3frac (problem 3.3.3)

Average Error: 9.7 → 0.3

Time: 13.1s

Precision: binary64

Cost: 6788

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -290000:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{elif}\;x \leq 90000000:\\ \;\;\;\;\frac{x \cdot x - \left(x - \left(x \cdot 2 + \left(-2 - x\right)\right) \cdot \left(-1 - x\right)\right)}{\left(x + 1\right) \cdot \left(x \cdot \left(x + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x} + \frac{4}{x \cdot x}}{\left(x + 1\right) \cdot \left(x \cdot \frac{x + -1}{x + -2}\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -290000.0)
   (/ 2.0 (pow x 3.0))
   (if (<= x 90000000.0)
     (/
      (- (* x x) (- x (* (+ (* x 2.0) (- -2.0 x)) (- -1.0 x))))
      (* (+ x 1.0) (* x (+ x -1.0))))
     (/
      (+ (/ 2.0 x) (/ 4.0 (* x x)))
      (* (+ x 1.0) (* x (/ (+ x -1.0) (+ x -2.0))))))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -290000.0) {
		tmp = 2.0 / pow(x, 3.0);
	} else if (x <= 90000000.0) {
		tmp = ((x * x) - (x - (((x * 2.0) + (-2.0 - x)) * (-1.0 - x)))) / ((x + 1.0) * (x * (x + -1.0)));
	} else {
		tmp = ((2.0 / x) + (4.0 / (x * x))) / ((x + 1.0) * (x * ((x + -1.0) / (x + -2.0))));
	}
	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) :: tmp
    if (x <= (-290000.0d0)) then
        tmp = 2.0d0 / (x ** 3.0d0)
    else if (x <= 90000000.0d0) then
        tmp = ((x * x) - (x - (((x * 2.0d0) + ((-2.0d0) - x)) * ((-1.0d0) - x)))) / ((x + 1.0d0) * (x * (x + (-1.0d0))))
    else
        tmp = ((2.0d0 / x) + (4.0d0 / (x * x))) / ((x + 1.0d0) * (x * ((x + (-1.0d0)) / (x + (-2.0d0)))))
    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 tmp;
	if (x <= -290000.0) {
		tmp = 2.0 / Math.pow(x, 3.0);
	} else if (x <= 90000000.0) {
		tmp = ((x * x) - (x - (((x * 2.0) + (-2.0 - x)) * (-1.0 - x)))) / ((x + 1.0) * (x * (x + -1.0)));
	} else {
		tmp = ((2.0 / x) + (4.0 / (x * x))) / ((x + 1.0) * (x * ((x + -1.0) / (x + -2.0))));
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -290000.0:
		tmp = 2.0 / math.pow(x, 3.0)
	elif x <= 90000000.0:
		tmp = ((x * x) - (x - (((x * 2.0) + (-2.0 - x)) * (-1.0 - x)))) / ((x + 1.0) * (x * (x + -1.0)))
	else:
		tmp = ((2.0 / x) + (4.0 / (x * x))) / ((x + 1.0) * (x * ((x + -1.0) / (x + -2.0))))
	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)
	tmp = 0.0
	if (x <= -290000.0)
		tmp = Float64(2.0 / (x ^ 3.0));
	elseif (x <= 90000000.0)
		tmp = Float64(Float64(Float64(x * x) - Float64(x - Float64(Float64(Float64(x * 2.0) + Float64(-2.0 - x)) * Float64(-1.0 - x)))) / Float64(Float64(x + 1.0) * Float64(x * Float64(x + -1.0))));
	else
		tmp = Float64(Float64(Float64(2.0 / x) + Float64(4.0 / Float64(x * x))) / Float64(Float64(x + 1.0) * Float64(x * Float64(Float64(x + -1.0) / Float64(x + -2.0)))));
	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)
	tmp = 0.0;
	if (x <= -290000.0)
		tmp = 2.0 / (x ^ 3.0);
	elseif (x <= 90000000.0)
		tmp = ((x * x) - (x - (((x * 2.0) + (-2.0 - x)) * (-1.0 - x)))) / ((x + 1.0) * (x * (x + -1.0)));
	else
		tmp = ((2.0 / x) + (4.0 / (x * x))) / ((x + 1.0) * (x * ((x + -1.0) / (x + -2.0))));
	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_] := If[LessEqual[x, -290000.0], N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 90000000.0], N[(N[(N[(x * x), $MachinePrecision] - N[(x - N[(N[(N[(x * 2.0), $MachinePrecision] + N[(-2.0 - x), $MachinePrecision]), $MachinePrecision] * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] * N[(x * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / x), $MachinePrecision] + N[(4.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] * N[(x * N[(N[(x + -1.0), $MachinePrecision] / N[(x + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	9.7
Target	0.3
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if x < -2.9e5

   1. Initial program 19.0

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

   2. Simplified 19.0

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

      [Start] 19.0	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 19.0	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 19.0	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 19.0	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 19.0	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 19.0	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 19.0	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 19.0	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 19.0	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 19.0	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Taylor expanded in x around inf 0.5

