2frac (problem 3.3.1)

Average Error: 14.9 → 0.5

Time: 1.6s

Precision: binary64

Cost: 13640

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{{x}^{3}} - \frac{1}{{x}^{2}}\\ \mathbf{if}\;x \leq -240000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 320000:\\ \;\;\;\;\frac{1}{x + 1} - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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)) (/ 1.0 (pow x 2.0)))))
   (if (<= x -240000.0)
     t_0
     (if (<= x 320000.0) (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)) 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)) - (1.0 / pow(x, 2.0));
	double tmp;
	if (x <= -240000.0) {
		tmp = t_0;
	} else if (x <= 320000.0) {
		tmp = (1.0 / (x + 1.0)) - (1.0 / x);
	} else {
		tmp = 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) :: tmp
    t_0 = (1.0d0 / (x ** 3.0d0)) - (1.0d0 / (x ** 2.0d0))
    if (x <= (-240000.0d0)) then
        tmp = t_0
    else if (x <= 320000.0d0) then
        tmp = (1.0d0 / (x + 1.0d0)) - (1.0d0 / x)
    else
        tmp = 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)) - (1.0 / Math.pow(x, 2.0));
	double tmp;
	if (x <= -240000.0) {
		tmp = t_0;
	} else if (x <= 320000.0) {
		tmp = (1.0 / (x + 1.0)) - (1.0 / x);
	} else {
		tmp = 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)) - (1.0 / math.pow(x, 2.0))
	tmp = 0
	if x <= -240000.0:
		tmp = t_0
	elif x <= 320000.0:
		tmp = (1.0 / (x + 1.0)) - (1.0 / x)
	else:
		tmp = 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(Float64(1.0 / (x ^ 3.0)) - Float64(1.0 / (x ^ 2.0)))
	tmp = 0.0
	if (x <= -240000.0)
		tmp = t_0;
	elseif (x <= 320000.0)
		tmp = Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(1.0 / x));
	else
		tmp = 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)) - (1.0 / (x ^ 2.0));
	tmp = 0.0;
	if (x <= -240000.0)
		tmp = t_0;
	elseif (x <= 320000.0)
		tmp = (1.0 / (x + 1.0)) - (1.0 / x);
	else
		tmp = 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[(N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -240000.0], t$95$0, If[LessEqual[x, 320000.0], N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4e5 or 3.2e5 < x

   1. Initial program 30.2

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

   2. Taylor expanded in x around inf 0.8

      \[\leadsto \color{blue}{\frac{1}{{x}^{3}} - \frac{1}{{x}^{2}}} \]

   if -2.4e5 < x < 3.2e5

   1. Initial program 0.1

      \[\frac{1}{x + 1} - \frac{1}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -240000:\\ \;\;\;\;\frac{1}{{x}^{3}} - \frac{1}{{x}^{2}}\\ \mathbf{elif}\;x \leq 320000:\\ \;\;\;\;\frac{1}{x + 1} - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{3}} - \frac{1}{{x}^{2}}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.0
Cost	7944

\[\begin{array}{l} t_0 := \frac{1}{x + 1} - \frac{1}{x}\\ \mathbf{if}\;t_0 \leq -0.01:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{-1}{{x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	16.1
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 3
Error	14.9
Cost	576

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

Alternative 4
Error	30.8
Cost	192

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

Alternative 5
Error	62.1
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023053 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))