2frac (problem 3.3.1)

Average Error: 14.8 → 0.4

Time: 8.1s

Precision: binary64

Cost: 80968

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{{x}^{3}}\\ t_1 := \frac{1}{{x}^{4}} + \frac{1}{{x}^{2}}\\ t_2 := t_0 - t_1\\ \mathbf{if}\;x \leq -1950:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + t_0\right) - t_1\\ \mathbf{elif}\;x \leq 14200:\\ \;\;\;\;\frac{1}{x + 1} - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2 - t_2 \cdot t_2}{1 + \left(t_1 - t_0\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (pow x 3.0)))
        (t_1 (+ (/ 1.0 (pow x 4.0)) (/ 1.0 (pow x 2.0))))
        (t_2 (- t_0 t_1)))
   (if (<= x -1950.0)
     (- (+ (/ 1.0 (pow x 5.0)) t_0) t_1)
     (if (<= x 14200.0)
       (- (/ 1.0 (+ x 1.0)) (/ 1.0 x))
       (/ (- t_2 (* t_2 t_2)) (+ 1.0 (- t_1 t_0)))))))

C

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

↓

double code(double x) {
	double t_0 = 1.0 / pow(x, 3.0);
	double t_1 = (1.0 / pow(x, 4.0)) + (1.0 / pow(x, 2.0));
	double t_2 = t_0 - t_1;
	double tmp;
	if (x <= -1950.0) {
		tmp = ((1.0 / pow(x, 5.0)) + t_0) - t_1;
	} else if (x <= 14200.0) {
		tmp = (1.0 / (x + 1.0)) - (1.0 / x);
	} else {
		tmp = (t_2 - (t_2 * t_2)) / (1.0 + (t_1 - t_0));
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 / (x ** 3.0d0)
    t_1 = (1.0d0 / (x ** 4.0d0)) + (1.0d0 / (x ** 2.0d0))
    t_2 = t_0 - t_1
    if (x <= (-1950.0d0)) then
        tmp = ((1.0d0 / (x ** 5.0d0)) + t_0) - t_1
    else if (x <= 14200.0d0) then
        tmp = (1.0d0 / (x + 1.0d0)) - (1.0d0 / x)
    else
        tmp = (t_2 - (t_2 * t_2)) / (1.0d0 + (t_1 - t_0))
    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 / Math.pow(x, 3.0);
	double t_1 = (1.0 / Math.pow(x, 4.0)) + (1.0 / Math.pow(x, 2.0));
	double t_2 = t_0 - t_1;
	double tmp;
	if (x <= -1950.0) {
		tmp = ((1.0 / Math.pow(x, 5.0)) + t_0) - t_1;
	} else if (x <= 14200.0) {
		tmp = (1.0 / (x + 1.0)) - (1.0 / x);
	} else {
		tmp = (t_2 - (t_2 * t_2)) / (1.0 + (t_1 - t_0));
	}
	return tmp;
}

Python

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

↓

def code(x):
	t_0 = 1.0 / math.pow(x, 3.0)
	t_1 = (1.0 / math.pow(x, 4.0)) + (1.0 / math.pow(x, 2.0))
	t_2 = t_0 - t_1
	tmp = 0
	if x <= -1950.0:
		tmp = ((1.0 / math.pow(x, 5.0)) + t_0) - t_1
	elif x <= 14200.0:
		tmp = (1.0 / (x + 1.0)) - (1.0 / x)
	else:
		tmp = (t_2 - (t_2 * t_2)) / (1.0 + (t_1 - t_0))
	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(1.0 / (x ^ 3.0))
	t_1 = Float64(Float64(1.0 / (x ^ 4.0)) + Float64(1.0 / (x ^ 2.0)))
	t_2 = Float64(t_0 - t_1)
	tmp = 0.0
	if (x <= -1950.0)
		tmp = Float64(Float64(Float64(1.0 / (x ^ 5.0)) + t_0) - t_1);
	elseif (x <= 14200.0)
		tmp = Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(1.0 / x));
	else
		tmp = Float64(Float64(t_2 - Float64(t_2 * t_2)) / Float64(1.0 + Float64(t_1 - t_0)));
	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 ^ 3.0);
	t_1 = (1.0 / (x ^ 4.0)) + (1.0 / (x ^ 2.0));
	t_2 = t_0 - t_1;
	tmp = 0.0;
	if (x <= -1950.0)
		tmp = ((1.0 / (x ^ 5.0)) + t_0) - t_1;
	elseif (x <= 14200.0)
		tmp = (1.0 / (x + 1.0)) - (1.0 / x);
	else
		tmp = (t_2 - (t_2 * t_2)) / (1.0 + (t_1 - t_0));
	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[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - t$95$1), $MachinePrecision]}, If[LessEqual[x, -1950.0], N[(N[(N[(1.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[x, 14200.0], N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 - N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$1 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1950