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

   if -2.9e5 < x < 9e7

   1. Initial program 0.6

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

   2. Simplified 0.6

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

      [Start] 0.6	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 0.6	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 0.6	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 0.6	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 0.6	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 0.6	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 0.6	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 0.6	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 0.6	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 0.6	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Applied egg-rr 0.2

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

   4. Simplified 0.2

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

      [Start] 0.2	\[ \frac{x \cdot x - \left(x + \left(1 + x\right) \cdot \left(-2 + \left(2 \cdot x - x\right)\right)\right)}{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
      +-commutative [=>] 0.2	\[ \frac{x \cdot x - \left(x + \color{blue}{\left(x + 1\right)} \cdot \left(-2 + \left(2 \cdot x - x\right)\right)\right)}{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
      +-commutative [=>] 0.2	\[ \frac{x \cdot x - \left(x + \left(x + 1\right) \cdot \color{blue}{\left(\left(2 \cdot x - x\right) + -2\right)}\right)}{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
      associate-+l- [=>] 0.2	\[ \frac{x \cdot x - \left(x + \left(x + 1\right) \cdot \color{blue}{\left(2 \cdot x - \left(x - -2\right)\right)}\right)}{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
      *-commutative [=>] 0.2	\[ \frac{x \cdot x - \left(x + \left(x + 1\right) \cdot \left(\color{blue}{x \cdot 2} - \left(x - -2\right)\right)\right)}{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, -x\right)} \]
      +-commutative [=>] 0.2	\[ \frac{x \cdot x - \left(x + \left(x + 1\right) \cdot \left(x \cdot 2 - \left(x - -2\right)\right)\right)}{\color{blue}{\left(x + 1\right)} \cdot \mathsf{fma}\left(x, x, -x\right)} \]
      fma-udef [=>] 0.2	\[ \frac{x \cdot x - \left(x + \left(x + 1\right) \cdot \left(x \cdot 2 - \left(x - -2\right)\right)\right)}{\left(x + 1\right) \cdot \color{blue}{\left(x \cdot x + \left(-x\right)\right)}} \]
      mul-1-neg [<=] 0.2	\[ \frac{x \cdot x - \left(x + \left(x + 1\right) \cdot \left(x \cdot 2 - \left(x - -2\right)\right)\right)}{\left(x + 1\right) \cdot \left(x \cdot x + \color{blue}{-1 \cdot x}\right)} \]
      distribute-rgt-in [<=] 0.2	\[ \frac{x \cdot x - \left(x + \left(x + 1\right) \cdot \left(x \cdot 2 - \left(x - -2\right)\right)\right)}{\left(x + 1\right) \cdot \color{blue}{\left(x \cdot \left(x + -1\right)\right)}} \]

   if 9e7 < x

   1. Initial program 19.7

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

   2. Simplified 19.7

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

      [Start] 19.7	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 19.7	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 19.7	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 19.7	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 19.7	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 19.7	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 19.7	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 19.7	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 19.7	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 19.7	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Applied egg-rr 19.8

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

   4. Simplified 19.8

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

      [Start] 19.8	\[ \frac{1}{1 + x} - \frac{\frac{-2 + \left(2 \cdot x - x\right)}{x}}{1} \cdot \frac{1}{x + -1} \]
      /-rgt-identity [=>] 19.8	\[ \frac{1}{1 + x} - \color{blue}{\frac{-2 + \left(2 \cdot x - x\right)}{x}} \cdot \frac{1}{x + -1} \]
      associate-*l/ [=>] 19.8	\[ \frac{1}{1 + x} - \color{blue}{\frac{\left(-2 + \left(2 \cdot x - x\right)\right) \cdot \frac{1}{x + -1}}{x}} \]
      +-commutative [=>] 19.8	\[ \frac{1}{1 + x} - \frac{\color{blue}{\left(\left(2 \cdot x - x\right) + -2\right)} \cdot \frac{1}{x + -1}}{x} \]
      associate-+l- [=>] 19.8	\[ \frac{1}{1 + x} - \frac{\color{blue}{\left(2 \cdot x - \left(x - -2\right)\right)} \cdot \frac{1}{x + -1}}{x} \]
      *-commutative [=>] 19.8	\[ \frac{1}{1 + x} - \frac{\left(\color{blue}{x \cdot 2} - \left(x - -2\right)\right) \cdot \frac{1}{x + -1}}{x} \]

