2frac (problem 3.3.1)

Average Error: 13.8 → 0.6

Time: 18.4s

Precision: binary64

Cost: 21384

Math

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

↓

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

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 -4e-6)
     (- (/ (/ 1.0 (* (+ x 1.0) (+ x 1.0))) t_0) (/ 1.0 x))
     (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) (/ 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);
	double t_1 = t_0 - (1.0 / x);
	double tmp;
	if (t_1 <= -4e-6) {
		tmp = ((1.0 / ((x + 1.0) * (x + 1.0))) / t_0) - (1.0 / x);
	} 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) - (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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 / (x + 1.0d0)
    t_1 = t_0 - (1.0d0 / x)
    if (t_1 <= (-4d-6)) then
        tmp = ((1.0d0 / ((x + 1.0d0) * (x + 1.0d0))) / t_0) - (1.0d0 / x)
    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) - (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);
	double t_1 = t_0 - (1.0 / x);
	double tmp;
	if (t_1 <= -4e-6) {
		tmp = ((1.0 / ((x + 1.0) * (x + 1.0))) / t_0) - (1.0 / x);
	} 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) - (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)
	t_1 = t_0 - (1.0 / x)
	tmp = 0
	if t_1 <= -4e-6:
		tmp = ((1.0 / ((x + 1.0) * (x + 1.0))) / t_0) - (1.0 / x)
	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) - (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(1.0 / Float64(x + 1.0))
	t_1 = Float64(t_0 - Float64(1.0 / x))
	tmp = 0.0
	if (t_1 <= -4e-6)
		tmp = Float64(Float64(Float64(1.0 / Float64(Float64(x + 1.0) * Float64(x + 1.0))) / t_0) - Float64(1.0 / x));
	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(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);
	t_1 = t_0 - (1.0 / x);
	tmp = 0.0;
	if (t_1 <= -4e-6)
		tmp = ((1.0 / ((x + 1.0) * (x + 1.0))) / t_0) - (1.0 / x);
	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) - (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[(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, -4e-6], N[(N[(N[(1.0 / N[(N[(x + 1.0), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] - N[(1.0 / x), $MachinePrecision]), $MachinePrecision], 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[(1.0 / x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 1 x)) < -3.99999999999999982e-6

   1. Initial program 0.1

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

   2. Applied egg-rr 0.1

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

   3. Simplified 0.1

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

      [Start] 0.1	\[ \left(1 + x\right) \cdot \frac{\frac{1}{1 + x}}{1 + x} - \frac{1}{x} \]
      rational.json-simplify-1 [=>] 0.1	\[ \color{blue}{\left(x + 1\right)} \cdot \frac{\frac{1}{1 + x}}{1 + x} - \frac{1}{x} \]
      rational.json-simplify-47 [=>] 0.1	\[ \left(x + 1\right) \cdot \color{blue}{\frac{1}{\left(1 + x\right) \cdot \left(1 + x\right)}} - \frac{1}{x} \]
      rational.json-simplify-1 [=>] 0.1	\[ \left(x + 1\right) \cdot \frac{1}{\color{blue}{\left(x + 1\right)} \cdot \left(1 + x\right)} - \frac{1}{x} \]
      rational.json-simplify-1 [=>] 0.1	\[ \left(x + 1\right) \cdot \frac{1}{\left(x + 1\right) \cdot \color{blue}{\left(x + 1\right)}} - \frac{1}{x} \]

   4. Applied egg-rr 0.1

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

   if -3.99999999999999982e-6 < (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 1 x)) < 0.0

   1. Initial program 28.6

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

   2. Taylor expanded in x around inf 0.9

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

   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 0.5

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

   3. Simplified 0.5

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

      [Start] 0.5	\[ \left(1 + -1 \cdot x\right) - \frac{1}{x} \]
      rational.json-simplify-1 [=>] 0.5	\[ \color{blue}{\left(-1 \cdot x + 1\right)} - \frac{1}{x} \]
      rational.json-simplify-2 [=>] 0.5	\[ \left(\color{blue}{x \cdot -1} + 1\right) - \frac{1}{x} \]
      rational.json-simplify-9 [=>] 0.5	\[ \left(\color{blue}{\left(-x\right)} + 1\right) - \frac{1}{x} \]
      rational.json-simplify-12 [=>] 0.5	\[ \left(\color{blue}{\left(0 - x\right)} + 1\right) - \frac{1}{x} \]
      metadata-eval [<=] 0.5	\[ \left(\left(0 - x\right) + \color{blue}{\left(1 - 0\right)}\right) - \frac{1}{x} \]
      rational.json-simplify-27 [=>] 0.5	\[ \color{blue}{\left(1 - x\right)} - \frac{1}{x} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.6

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

Alternatives

Alternative 1
Error	0.7
Cost	14664

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

Alternative 2
Error	0.8
Cost	7944

\[\begin{array}{l} t_0 := \frac{1}{x + 1}\\ t_1 := t_0 - \frac{1}{x}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{1}{\left(x + 1\right) \cdot \left(x + 1\right)}}{t_0} - \frac{1}{x}\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{-1}{{x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) - \frac{1}{x}\\ \end{array} \]

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	13.8
Cost	576

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

Alternative 5
Error	30.2
Cost	192

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

Reproduce

herbie shell --seed 2023064 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))