   1. Initial program 29.5

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

   2. Taylor expanded in x around inf 0.8

      \[\leadsto \color{blue}{\left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{3}}\right) - \left(\frac{1}{{x}^{4}} + \frac{1}{{x}^{2}}\right)} \]

   if -1950 < x < 14200

   1. Initial program 0.1

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

   if 14200 < x

   1. Initial program 30.0

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

   2. Taylor expanded in x around inf 0.7

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

   3. Applied egg-rr 0.7

      \[\leadsto \color{blue}{\frac{\left(\frac{1}{{x}^{3}} - \left(\frac{1}{{x}^{4}} + \frac{1}{{x}^{2}}\right)\right) - \left(\frac{1}{{x}^{3}} - \left(\frac{1}{{x}^{4}} + \frac{1}{{x}^{2}}\right)\right) \cdot \left(\frac{1}{{x}^{3}} - \left(\frac{1}{{x}^{4}} + \frac{1}{{x}^{2}}\right)\right)}{1 + \left(\left(\frac{1}{{x}^{4}} + \frac{1}{{x}^{2}}\right) - \frac{1}{{x}^{3}}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1950:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{3}}\right) - \left(\frac{1}{{x}^{4}} + \frac{1}{{x}^{2}}\right)\\ \mathbf{elif}\;x \leq 14200:\\ \;\;\;\;\frac{1}{x + 1} - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{{x}^{3}} - \left(\frac{1}{{x}^{4}} + \frac{1}{{x}^{2}}\right)\right) - \left(\frac{1}{{x}^{3}} - \left(\frac{1}{{x}^{4}} + \frac{1}{{x}^{2}}\right)\right) \cdot \left(\frac{1}{{x}^{3}} - \left(\frac{1}{{x}^{4}} + \frac{1}{{x}^{2}}\right)\right)}{1 + \left(\left(\frac{1}{{x}^{4}} + \frac{1}{{x}^{2}}\right) - \frac{1}{{x}^{3}}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	28104

\[\begin{array}{l} t_0 := \frac{1}{x + 1} - \frac{1}{x}\\ \mathbf{if}\;t_0 \leq -0.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{3}}\right) - \left(\frac{1}{{x}^{4}} + \frac{1}{{x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) - \frac{1}{x}\\ \end{array} \]

Alternative 2
Error	0.7
Cost	21384

\[\begin{array}{l} t_0 := \frac{1}{x + 1} - \frac{1}{x}\\ \mathbf{if}\;t_0 \leq -0.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{1}{{x}^{3}} - \left(\frac{1}{{x}^{4}} + \frac{1}{{x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) - \frac{1}{x}\\ \end{array} \]

Alternative 3
Error	0.5
Cost	14664

\[\begin{array}{l} t_0 := \frac{1}{x + 1} - \frac{1}{x}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{1}{{x}^{3}} - \frac{1}{{x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) - \frac{1}{x}\\ \end{array} \]

Alternative 4
Error	0.6
Cost	6920

\[\begin{array}{l} t_0 := \frac{-1}{{x}^{2}}\\ \mathbf{if}\;x \leq -110000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 250000000:\\ \;\;\;\;\frac{1}{x + 1} - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	15.8
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 6
Error	14.8
Cost	576

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

Alternative 7
Error	30.9
Cost	192

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

Reproduce

herbie shell --seed 2023104 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))