3frac (problem 3.3.3)

Average Error: 15.51% → 0.36%

Time: 12.0s

Precision: binary64

Cost: 13508

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot \left(x + 1\right)\\ t_1 := \frac{2}{{x}^{3}}\\ \mathbf{if}\;x \leq -40000000:\\ \;\;\;\;\frac{2}{{x}^{5}} + t_1\\ \mathbf{elif}\;x \leq 100000000:\\ \;\;\;\;\frac{t_0 + \left(x + 2 \cdot \left(-1 - x\right)\right) \cdot \left(x + -1\right)}{t_0 \cdot \left(x + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (* x (+ x 1.0))) (t_1 (/ 2.0 (pow x 3.0))))
   (if (<= x -40000000.0)
     (+ (/ 2.0 (pow x 5.0)) t_1)
     (if (<= x 100000000.0)
       (/ (+ t_0 (* (+ x (* 2.0 (- -1.0 x))) (+ x -1.0))) (* t_0 (+ x -1.0)))
       t_1))))

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 = x * (x + 1.0);
	double t_1 = 2.0 / pow(x, 3.0);
	double tmp;
	if (x <= -40000000.0) {
		tmp = (2.0 / pow(x, 5.0)) + t_1;
	} else if (x <= 100000000.0) {
		tmp = (t_0 + ((x + (2.0 * (-1.0 - x))) * (x + -1.0))) / (t_0 * (x + -1.0));
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = x * (x + 1.0d0)
    t_1 = 2.0d0 / (x ** 3.0d0)
    if (x <= (-40000000.0d0)) then
        tmp = (2.0d0 / (x ** 5.0d0)) + t_1
    else if (x <= 100000000.0d0) then
        tmp = (t_0 + ((x + (2.0d0 * ((-1.0d0) - x))) * (x + (-1.0d0)))) / (t_0 * (x + (-1.0d0)))
    else
        tmp = t_1
    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 = x * (x + 1.0);
	double t_1 = 2.0 / Math.pow(x, 3.0);
	double tmp;
	if (x <= -40000000.0) {
		tmp = (2.0 / Math.pow(x, 5.0)) + t_1;
	} else if (x <= 100000000.0) {
		tmp = (t_0 + ((x + (2.0 * (-1.0 - x))) * (x + -1.0))) / (t_0 * (x + -1.0));
	} else {
		tmp = t_1;
	}
	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 = x * (x + 1.0)
	t_1 = 2.0 / math.pow(x, 3.0)
	tmp = 0
	if x <= -40000000.0:
		tmp = (2.0 / math.pow(x, 5.0)) + t_1
	elif x <= 100000000.0:
		tmp = (t_0 + ((x + (2.0 * (-1.0 - x))) * (x + -1.0))) / (t_0 * (x + -1.0))
	else:
		tmp = t_1
	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(x * Float64(x + 1.0))
	t_1 = Float64(2.0 / (x ^ 3.0))
	tmp = 0.0
	if (x <= -40000000.0)
		tmp = Float64(Float64(2.0 / (x ^ 5.0)) + t_1);
	elseif (x <= 100000000.0)
		tmp = Float64(Float64(t_0 + Float64(Float64(x + Float64(2.0 * Float64(-1.0 - x))) * Float64(x + -1.0))) / Float64(t_0 * Float64(x + -1.0)));
	else
		tmp = t_1;
	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 = x * (x + 1.0);
	t_1 = 2.0 / (x ^ 3.0);
	tmp = 0.0;
	if (x <= -40000000.0)
		tmp = (2.0 / (x ^ 5.0)) + t_1;
	elseif (x <= 100000000.0)
		tmp = (t_0 + ((x + (2.0 * (-1.0 - x))) * (x + -1.0))) / (t_0 * (x + -1.0));
	else
		tmp = t_1;
	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[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -40000000.0], N[(N[(2.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[x, 100000000.0], N[(N[(t$95$0 + N[(N[(x + N[(2.0 * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Target

Original	15.51%
Target	0.37%
Herbie	0.36%

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

Derivation

1. Split input into 3 regimes

2. if x < -4e7

   1. Initial program 30.31

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

   2. Taylor expanded in x around inf 0.67

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{3}}} \]

   3. Simplified 0.67

      \[\leadsto \color{blue}{\frac{2}{{x}^{5}} + \frac{2}{{x}^{3}}} \]
      Proof

      [Start] 0.67	\[ 2 \cdot \frac{1}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{3}} \]
      associate-*r/ [=>] 0.67	\[ \color{blue}{\frac{2 \cdot 1}{{x}^{5}}} + 2 \cdot \frac{1}{{x}^{3}} \]
      metadata-eval [=>] 0.67	\[ \frac{\color{blue}{2}}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{3}} \]
      associate-*r/ [=>] 0.67	\[ \frac{2}{{x}^{5}} + \color{blue}{\frac{2 \cdot 1}{{x}^{3}}} \]
      metadata-eval [=>] 0.67	\[ \frac{2}{{x}^{5}} + \frac{\color{blue}{2}}{{x}^{3}} \]

   if -4e7 < x < 1e8

   1. Initial program 0.87

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

   2. Applied egg-rr 0.06

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

   if 1e8 < x

   1. Initial program 31.19

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

   2. Taylor expanded in x around inf 0.69

      \[\leadsto \color{blue}{\frac{2}{{x}^{3}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.36

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -40000000:\\ \;\;\;\;\frac{2}{{x}^{5}} + \frac{2}{{x}^{3}}\\ \mathbf{elif}\;x \leq 100000000:\\ \;\;\;\;\frac{x \cdot \left(x + 1\right) + \left(x + 2 \cdot \left(-1 - x\right)\right) \cdot \left(x + -1\right)}{\left(x \cdot \left(x + 1\right)\right) \cdot \left(x + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.36%
Cost	6921

\[\begin{array}{l} t_0 := x \cdot \left(x + 1\right)\\ \mathbf{if}\;x \leq -90000000 \lor \neg \left(x \leq 100000000\right):\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + \left(x + 2 \cdot \left(-1 - x\right)\right) \cdot \left(x + -1\right)}{t_0 \cdot \left(x + -1\right)}\\ \end{array} \]

Alternative 2
Error	15.51%
Cost	960

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

Alternative 3
Error	15.52%
Cost	960

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

Alternative 4
Error	15.52%
Cost	960

\[\frac{\frac{-2 - x}{x + 1}}{x} + \frac{1}{x + -1} \]

Alternative 5
Error	16.89%
Cost	456

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

Alternative 6
Error	16.93%
Cost	448

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

Alternative 7
Error	64.94%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023115 
(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))))