3frac (problem 3.3.3)

Average Error: 9.8 → 0.6

Time: 26.0s

Precision: binary64

Cost: 22280

Math

\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

↓

\[\begin{array}{l} t_0 := \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{if}\;t_0 \leq -400000:\\ \;\;\;\;-2 \cdot x - 2 \cdot \frac{1}{x}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-12}:\\ \;\;\;\;2 \cdot \left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{3}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0)))))
   (if (<= t_0 -400000.0)
     (- (* -2.0 x) (* 2.0 (/ 1.0 x)))
     (if (<= t_0 5e-12)
       (*
        2.0
        (+ (/ 1.0 (pow x 7.0)) (+ (/ 1.0 (pow x 5.0)) (/ 1.0 (pow x 3.0)))))
       t_0))))

C

double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

↓

double code(double x) {
	double t_0 = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
	double tmp;
	if (t_0 <= -400000.0) {
		tmp = (-2.0 * x) - (2.0 * (1.0 / x));
	} else if (t_0 <= 5e-12) {
		tmp = 2.0 * ((1.0 / pow(x, 7.0)) + ((1.0 / pow(x, 5.0)) + (1.0 / pow(x, 3.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((1.0d0 / (x + 1.0d0)) - (2.0d0 / x)) + (1.0d0 / (x - 1.0d0))
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)) - (2.0d0 / x)) + (1.0d0 / (x - 1.0d0))
    if (t_0 <= (-400000.0d0)) then
        tmp = ((-2.0d0) * x) - (2.0d0 * (1.0d0 / x))
    else if (t_0 <= 5d-12) then
        tmp = 2.0d0 * ((1.0d0 / (x ** 7.0d0)) + ((1.0d0 / (x ** 5.0d0)) + (1.0d0 / (x ** 3.0d0))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

↓

public static double code(double x) {
	double t_0 = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
	double tmp;
	if (t_0 <= -400000.0) {
		tmp = (-2.0 * x) - (2.0 * (1.0 / x));
	} else if (t_0 <= 5e-12) {
		tmp = 2.0 * ((1.0 / Math.pow(x, 7.0)) + ((1.0 / Math.pow(x, 5.0)) + (1.0 / Math.pow(x, 3.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0))

↓

def code(x):
	t_0 = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0))
	tmp = 0
	if t_0 <= -400000.0:
		tmp = (-2.0 * x) - (2.0 * (1.0 / x))
	elif t_0 <= 5e-12:
		tmp = 2.0 * ((1.0 / math.pow(x, 7.0)) + ((1.0 / math.pow(x, 5.0)) + (1.0 / math.pow(x, 3.0))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x - 1.0)))
end

↓

function code(x)
	t_0 = Float64(Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x - 1.0)))
	tmp = 0.0
	if (t_0 <= -400000.0)
		tmp = Float64(Float64(-2.0 * x) - Float64(2.0 * Float64(1.0 / x)));
	elseif (t_0 <= 5e-12)
		tmp = Float64(2.0 * Float64(Float64(1.0 / (x ^ 7.0)) + Float64(Float64(1.0 / (x ^ 5.0)) + Float64(1.0 / (x ^ 3.0)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
end

↓

function tmp_2 = code(x)
	t_0 = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
	tmp = 0.0;
	if (t_0 <= -400000.0)
		tmp = (-2.0 * x) - (2.0 * (1.0 / x));
	elseif (t_0 <= 5e-12)
		tmp = 2.0 * ((1.0 / (x ^ 7.0)) + ((1.0 / (x ^ 5.0)) + (1.0 / (x ^ 3.0))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := N[(N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := Block[{t$95$0 = N[(N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -400000.0], N[(N[(-2.0 * x), $MachinePrecision] - N[(2.0 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-12], N[(2.0 * N[(N[(1.0 / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Target

Original	9.8
Target	0.2
Herbie	0.6

\[\frac{2}{x \cdot \left(x \cdot x - 1\right)} \]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -4e5

   1. Initial program 0.0

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

   2. Taylor expanded in x around 0 0.0

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

   if -4e5 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 4.9999999999999997e-12

   1. Initial program 19.3

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

   2. Taylor expanded in x around inf 1.1

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{5}} + \left(2 \cdot \frac{1}{{x}^{7}} + 2 \cdot \frac{1}{{x}^{3}}\right)} \]

   3. Simplified 1.1

      \[\leadsto \color{blue}{2 \cdot \left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{3}}\right)\right)} \]
      Proof

      [Start] 1.1	\[ 2 \cdot \frac{1}{{x}^{5}} + \left(2 \cdot \frac{1}{{x}^{7}} + 2 \cdot \frac{1}{{x}^{3}}\right) \]
      rational.json-simplify-41 [=>] 1.1	\[ \color{blue}{2 \cdot \frac{1}{{x}^{7}} + \left(2 \cdot \frac{1}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{5}}\right)} \]
      rational.json-simplify-2 [=>] 1.1	\[ 2 \cdot \frac{1}{{x}^{7}} + \left(2 \cdot \frac{1}{{x}^{3}} + \color{blue}{\frac{1}{{x}^{5}} \cdot 2}\right) \]
      rational.json-simplify-51 [=>] 1.1	\[ 2 \cdot \frac{1}{{x}^{7}} + \color{blue}{2 \cdot \left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{3}}\right)} \]
      rational.json-simplify-2 [=>] 1.1	\[ 2 \cdot \frac{1}{{x}^{7}} + \color{blue}{\left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{3}}\right) \cdot 2} \]
      rational.json-simplify-51 [=>] 1.1	\[ \color{blue}{2 \cdot \left(\left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{3}}\right) + \frac{1}{{x}^{7}}\right)} \]
      rational.json-simplify-1 [<=] 1.1	\[ 2 \cdot \color{blue}{\left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{3}}\right)\right)} \]

   if 4.9999999999999997e-12 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 0.1

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \leq -400000:\\ \;\;\;\;-2 \cdot x - 2 \cdot \frac{1}{x}\\ \mathbf{elif}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;2 \cdot \left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{3}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	15560

\[\begin{array}{l} t_0 := \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{if}\;t_0 \leq -400000:\\ \;\;\;\;-2 \cdot x - 2 \cdot \frac{1}{x}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-12}:\\ \;\;\;\;2 \cdot \left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.5
Cost	8712

\[\begin{array}{l} t_0 := \frac{1}{x + 1} - \frac{2}{x}\\ t_1 := \frac{1}{x - 1}\\ t_2 := t_0 + t_1\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{-10}:\\ \;\;\;\;t_0 + \frac{1 - x}{\left(x + -1\right) \cdot \left(1 - x\right)}\\ \mathbf{elif}\;t_2 \leq 10^{-25}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\left(1 + x\right) \cdot \left(1 + x\right)} \cdot \left(1 + x\right) - \frac{2}{x}\right) + t_1\\ \end{array} \]

Alternative 3
Error	9.8
Cost	960

\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

Alternative 4
Error	10.7
Cost	448

\[\left(1 - \frac{2}{x}\right) + -1 \]

Alternative 5
Error	31.0
Cost	192

\[\frac{-2}{x} \]

Alternative 6
Error	61.9
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023075 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))