2frac (problem 3.3.1)

Average Error: 14.1 → 0.6

Time: 21.6s

Precision: binary64

Cost: 21384

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{x + 1} - \frac{1}{x}\\ \mathbf{if}\;t_0 \leq -0.001:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\left(\frac{1}{{x}^{3}} - \frac{1}{{x}^{4}}\right) - \frac{1}{{x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 (+ x 1.0)) (/ 1.0 x))))
   (if (<= t_0 -0.001)
     t_0
     (if (<= t_0 0.0)
       (- (- (/ 1.0 (pow x 3.0)) (/ 1.0 (pow x 4.0))) (/ 1.0 (pow x 2.0)))
       (- 1.0 (/ 1.0 x))))))

C

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

↓

double code(double x) {
	double t_0 = (1.0 / (x + 1.0)) - (1.0 / x);
	double tmp;
	if (t_0 <= -0.001) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = ((1.0 / pow(x, 3.0)) - (1.0 / pow(x, 4.0))) - (1.0 / pow(x, 2.0));
	} else {
		tmp = 1.0 - (1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / (x + 1.0d0)) - (1.0d0 / x)
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / (x + 1.0d0)) - (1.0d0 / x)
    if (t_0 <= (-0.001d0)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = ((1.0d0 / (x ** 3.0d0)) - (1.0d0 / (x ** 4.0d0))) - (1.0d0 / (x ** 2.0d0))
    else
        tmp = 1.0d0 - (1.0d0 / x)
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x) {
	double t_0 = (1.0 / (x + 1.0)) - (1.0 / x);
	double tmp;
	if (t_0 <= -0.001) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = ((1.0 / Math.pow(x, 3.0)) - (1.0 / Math.pow(x, 4.0))) - (1.0 / Math.pow(x, 2.0));
	} else {
		tmp = 1.0 - (1.0 / x);
	}
	return tmp;
}

Python

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

↓

def code(x):
	t_0 = (1.0 / (x + 1.0)) - (1.0 / x)
	tmp = 0
	if t_0 <= -0.001:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = ((1.0 / math.pow(x, 3.0)) - (1.0 / math.pow(x, 4.0))) - (1.0 / math.pow(x, 2.0))
	else:
		tmp = 1.0 - (1.0 / x)
	return tmp

Julia

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

↓

function code(x)
	t_0 = Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(1.0 / x))
	tmp = 0.0
	if (t_0 <= -0.001)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(1.0 / (x ^ 3.0)) - Float64(1.0 / (x ^ 4.0))) - Float64(1.0 / (x ^ 2.0)));
	else
		tmp = Float64(1.0 - Float64(1.0 / x));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x)
	t_0 = (1.0 / (x + 1.0)) - (1.0 / x);
	tmp = 0.0;
	if (t_0 <= -0.001)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = ((1.0 / (x ^ 3.0)) - (1.0 / (x ^ 4.0))) - (1.0 / (x ^ 2.0));
	else
		tmp = 1.0 - (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_] := Block[{t$95$0 = N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.001], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 1 x)) < -1e-3

   1. Initial program 0.0

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

   if -1e-3 < (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 1 x)) < 0.0

   1. Initial program 28.7

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

   2. Applied egg-rr 58.7

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

   3. Taylor expanded in x around inf 0.9

      \[\leadsto \color{blue}{\frac{1}{{x}^{3}} - \left(\frac{1}{{x}^{4}} + \frac{1}{{x}^{2}}\right)} \]

   4. Simplified 0.9

      \[\leadsto \color{blue}{\left(\frac{1}{{x}^{3}} - \frac{1}{{x}^{4}}\right) - \frac{1}{{x}^{2}}} \]
      Proof

      [Start] 0.9	\[ \frac{1}{{x}^{3}} - \left(\frac{1}{{x}^{4}} + \frac{1}{{x}^{2}}\right) \]
      rational.json-simplify-46 [=>] 0.9	\[ \color{blue}{\left(\frac{1}{{x}^{3}} - \frac{1}{{x}^{4}}\right) - \frac{1}{{x}^{2}}} \]

   if 0.0 < (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 1 x))

   1. Initial program 0.0

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

   2. Taylor expanded in x around 0 0.7

      \[\leadsto \color{blue}{1 - \frac{1}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{x + 1} - \frac{1}{x} \leq -0.001:\\ \;\;\;\;\frac{1}{x + 1} - \frac{1}{x}\\ \mathbf{elif}\;\frac{1}{x + 1} - \frac{1}{x} \leq 0:\\ \;\;\;\;\left(\frac{1}{{x}^{3}} - \frac{1}{{x}^{4}}\right) - \frac{1}{{x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	14664

\[\begin{array}{l} t_0 := \frac{1}{x + 1} - \frac{1}{x}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{1}{{x}^{3}} - \frac{1}{{x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 2
Error	0.6
Cost	6920

\[\begin{array}{l} t_0 := \frac{1}{1 + x}\\ t_1 := \frac{-1}{{x}^{2}}\\ \mathbf{if}\;x \leq -76000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 155000000:\\ \;\;\;\;\frac{1}{t_0} \cdot \left(t_0 \cdot t_0\right) - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	15.1
Cost	712

\[\begin{array}{l} t_0 := \frac{1}{x} - \frac{1}{x}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+61}:\\ \;\;\;\;\left(1 - x\right) - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	14.1
Cost	576

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

Alternative 5
Error	30.3
Cost	192

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

Alternative 6
Error	62.0
Cost	128

\[-x \]

Reproduce

herbie shell --seed 2023077 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))