3frac (problem 3.3.3)

Average Error: 9.7 → 0.6

Time: 12.6s

Precision: binary64

Cost: 8712

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{x + 1}\\ t_1 := \left(t_0 + \frac{-2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_1 \leq -10:\\ \;\;\;\;t_0 - \left(-1 - x\right) \cdot \frac{\left(-2 - x\right) - x \cdot -2}{x \cdot \left(1 - x \cdot x\right)}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(\frac{-2}{x} - \frac{x}{x \cdot \left(1 - x\right)}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ x 1.0)))
        (t_1 (+ (+ t_0 (/ -2.0 x)) (/ 1.0 (+ x -1.0)))))
   (if (<= t_1 -10.0)
     (- t_0 (* (- -1.0 x) (/ (- (- -2.0 x) (* x -2.0)) (* x (- 1.0 (* x x))))))
     (if (<= t_1 5e-22)
       (/ 2.0 (pow x 3.0))
       (+ t_0 (- (/ -2.0 x) (/ x (* 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 / (x + 1.0);
	double t_1 = (t_0 + (-2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if (t_1 <= -10.0) {
		tmp = t_0 - ((-1.0 - x) * (((-2.0 - x) - (x * -2.0)) / (x * (1.0 - (x * x)))));
	} else if (t_1 <= 5e-22) {
		tmp = 2.0 / pow(x, 3.0);
	} else {
		tmp = t_0 + ((-2.0 / x) - (x / (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) :: tmp
    t_0 = 1.0d0 / (x + 1.0d0)
    t_1 = (t_0 + ((-2.0d0) / x)) + (1.0d0 / (x + (-1.0d0)))
    if (t_1 <= (-10.0d0)) then
        tmp = t_0 - (((-1.0d0) - x) * ((((-2.0d0) - x) - (x * (-2.0d0))) / (x * (1.0d0 - (x * x)))))
    else if (t_1 <= 5d-22) then
        tmp = 2.0d0 / (x ** 3.0d0)
    else
        tmp = t_0 + (((-2.0d0) / x) - (x / (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 / (x + 1.0);
	double t_1 = (t_0 + (-2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if (t_1 <= -10.0) {
		tmp = t_0 - ((-1.0 - x) * (((-2.0 - x) - (x * -2.0)) / (x * (1.0 - (x * x)))));
	} else if (t_1 <= 5e-22) {
		tmp = 2.0 / Math.pow(x, 3.0);
	} else {
		tmp = t_0 + ((-2.0 / x) - (x / (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 / (x + 1.0)
	t_1 = (t_0 + (-2.0 / x)) + (1.0 / (x + -1.0))
	tmp = 0
	if t_1 <= -10.0:
		tmp = t_0 - ((-1.0 - x) * (((-2.0 - x) - (x * -2.0)) / (x * (1.0 - (x * x)))))
	elif t_1 <= 5e-22:
		tmp = 2.0 / math.pow(x, 3.0)
	else:
		tmp = t_0 + ((-2.0 / x) - (x / (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(x + 1.0))
	t_1 = Float64(Float64(t_0 + Float64(-2.0 / x)) + Float64(1.0 / Float64(x + -1.0)))
	tmp = 0.0
	if (t_1 <= -10.0)
		tmp = Float64(t_0 - Float64(Float64(-1.0 - x) * Float64(Float64(Float64(-2.0 - x) - Float64(x * -2.0)) / Float64(x * Float64(1.0 - Float64(x * x))))));
	elseif (t_1 <= 5e-22)
		tmp = Float64(2.0 / (x ^ 3.0));
	else
		tmp = Float64(t_0 + Float64(Float64(-2.0 / x) - Float64(x / Float64(x * Float64(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 / (x + 1.0);
	t_1 = (t_0 + (-2.0 / x)) + (1.0 / (x + -1.0));
	tmp = 0.0;
	if (t_1 <= -10.0)
		tmp = t_0 - ((-1.0 - x) * (((-2.0 - x) - (x * -2.0)) / (x * (1.0 - (x * x)))));
	elseif (t_1 <= 5e-22)
		tmp = 2.0 / (x ^ 3.0);
	else
		tmp = t_0 + ((-2.0 / x) - (x / (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[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 + N[(-2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -10.0], N[(t$95$0 - N[(N[(-1.0 - x), $MachinePrecision] * N[(N[(N[(-2.0 - x), $MachinePrecision] - N[(x * -2.0), $MachinePrecision]), $MachinePrecision] / N[(x * N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-22], N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(N[(-2.0 / x), $MachinePrecision] - N[(x / N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	9.7
Target	0.3
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))) < -10

