3frac (problem 3.3.3)

Average Error: 9.7 → 0.1

Time: 10.8s

Precision: binary64

Cost: 22024

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{1 + x}\\ t_1 := \frac{1}{x + -1}\\ t_2 := \left(t_0 + \frac{-2}{x}\right) + t_1\\ \mathbf{if}\;t_2 \leq -20:\\ \;\;\;\;t_0 + \left(t_1 + \frac{-2}{x}\right)\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{{x}^{5}} + 2 \cdot \left({x}^{-3} + {x}^{-7}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{\frac{x + \left(x + -1\right) \cdot -2}{x + -1}}{x}\\ \end{array} \]

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 (+ 1.0 x)))
        (t_1 (/ 1.0 (+ x -1.0)))
        (t_2 (+ (+ t_0 (/ -2.0 x)) t_1)))
   (if (<= t_2 -20.0)
     (+ t_0 (+ t_1 (/ -2.0 x)))
     (if (<= t_2 5e-8)
       (+ (/ 2.0 (pow x 5.0)) (* 2.0 (+ (pow x -3.0) (pow x -7.0))))
       (+ t_0 (/ (/ (+ x (* (+ x -1.0) -2.0)) (+ x -1.0)) x))))))

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 / (1.0 + x);
	double t_1 = 1.0 / (x + -1.0);
	double t_2 = (t_0 + (-2.0 / x)) + t_1;
	double tmp;
	if (t_2 <= -20.0) {
		tmp = t_0 + (t_1 + (-2.0 / x));
	} else if (t_2 <= 5e-8) {
		tmp = (2.0 / pow(x, 5.0)) + (2.0 * (pow(x, -3.0) + pow(x, -7.0)));
	} else {
		tmp = t_0 + (((x + ((x + -1.0) * -2.0)) / (x + -1.0)) / x);
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 / (1.0d0 + x)
    t_1 = 1.0d0 / (x + (-1.0d0))
    t_2 = (t_0 + ((-2.0d0) / x)) + t_1
    if (t_2 <= (-20.0d0)) then
        tmp = t_0 + (t_1 + ((-2.0d0) / x))
    else if (t_2 <= 5d-8) then
        tmp = (2.0d0 / (x ** 5.0d0)) + (2.0d0 * ((x ** (-3.0d0)) + (x ** (-7.0d0))))
    else
        tmp = t_0 + (((x + ((x + (-1.0d0)) * (-2.0d0))) / (x + (-1.0d0))) / x)
    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 / (1.0 + x);
	double t_1 = 1.0 / (x + -1.0);
	double t_2 = (t_0 + (-2.0 / x)) + t_1;
	double tmp;
	if (t_2 <= -20.0) {
		tmp = t_0 + (t_1 + (-2.0 / x));
	} else if (t_2 <= 5e-8) {
		tmp = (2.0 / Math.pow(x, 5.0)) + (2.0 * (Math.pow(x, -3.0) + Math.pow(x, -7.0)));
	} else {
		tmp = t_0 + (((x + ((x + -1.0) * -2.0)) / (x + -1.0)) / x);
	}
	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 / (1.0 + x)
	t_1 = 1.0 / (x + -1.0)
	t_2 = (t_0 + (-2.0 / x)) + t_1
	tmp = 0
	if t_2 <= -20.0:
		tmp = t_0 + (t_1 + (-2.0 / x))
	elif t_2 <= 5e-8:
		tmp = (2.0 / math.pow(x, 5.0)) + (2.0 * (math.pow(x, -3.0) + math.pow(x, -7.0)))
	else:
		tmp = t_0 + (((x + ((x + -1.0) * -2.0)) / (x + -1.0)) / x)
	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(1.0 / Float64(1.0 + x))
	t_1 = Float64(1.0 / Float64(x + -1.0))
	t_2 = Float64(Float64(t_0 + Float64(-2.0 / x)) + t_1)
	tmp = 0.0
	if (t_2 <= -20.0)
		tmp = Float64(t_0 + Float64(t_1 + Float64(-2.0 / x)));
	elseif (t_2 <= 5e-8)
		tmp = Float64(Float64(2.0 / (x ^ 5.0)) + Float64(2.0 * Float64((x ^ -3.0) + (x ^ -7.0))));
	else
		tmp = Float64(t_0 + Float64(Float64(Float64(x + Float64(Float64(x + -1.0) * -2.0)) / Float64(x + -1.0)) / x));
	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 / (1.0 + x);
	t_1 = 1.0 / (x + -1.0);
	t_2 = (t_0 + (-2.0 / x)) + t_1;
	tmp = 0.0;
	if (t_2 <= -20.0)
		tmp = t_0 + (t_1 + (-2.0 / x));
	elseif (t_2 <= 5e-8)
		tmp = (2.0 / (x ^ 5.0)) + (2.0 * ((x ^ -3.0) + (x ^ -7.0)));
	else
		tmp = t_0 + (((x + ((x + -1.0) * -2.0)) / (x + -1.0)) / x);
	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[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 + N[(-2.0 / x), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -20.0], N[(t$95$0 + N[(t$95$1 + N[(-2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-8], N[(N[(2.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[Power[x, -3.0], $MachinePrecision] + N[Power[x, -7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(N[(N[(x + N[(N[(x + -1.0), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	9.7
Target	0.2
Herbie	0.1