   5. Applied egg-rr 30.2

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

   6. Taylor expanded in x around inf 0.1

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

   7. Simplified 0.1

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

      [Start] 0.1	\[ \frac{4 \cdot \frac{1}{{x}^{2}} + 2 \cdot \frac{1}{x}}{\left(1 + x\right) \cdot \left(x \cdot \frac{x + -1}{x + -2}\right)} \]
      +-commutative [=>] 0.1	\[ \frac{\color{blue}{2 \cdot \frac{1}{x} + 4 \cdot \frac{1}{{x}^{2}}}}{\left(1 + x\right) \cdot \left(x \cdot \frac{x + -1}{x + -2}\right)} \]
      associate-*r/ [=>] 0.1	\[ \frac{\color{blue}{\frac{2 \cdot 1}{x}} + 4 \cdot \frac{1}{{x}^{2}}}{\left(1 + x\right) \cdot \left(x \cdot \frac{x + -1}{x + -2}\right)} \]
      metadata-eval [=>] 0.1	\[ \frac{\frac{\color{blue}{2}}{x} + 4 \cdot \frac{1}{{x}^{2}}}{\left(1 + x\right) \cdot \left(x \cdot \frac{x + -1}{x + -2}\right)} \]
      associate-*r/ [=>] 0.1	\[ \frac{\frac{2}{x} + \color{blue}{\frac{4 \cdot 1}{{x}^{2}}}}{\left(1 + x\right) \cdot \left(x \cdot \frac{x + -1}{x + -2}\right)} \]
      metadata-eval [=>] 0.1	\[ \frac{\frac{2}{x} + \frac{\color{blue}{4}}{{x}^{2}}}{\left(1 + x\right) \cdot \left(x \cdot \frac{x + -1}{x + -2}\right)} \]
      unpow2 [=>] 0.1	\[ \frac{\frac{2}{x} + \frac{4}{\color{blue}{x \cdot x}}}{\left(1 + x\right) \cdot \left(x \cdot \frac{x + -1}{x + -2}\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -290000:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{elif}\;x \leq 90000000:\\ \;\;\;\;\frac{x \cdot x - \left(x - \left(x \cdot 2 + \left(-2 - x\right)\right) \cdot \left(-1 - x\right)\right)}{\left(x + 1\right) \cdot \left(x \cdot \left(x + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x} + \frac{4}{x \cdot x}}{\left(x + 1\right) \cdot \left(x \cdot \frac{x + -1}{x + -2}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	3529

\[\begin{array}{l} t_0 := \left(\frac{1}{x + 1} + \frac{-2}{x}\right) + \frac{1}{x + -1}\\ t_1 := x \cdot \left(x + -1\right)\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-15} \lor \neg \left(t_0 \leq 10^{-25}\right):\\ \;\;\;\;\frac{\frac{t_1 + \left(x + 1\right) \cdot \left(2 - x\right)}{x + 1}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x} + \frac{4}{x \cdot x}}{\left(x + 1\right) \cdot \left(x \cdot \frac{x + -1}{x + -2}\right)}\\ \end{array} \]

Alternative 2
Error	0.6
Cost	3528

\[\begin{array}{l} t_0 := \left(\frac{1}{x + 1} + \frac{-2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_0 \leq -200:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 10^{-25}:\\ \;\;\;\;\frac{\frac{2}{x} + \frac{4}{x \cdot x}}{\left(x + 1\right) \cdot \left(x \cdot \frac{x + -1}{x + -2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{x + 1}{x} \cdot \frac{2 - x}{x + -1}}{x + 1}\\ \end{array} \]

Alternative 3
Error	0.6
Cost	3017

\[\begin{array}{l} t_0 := \left(\frac{1}{x + 1} + \frac{-2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-15} \lor \neg \left(t_0 \leq 10^{-25}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x \cdot x}}{x + 1}\\ \end{array} \]

Alternative 4
Error	0.6
Cost	3016

\[\begin{array}{l} t_0 := \frac{1}{x + 1}\\ t_1 := \left(t_0 + \frac{-2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 10^{-25}:\\ \;\;\;\;\frac{\frac{2}{x \cdot x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{2 - x}{x \cdot \left(x + -1\right)}\\ \end{array} \]

Alternative 5
Error	0.2
Cost	1993

\[\begin{array}{l} \mathbf{if}\;x \leq -290000 \lor \neg \left(x \leq 90000000\right):\\ \;\;\;\;\frac{\frac{2}{x} + \frac{4}{x \cdot x}}{\left(x + 1\right) \cdot \left(x \cdot \frac{x + -1}{x + -2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x - \left(x - \left(x \cdot 2 + \left(-2 - x\right)\right) \cdot \left(-1 - x\right)\right)}{\left(x + 1\right) \cdot \left(x \cdot \left(x + -1\right)\right)}\\ \end{array} \]

Alternative 6
Error	0.5
Cost	1224

\[\begin{array}{l} \mathbf{if}\;x \leq -720000:\\ \;\;\;\;\frac{\frac{2}{x}}{\left(x + 1\right) \cdot \left(x \cdot \frac{x + -1}{x + -2}\right)}\\ \mathbf{elif}\;x \leq 300000:\\ \;\;\;\;\frac{1}{x + 1} + \frac{2 - x}{x \cdot \left(x + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x \cdot x}}{x + 1}\\ \end{array} \]

Alternative 7
Error	1.0
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -0.86 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{\frac{2}{x \cdot x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2 + \frac{-2}{x}\\ \end{array} \]

Alternative 8
Error	15.4
Cost	585

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

Alternative 9
Error	10.7
Cost	448

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

Alternative 10
Error	30.8
Cost	192

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

Alternative 11
Error	61.9
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023016 
(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))))