   1. Initial program 0.0

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

   2. Simplified 0.0

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

      [Start] 0.0	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 0.0	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 0.0	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 0.0	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 0.0	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 0.0	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 0.0	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 0.0	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 0.0	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 0.0	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Applied egg-rr 0.0

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

   4. Simplified 0.0

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

      [Start] 0.0	\[ \frac{1}{1 + x} - \frac{\frac{-2 + \left(2 \cdot x - x\right)}{x}}{1 - x \cdot x} \cdot \left(-1 - x\right) \]
      *-commutative [=>] 0.0	\[ \frac{1}{1 + x} - \color{blue}{\left(-1 - x\right) \cdot \frac{\frac{-2 + \left(2 \cdot x - x\right)}{x}}{1 - x \cdot x}} \]
      associate-/l/ [=>] 0.0	\[ \frac{1}{1 + x} - \left(-1 - x\right) \cdot \color{blue}{\frac{-2 + \left(2 \cdot x - x\right)}{\left(1 - x \cdot x\right) \cdot x}} \]
      +-commutative [=>] 0.0	\[ \frac{1}{1 + x} - \left(-1 - x\right) \cdot \frac{\color{blue}{\left(2 \cdot x - x\right) + -2}}{\left(1 - x \cdot x\right) \cdot x} \]
      associate-+l- [=>] 0.0	\[ \frac{1}{1 + x} - \left(-1 - x\right) \cdot \frac{\color{blue}{2 \cdot x - \left(x - -2\right)}}{\left(1 - x \cdot x\right) \cdot x} \]
      *-commutative [=>] 0.0	\[ \frac{1}{1 + x} - \left(-1 - x\right) \cdot \frac{\color{blue}{x \cdot 2} - \left(x - -2\right)}{\left(1 - x \cdot x\right) \cdot x} \]
      *-commutative [=>] 0.0	\[ \frac{1}{1 + x} - \left(-1 - x\right) \cdot \frac{x \cdot 2 - \left(x - -2\right)}{\color{blue}{x \cdot \left(1 - x \cdot x\right)}} \]

   if -10 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 4.99999999999999954e-22

   1. Initial program 19.6

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

   2. Simplified 19.6

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

      [Start] 19.6	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 19.6	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 19.6	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 19.6	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 19.6	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 19.6	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 19.6	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 19.6	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 19.6	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 19.6	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Taylor expanded in x around inf 0.9

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

   if 4.99999999999999954e-22 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 0.5

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

   2. Simplified 0.5

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

      [Start] 0.5	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 0.5	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 0.5	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 0.5	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 0.5	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 0.5	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 0.5	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 0.5	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 0.5	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 0.5	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Applied egg-rr 0.5