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -20

   1. Initial program 0.0

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

   2. Simplified 0.0

      \[\leadsto \color{blue}{\frac{1}{1 + x} + \left(\frac{1}{x + -1} + \frac{-2}{x}\right)} \]
      Proof
      (+.f64 (/.f64 1 (+.f64 1 x)) (+.f64 (/.f64 1 (+.f64 x -1)) (/.f64 -2 x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 (Rewrite<= +-commutative_binary64 (+.f64 x 1))) (+.f64 (/.f64 1 (+.f64 x -1)) (/.f64 -2 x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 (+.f64 x 1)) (+.f64 (/.f64 1 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1)))) (/.f64 -2 x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 (+.f64 x 1)) (+.f64 (/.f64 1 (Rewrite<= sub-neg_binary64 (-.f64 x 1))) (/.f64 -2 x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 (+.f64 x 1)) (+.f64 (/.f64 1 (-.f64 x 1)) (/.f64 (Rewrite<= metadata-eval (neg.f64 2)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 (+.f64 x 1)) (+.f64 (/.f64 1 (-.f64 x 1)) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 2 x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 (+.f64 x 1)) (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 (/.f64 2 x)) (/.f64 1 (-.f64 x 1))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (/.f64 1 (+.f64 x 1)) (neg.f64 (/.f64 2 x))) (/.f64 1 (-.f64 x 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x))) (/.f64 1 (-.f64 x 1))): 0 points increase in error, 0 points decrease in error

   if -20 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 4.9999999999999998e-8

   1. Initial program 19.1

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

   2. Simplified 19.1

      \[\leadsto \color{blue}{\frac{1}{1 + x} + \left(\frac{1}{x + -1} + \frac{-2}{x}\right)} \]
      Proof
      (+.f64 (/.f64 1 (+.f64 1 x)) (+.f64 (/.f64 1 (+.f64 x -1)) (/.f64 -2 x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 (Rewrite<= +-commutative_binary64 (+.f64 x 1))) (+.f64 (/.f64 1 (+.f64 x -1)) (/.f64 -2 x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 (+.f64 x 1)) (+.f64 (/.f64 1 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1)))) (/.f64 -2 x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 (+.f64 x 1)) (+.f64 (/.f64 1 (Rewrite<= sub-neg_binary64 (-.f64 x 1))) (/.f64 -2 x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 (+.f64 x 1)) (+.f64 (/.f64 1 (-.f64 x 1)) (/.f64 (Rewrite<= metadata-eval (neg.f64 2)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 (+.f64 x 1)) (+.f64 (/.f64 1 (-.f64 x 1)) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 2 x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 (+.f64 x 1)) (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 (/.f64 2 x)) (/.f64 1 (-.f64 x 1))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (/.f64 1 (+.f64 x 1)) (neg.f64 (/.f64 2 x))) (/.f64 1 (-.f64 x 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x))) (/.f64 1 (-.f64 x 1))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in x around inf 0.7

