
      distribute-rgt-in [=>] 100.0	\[ \frac{\color{blue}{\left(x \cdot x\right) \cdot x + -1 \cdot x}}{{x}^{4} + -1} \]
      neg-mul-1 [<=] 100.0	\[ \frac{\left(x \cdot x\right) \cdot x + \color{blue}{\left(-x\right)}}{{x}^{4} + -1} \]
      unpow3 [<=] 100.0	\[ \frac{\color{blue}{{x}^{3}} + \left(-x\right)}{{x}^{4} + -1} \]
      unsub-neg [=>] 100.0	\[ \frac{\color{blue}{{x}^{3} - x}}{{x}^{4} + -1} \]
      +-commutative [=>] 100.0	\[ \frac{{x}^{3} - x}{\color{blue}{-1 + {x}^{4}}} \]

   if 2e3 < x

   1. Initial program 53.0%

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

   2. Taylor expanded in x around inf 100.0%

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

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(1 - {x}^{-2}\right)} \]
      Proof

      [Start] 100.0	\[ \frac{1}{x} - \frac{1}{{x}^{3}} \]
      *-un-lft-identity [=>] 100.0	\[ \color{blue}{1 \cdot \frac{1}{x}} - \frac{1}{{x}^{3}} \]
      metadata-eval [<=] 100.0	\[ 1 \cdot \frac{1}{x} - \frac{\color{blue}{{1}^{3}}}{{x}^{3}} \]
      cube-div [<=] 100.0	\[ 1 \cdot \frac{1}{x} - \color{blue}{{\left(\frac{1}{x}\right)}^{3}} \]
      unpow3 [=>] 100.0	\[ 1 \cdot \frac{1}{x} - \color{blue}{\left(\frac{1}{x} \cdot \frac{1}{x}\right) \cdot \frac{1}{x}} \]
      distribute-rgt-out-- [=>] 100.0	\[ \color{blue}{\frac{1}{x} \cdot \left(1 - \frac{1}{x} \cdot \frac{1}{x}\right)} \]
      inv-pow [=>] 100.0	\[ \frac{1}{x} \cdot \left(1 - \color{blue}{{x}^{-1}} \cdot \frac{1}{x}\right) \]
      metadata-eval [<=] 100.0	\[ \frac{1}{x} \cdot \left(1 - {x}^{\color{blue}{\left(-1\right)}} \cdot \frac{1}{x}\right) \]
      inv-pow [=>] 100.0	\[ \frac{1}{x} \cdot \left(1 - {x}^{\left(-1\right)} \cdot \color{blue}{{x}^{-1}}\right) \]
      metadata-eval [<=] 100.0	\[ \frac{1}{x} \cdot \left(1 - {x}^{\left(-1\right)} \cdot {x}^{\color{blue}{\left(-1\right)}}\right) \]
      pow-sqr [=>] 100.0	\[ \frac{1}{x} \cdot \left(1 - \color{blue}{{x}^{\left(2 \cdot \left(-1\right)\right)}}\right) \]
      metadata-eval [=>] 100.0	\[ \frac{1}{x} \cdot \left(1 - {x}^{\left(2 \cdot \color{blue}{-1}\right)}\right) \]
      metadata-eval [=>] 100.0	\[ \frac{1}{x} \cdot \left(1 - {x}^{\color{blue}{-2}}\right) \]

   4. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1 - {x}^{-2}}{x}} \]
      Proof

      [Start] 100.0	\[ \frac{1}{x} \cdot \left(1 - {x}^{-2}\right) \]
      sub-neg [=>] 100.0	\[ \frac{1}{x} \cdot \color{blue}{\left(1 + \left(-{x}^{-2}\right)\right)} \]
      associate-*l/ [=>] 100.0	\[ \color{blue}{\frac{1 \cdot \left(1 + \left(-{x}^{-2}\right)\right)}{x}} \]
      *-lft-identity [=>] 100.0	\[ \frac{\color{blue}{1 + \left(-{x}^{-2}\right)}}{x} \]
      sub-neg [<=] 100.0	\[ \frac{\color{blue}{1 - {x}^{-2}}}{x} \]

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{1 - \color{blue}{\frac{1}{{x}^{2}}}}{x} \]

   6. Simplified 100.0%

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

      [Start] 100.0	\[ \frac{1 - \frac{1}{{x}^{2}}}{x} \]
      unpow2 [=>] 100.0	\[ \frac{1 - \frac{1}{\color{blue}{x \cdot x}}}{x} \]

3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 2000:\\ \;\;\;\;\frac{{x}^{3} - x}{-1 + {x}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{1}{x \cdot x}}{x}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	100.0%
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 2000:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{1}{x \cdot x}}{x}\\ \end{array} \]

Alternative 2
Accuracy	100.0%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{elif}\;x \leq 200000:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 3
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 4
Accuracy	51.0%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023136 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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