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

   4. Simplified 0.5

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

      [Start] 0.5	\[ \left(\frac{1}{1 + x} - \frac{2}{x}\right) + \left(\mathsf{fma}\left(-\frac{2}{x}, 1, \frac{2}{x}\right) - \frac{1}{1 - x}\right) \]
      sub-neg [=>] 0.5	\[ \color{blue}{\left(\frac{1}{1 + x} + \left(-\frac{2}{x}\right)\right)} + \left(\mathsf{fma}\left(-\frac{2}{x}, 1, \frac{2}{x}\right) - \frac{1}{1 - x}\right) \]
      mul-1-neg [<=] 0.5	\[ \left(\frac{1}{1 + x} + \color{blue}{-1 \cdot \frac{2}{x}}\right) + \left(\mathsf{fma}\left(-\frac{2}{x}, 1, \frac{2}{x}\right) - \frac{1}{1 - x}\right) \]
      associate-+l+ [=>] 0.5	\[ \color{blue}{\frac{1}{1 + x} + \left(-1 \cdot \frac{2}{x} + \left(\mathsf{fma}\left(-\frac{2}{x}, 1, \frac{2}{x}\right) - \frac{1}{1 - x}\right)\right)} \]
      +-commutative [=>] 0.5	\[ \frac{1}{\color{blue}{x + 1}} + \left(-1 \cdot \frac{2}{x} + \left(\mathsf{fma}\left(-\frac{2}{x}, 1, \frac{2}{x}\right) - \frac{1}{1 - x}\right)\right) \]
      associate-*r/ [=>] 0.5	\[ \frac{1}{x + 1} + \left(\color{blue}{\frac{-1 \cdot 2}{x}} + \left(\mathsf{fma}\left(-\frac{2}{x}, 1, \frac{2}{x}\right) - \frac{1}{1 - x}\right)\right) \]
      metadata-eval [=>] 0.5	\[ \frac{1}{x + 1} + \left(\frac{\color{blue}{-2}}{x} + \left(\mathsf{fma}\left(-\frac{2}{x}, 1, \frac{2}{x}\right) - \frac{1}{1 - x}\right)\right) \]
      mul-1-neg [<=] 0.5	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \left(\mathsf{fma}\left(\color{blue}{-1 \cdot \frac{2}{x}}, 1, \frac{2}{x}\right) - \frac{1}{1 - x}\right)\right) \]
      fma-udef [=>] 0.5	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \left(\color{blue}{\left(\left(-1 \cdot \frac{2}{x}\right) \cdot 1 + \frac{2}{x}\right)} - \frac{1}{1 - x}\right)\right) \]
      associate--l+ [=>] 0.5	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \color{blue}{\left(\left(-1 \cdot \frac{2}{x}\right) \cdot 1 + \left(\frac{2}{x} - \frac{1}{1 - x}\right)\right)}\right) \]
      associate-*r/ [=>] 0.5	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \left(\color{blue}{\frac{-1 \cdot 2}{x}} \cdot 1 + \left(\frac{2}{x} - \frac{1}{1 - x}\right)\right)\right) \]
      metadata-eval [=>] 0.5	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \left(\frac{\color{blue}{-2}}{x} \cdot 1 + \left(\frac{2}{x} - \frac{1}{1 - x}\right)\right)\right) \]
      associate-*l/ [=>] 0.5	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \left(\color{blue}{\frac{-2 \cdot 1}{x}} + \left(\frac{2}{x} - \frac{1}{1 - x}\right)\right)\right) \]
      metadata-eval [=>] 0.5	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \left(\frac{\color{blue}{-2}}{x} + \left(\frac{2}{x} - \frac{1}{1 - x}\right)\right)\right) \]
      sub-neg [=>] 0.5	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \left(\frac{-2}{x} + \color{blue}{\left(\frac{2}{x} + \left(-\frac{1}{1 - x}\right)\right)}\right)\right) \]
      distribute-neg-frac [=>] 0.5	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \left(\frac{-2}{x} + \left(\frac{2}{x} + \color{blue}{\frac{-1}{1 - x}}\right)\right)\right) \]
      metadata-eval [=>] 0.5	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \left(\frac{-2}{x} + \left(\frac{2}{x} + \frac{\color{blue}{-1}}{1 - x}\right)\right)\right) \]