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

   4. Simplified 0.7

      \[\leadsto \color{blue}{\frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{3}} + \frac{2}{{x}^{7}}\right)} \]
      Proof
      (+.f64 (/.f64 2 (pow.f64 x 5)) (+.f64 (/.f64 2 (pow.f64 x 3)) (/.f64 2 (pow.f64 x 7)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 2 1)) (pow.f64 x 5)) (+.f64 (/.f64 2 (pow.f64 x 3)) (/.f64 2 (pow.f64 x 7)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 1 (pow.f64 x 5)))) (+.f64 (/.f64 2 (pow.f64 x 3)) (/.f64 2 (pow.f64 x 7)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 2 (/.f64 1 (pow.f64 x 5))) (+.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 2 1)) (pow.f64 x 3)) (/.f64 2 (pow.f64 x 7)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 2 (/.f64 1 (pow.f64 x 5))) (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 1 (pow.f64 x 3)))) (/.f64 2 (pow.f64 x 7)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 2 (/.f64 1 (pow.f64 x 5))) (+.f64 (*.f64 2 (/.f64 1 (pow.f64 x 3))) (/.f64 (Rewrite<= metadata-eval (*.f64 2 1)) (pow.f64 x 7)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 2 (/.f64 1 (pow.f64 x 5))) (+.f64 (*.f64 2 (/.f64 1 (pow.f64 x 3))) (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 1 (pow.f64 x 7)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 2 (/.f64 1 (pow.f64 x 5))) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 x 7))) (*.f64 2 (/.f64 1 (pow.f64 x 3)))))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 0.2

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

   if 4.9999999999999998e-8 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 0.1

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{\frac{1}{1 + x} + \left(\frac{1}{x + -1} + \frac{-2}{x}\right)} \]
      Proof
      (+.f64 (/.f64 1 (+.f64 1 x)) (+.f64 (/.f64 1 (+.f64 x -1)) (/.f64 -2 x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 (Rewrite<= +-commutative_binary64 (+.f64 x 1))) (+.f64 (/.f64 1 (+.f64 x -1)) (/.f64 -2 x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 (+.f64 x 1)) (+.f64 (/.f64 1 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1)))) (/.f64 -2 x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 (+.f64 x 1)) (+.f64 (/.f64 1 (Rewrite<= sub-neg_binary64 (-.f64 x 1))) (/.f64 -2 x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 (+.f64 x 1)) (+.f64 (/.f64 1 (-.f64 x 1)) (/.f64 (Rewrite<= metadata-eval (neg.f64 2)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 (+.f64 x 1)) (+.f64 (/.f64 1 (-.f64 x 1)) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 2 x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 (+.f64 x 1)) (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 (/.f64 2 x)) (/.f64 1 (-.f64 x 1))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (/.f64 1 (+.f64 x 1)) (neg.f64 (/.f64 2 x))) (/.f64 1 (-.f64 x 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x))) (/.f64 1 (-.f64 x 1))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 0.1

