2frac (problem 3.3.1)

Average Error: 14.7 → 0.4

Time: 1.6min

Precision: binary64

Cost: 21384

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{x + 1}\\ t_1 := t_0 - \frac{1}{x}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{1}{{x}^{3}} - \left(\frac{1}{{x}^{4}} + \frac{1}{{x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(t_0 + \frac{1}{x} \cdot -2\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ x 1.0))) (t_1 (- t_0 (/ 1.0 x))))
   (if (<= t_1 -2e-6)
     t_1
     (if (<= t_1 0.0)
       (- (/ 1.0 (pow x 3.0)) (+ (/ 1.0 (pow x 4.0)) (/ 1.0 (pow x 2.0))))
       (+ (/ 1.0 x) (+ t_0 (* (/ 1.0 x) -2.0)))))))

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);
	double t_1 = t_0 - (1.0 / x);
	double tmp;
	if (t_1 <= -2e-6) {
		tmp = t_1;
	} else if (t_1 <= 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 / x) + (t_0 + ((1.0 / x) * -2.0));
	}
	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) :: tmp
    t_0 = 1.0d0 / (x + 1.0d0)
    t_1 = t_0 - (1.0d0 / x)
    if (t_1 <= (-2d-6)) then
        tmp = t_1
    else if (t_1 <= 0.0d0) then
        tmp = (1.0d0 / (x ** 3.0d0)) - ((1.0d0 / (x ** 4.0d0)) + (1.0d0 / (x ** 2.0d0)))
    else
        tmp = (1.0d0 / x) + (t_0 + ((1.0d0 / x) * (-2.0d0)))
    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);
	double t_1 = t_0 - (1.0 / x);
	double tmp;
	if (t_1 <= -2e-6) {
		tmp = t_1;
	} else if (t_1 <= 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 / x) + (t_0 + ((1.0 / x) * -2.0));
	}
	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)
	t_1 = t_0 - (1.0 / x)
	tmp = 0
	if t_1 <= -2e-6:
		tmp = t_1
	elif t_1 <= 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 / x) + (t_0 + ((1.0 / x) * -2.0))
	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 / Float64(x + 1.0))
	t_1 = Float64(t_0 - Float64(1.0 / x))
	tmp = 0.0
	if (t_1 <= -2e-6)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(1.0 / (x ^ 3.0)) - Float64(Float64(1.0 / (x ^ 4.0)) + Float64(1.0 / (x ^ 2.0))));
	else
		tmp = Float64(Float64(1.0 / x) + Float64(t_0 + Float64(Float64(1.0 / x) * -2.0)));
	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 + 1.0);
	t_1 = t_0 - (1.0 / x);
	tmp = 0.0;
	if (t_1 <= -2e-6)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = (1.0 / (x ^ 3.0)) - ((1.0 / (x ^ 4.0)) + (1.0 / (x ^ 2.0)));
	else
		tmp = (1.0 / x) + (t_0 + ((1.0 / x) * -2.0));
	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[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-6], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] + N[(t$95$0 + N[(N[(1.0 / x), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 1 x)) < -1.99999999999999991e-6

   1. Initial program 0.1

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

   if -1.99999999999999991e-6 < (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 1 x)) < 0.0

   1. Initial program 29.5

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

   2. Taylor expanded in x around inf 0.8

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

   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. Applied egg-rr 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{1 + x} + \frac{1}{x}\right) + \left(0 - \frac{1}{x} \cdot 2\right)} \]

   3. Simplified 0.0

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

      [Start] 0.0	\[ \left(\frac{1}{1 + x} + \frac{1}{x}\right) + \left(0 - \frac{1}{x} \cdot 2\right) \]
      rational_best.json-simplify-1 [=>] 0.0	\[ \color{blue}{\left(0 - \frac{1}{x} \cdot 2\right) + \left(\frac{1}{1 + x} + \frac{1}{x}\right)} \]
      rational_best.json-simplify-43 [=>] 0.0	\[ \color{blue}{\frac{1}{x} + \left(\frac{1}{1 + x} + \left(0 - \frac{1}{x} \cdot 2\right)\right)} \]
      rational_best.json-simplify-1 [=>] 0.0	\[ \frac{1}{x} + \color{blue}{\left(\left(0 - \frac{1}{x} \cdot 2\right) + \frac{1}{1 + x}\right)} \]
      rational_best.json-simplify-1 [<=] 0.0	\[ \frac{1}{x} + \color{blue}{\left(\frac{1}{1 + x} + \left(0 - \frac{1}{x} \cdot 2\right)\right)} \]
      rational_best.json-simplify-1 [=>] 0.0	\[ \frac{1}{x} + \left(\frac{1}{\color{blue}{x + 1}} + \left(0 - \frac{1}{x} \cdot 2\right)\right) \]
      rational_best.json-simplify-10 [=>] 0.0	\[ \frac{1}{x} + \left(\frac{1}{x + 1} + \color{blue}{\left(-\frac{1}{x} \cdot 2\right)}\right) \]

   4. Applied egg-rr 0.0

      \[\leadsto \frac{1}{x} + \color{blue}{\left(\left(\frac{1}{1 + x} + \frac{1}{x} \cdot -2\right) - 0\right)} \]

   5. Simplified 0.0

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

      [Start] 0.0	\[ \frac{1}{x} + \left(\left(\frac{1}{1 + x} + \frac{1}{x} \cdot -2\right) - 0\right) \]
      rational_best.json-simplify-6 [=>] 0.0	\[ \frac{1}{x} + \color{blue}{\left(\frac{1}{1 + x} + \frac{1}{x} \cdot -2\right)} \]
      rational_best.json-simplify-1 [=>] 0.0	\[ \frac{1}{x} + \left(\frac{1}{\color{blue}{x + 1}} + \frac{1}{x} \cdot -2\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{x + 1} - \frac{1}{x} \leq -2 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x + 1} - \frac{1}{x}\\ \mathbf{elif}\;\frac{1}{x + 1} - \frac{1}{x} \leq 0:\\ \;\;\;\;\frac{1}{{x}^{3}} - \left(\frac{1}{{x}^{4}} + \frac{1}{{x}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{x + 1} + \frac{1}{x} \cdot -2\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	14664

\[\begin{array}{l} t_0 := \frac{1}{x + 1}\\ t_1 := t_0 - \frac{1}{x}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{1}{{x}^{3}} - \frac{1}{{x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(t_0 + \frac{1}{x} \cdot -2\right)\\ \end{array} \]

Alternative 2
Error	0.7
Cost	7944

\[\begin{array}{l} t_0 := \frac{1}{x + 1}\\ t_1 := t_0 - \frac{1}{x}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{-1}{{x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(t_0 + \frac{1}{x} \cdot -2\right)\\ \end{array} \]

Alternative 3
Error	15.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+61}:\\ \;\;\;\;1 - \left(x + \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4
Error	16.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5
Error	14.7
Cost	576

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

Alternative 6
Error	16.2
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+102}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7
Error	46.7
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023088 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))