   5. Applied egg-rr 31.0

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

   6. Simplified 31.0

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

      [Start] 31.0	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \frac{-2 \cdot \left(x \cdot \left(1 - x\right)\right) + x \cdot \mathsf{fma}\left(2, 1 - x, -x\right)}{x \cdot \left(x \cdot \left(1 - x\right)\right)}\right) \]
      +-commutative [=>] 31.0	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \frac{\color{blue}{x \cdot \mathsf{fma}\left(2, 1 - x, -x\right) + -2 \cdot \left(x \cdot \left(1 - x\right)\right)}}{x \cdot \left(x \cdot \left(1 - x\right)\right)}\right) \]
      *-commutative [<=] 31.0	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \frac{x \cdot \mathsf{fma}\left(2, 1 - x, -x\right) + \color{blue}{\left(x \cdot \left(1 - x\right)\right) \cdot -2}}{x \cdot \left(x \cdot \left(1 - x\right)\right)}\right) \]
      associate-*l* [=>] 31.0	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \frac{x \cdot \mathsf{fma}\left(2, 1 - x, -x\right) + \color{blue}{x \cdot \left(\left(1 - x\right) \cdot -2\right)}}{x \cdot \left(x \cdot \left(1 - x\right)\right)}\right) \]
      distribute-lft-out [=>] 31.0	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \frac{\color{blue}{x \cdot \left(\mathsf{fma}\left(2, 1 - x, -x\right) + \left(1 - x\right) \cdot -2\right)}}{x \cdot \left(x \cdot \left(1 - x\right)\right)}\right) \]
      fma-udef [=>] 31.0	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \frac{x \cdot \left(\color{blue}{\left(2 \cdot \left(1 - x\right) + \left(-x\right)\right)} + \left(1 - x\right) \cdot -2\right)}{x \cdot \left(x \cdot \left(1 - x\right)\right)}\right) \]
      unsub-neg [=>] 31.0	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \frac{x \cdot \left(\color{blue}{\left(2 \cdot \left(1 - x\right) - x\right)} + \left(1 - x\right) \cdot -2\right)}{x \cdot \left(x \cdot \left(1 - x\right)\right)}\right) \]
      *-commutative [=>] 31.0	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \frac{x \cdot \left(\left(2 \cdot \left(1 - x\right) - x\right) + \color{blue}{-2 \cdot \left(1 - x\right)}\right)}{x \cdot \left(x \cdot \left(1 - x\right)\right)}\right) \]

   7. Applied egg-rr 1.4

      \[\leadsto \frac{1}{x + 1} + \left(\frac{-2}{x} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{2 \cdot \left(1 - x\right) - \mathsf{fma}\left(2, 1 - x, x\right)}{x \cdot \left(1 - x\right)}\right)} - 1\right)}\right) \]

   8. Simplified 0.5

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

      [Start] 1.4	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \left(e^{\mathsf{log1p}\left(\frac{2 \cdot \left(1 - x\right) - \mathsf{fma}\left(2, 1 - x, x\right)}{x \cdot \left(1 - x\right)}\right)} - 1\right)\right) \]
      expm1-def [=>] 1.2	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot \left(1 - x\right) - \mathsf{fma}\left(2, 1 - x, x\right)}{x \cdot \left(1 - x\right)}\right)\right)}\right) \]
      expm1-log1p [=>] 0.5	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \color{blue}{\frac{2 \cdot \left(1 - x\right) - \mathsf{fma}\left(2, 1 - x, x\right)}{x \cdot \left(1 - x\right)}}\right) \]
      fma-udef [=>] 0.5	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \frac{2 \cdot \left(1 - x\right) - \color{blue}{\left(2 \cdot \left(1 - x\right) + x\right)}}{x \cdot \left(1 - x\right)}\right) \]
      associate--r+ [=>] 0.5	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \frac{\color{blue}{\left(2 \cdot \left(1 - x\right) - 2 \cdot \left(1 - x\right)\right) - x}}{x \cdot \left(1 - x\right)}\right) \]
      +-inverses [=>] 0.5	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \frac{\color{blue}{0} - x}{x \cdot \left(1 - x\right)}\right) \]
      neg-sub0 [<=] 0.5	\[ \frac{1}{x + 1} + \left(\frac{-2}{x} + \frac{\color{blue}{-x}}{x \cdot \left(1 - x\right)}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{x + 1} + \frac{-2}{x}\right) + \frac{1}{x + -1} \leq -10:\\ \;\;\;\;\frac{1}{x + 1} - \left(-1 - x\right) \cdot \frac{\left(-2 - x\right) - x \cdot -2}{x \cdot \left(1 - x \cdot x\right)}\\ \mathbf{elif}\;\left(\frac{1}{x + 1} + \frac{-2}{x}\right) + \frac{1}{x + -1} \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + 1} + \left(\frac{-2}{x} - \frac{x}{x \cdot \left(1 - x\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	7040

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

Alternative 2
Error	9.7
Cost	960

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

Alternative 3
Error	9.7
Cost	960

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

Alternative 4
Error	15.1
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.38\right):\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x}\\ \end{array} \]

Alternative 5
Error	14.9
Cost	584

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

Alternative 6
Error	10.5
Cost	448

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

Alternative 7
Error	30.4
Cost	192

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

Alternative 8
Error	61.9
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023237 
(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))))