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

   4. Applied egg-rr 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} + \frac{-2}{x}\right) + \frac{1}{x + -1} \leq -20:\\ \;\;\;\;\frac{1}{1 + x} + \left(\frac{1}{x + -1} + \frac{-2}{x}\right)\\ \mathbf{elif}\;\left(\frac{1}{1 + x} + \frac{-2}{x}\right) + \frac{1}{x + -1} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{{x}^{5}} + 2 \cdot \left({x}^{-3} + {x}^{-7}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + x} + \frac{\frac{x + \left(x + -1\right) \cdot -2}{x + -1}}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	13640

\[\begin{array}{l} \mathbf{if}\;x \leq -51841659923.4166:\\ \;\;\;\;2 \cdot {x}^{-3}\\ \mathbf{elif}\;x \leq 365.63712801122784:\\ \;\;\;\;\frac{1}{1 + x} + \frac{\frac{x + \left(x + -1\right) \cdot -2}{x + -1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{5}} + \frac{2}{{x}^{3}}\\ \end{array} \]

Alternative 2
Error	0.2
Cost	8712

\[\begin{array}{l} t_0 := \frac{1}{1 + x}\\ t_1 := \left(t_0 + \frac{-2}{x}\right) + \frac{1}{x + -1}\\ t_2 := t_0 + \frac{\frac{x + \left(x + -1\right) \cdot -2}{x + -1}}{x}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-10}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-14}:\\ \;\;\;\;2 \cdot {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	0.4
Cost	8456

\[\begin{array}{l} \mathbf{if}\;x \leq -51841659923.4166:\\ \;\;\;\;2 \cdot {x}^{-3}\\ \mathbf{elif}\;x \leq 365.63712801122784:\\ \;\;\;\;\frac{1}{1 + x} + \frac{\frac{x + \left(x + -1\right) \cdot -2}{x + -1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-4}{x \cdot x} + \left(\frac{-2}{x} + \frac{-8}{{x}^{3}}\right)}{\left(1 + x\right) \cdot \left(x \cdot \frac{x + -1}{2 - x}\right)}\\ \end{array} \]

Alternative 4
Error	0.4
Cost	3016

\[\begin{array}{l} t_0 := \frac{1}{1 + x}\\ t_1 := \left(t_0 + \frac{-2}{x}\right) + \frac{1}{x + -1}\\ t_2 := t_0 + \frac{\frac{2 - x}{x + -1}}{x}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-10}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	0.6
Cost	1480

\[\begin{array}{l} \mathbf{if}\;x \leq -51841659923.4166:\\ \;\;\;\;\frac{\frac{-2}{x}}{\left(1 + x\right) \cdot \left(x \cdot \frac{x + -1}{2 - x}\right)}\\ \mathbf{elif}\;x \leq 13305.34583500944:\\ \;\;\;\;\frac{1}{1 + x} + \frac{\frac{x + \left(x + -1\right) \cdot -2}{x + -1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \left(x \cdot x\right)}\\ \end{array} \]

Alternative 6
Error	0.6
Cost	1224

\[\begin{array}{l} \mathbf{if}\;x \leq -51841659923.4166:\\ \;\;\;\;\frac{\frac{-2}{x}}{\left(1 + x\right) \cdot \left(x \cdot \frac{x + -1}{2 - x}\right)}\\ \mathbf{elif}\;x \leq 13305.34583500944:\\ \;\;\;\;\frac{1}{1 + x} + \frac{\frac{2 - x}{x + -1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot \left(x \cdot x\right)}\\ \end{array} \]

Alternative 7
Error	0.8
Cost	840

\[\begin{array}{l} t_0 := \frac{2}{x \cdot \left(x \cdot x\right)}\\ \mathbf{if}\;x \leq -1972.7549491055456:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.05591673386039308:\\ \;\;\;\;x \cdot -2 + 2 \cdot \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	1.0
Cost	712

\[\begin{array}{l} t_0 := \frac{2}{x \cdot \left(x \cdot x\right)}\\ \mathbf{if}\;x \leq -1972.7549491055456:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.05591673386039308:\\ \;\;\;\;\frac{-2}{x} - x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	31.1
Cost	192

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

Alternative 10
Error	61.9
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022300 
(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))))