3frac (problem 3.3.3)

Percentage Accurate: 84.2% → 99.7%

Time: 9.9s

Precision: binary64

Cost: 16392

Math

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

↓

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x + -1\right)\\ t_1 := \left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_0 - \left(-1 - x\right) \cdot \left(x - 2 \cdot \left(x + -1\right)\right)}{t_0 \cdot \left(1 + x\right)}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-28}:\\ \;\;\;\;2 \cdot {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + \mathsf{fma}\left(2, x + -1, -x\right) \cdot \left(-1 - x\right)}{\left(x + -1\right) \cdot \mathsf{fma}\left(x, x, x\right)}\\ \end{array} \end{array} \]

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 (+ (- (/ 1.0 (+ 1.0 x)) (/ 2.0 x)) (/ 1.0 (+ x -1.0)))))
   (if (<= t_1 -1e-6)
     (/ (- t_0 (* (- -1.0 x) (- x (* 2.0 (+ x -1.0))))) (* t_0 (+ 1.0 x)))
     (if (<= t_1 2e-28)
       (* 2.0 (pow x -3.0))
       (/
        (+ t_0 (* (fma 2.0 (+ x -1.0) (- x)) (- -1.0 x)))
        (* (+ x -1.0) (fma x x 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 = x * (x + -1.0);
	double t_1 = ((1.0 / (1.0 + x)) - (2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if (t_1 <= -1e-6) {
		tmp = (t_0 - ((-1.0 - x) * (x - (2.0 * (x + -1.0))))) / (t_0 * (1.0 + x));
	} else if (t_1 <= 2e-28) {
		tmp = 2.0 * pow(x, -3.0);
	} else {
		tmp = (t_0 + (fma(2.0, (x + -1.0), -x) * (-1.0 - x))) / ((x + -1.0) * fma(x, x, 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(x * Float64(x + -1.0))
	t_1 = Float64(Float64(Float64(1.0 / Float64(1.0 + x)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x + -1.0)))
	tmp = 0.0
	if (t_1 <= -1e-6)
		tmp = Float64(Float64(t_0 - Float64(Float64(-1.0 - x) * Float64(x - Float64(2.0 * Float64(x + -1.0))))) / Float64(t_0 * Float64(1.0 + x)));
	elseif (t_1 <= 2e-28)
		tmp = Float64(2.0 * (x ^ -3.0));
	else
		tmp = Float64(Float64(t_0 + Float64(fma(2.0, Float64(x + -1.0), Float64(-x)) * Float64(-1.0 - x))) / Float64(Float64(x + -1.0) * fma(x, x, x)));
	end
	return 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[(N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-6], N[(N[(t$95$0 - N[(N[(-1.0 - x), $MachinePrecision] * N[(x - N[(2.0 * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-28], N[(2.0 * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 + N[(N[(2.0 * N[(x + -1.0), $MachinePrecision] + (-x)), $MachinePrecision] * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x + -1.0), $MachinePrecision] * N[(x * x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	84.2%
Target	99.6%
Herbie	99.7%

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -9.99999999999999955e-7

   1. Initial program 99.7%

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

   2. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right)} \]
      Step-by-step derivation

      [Start] 99.7%	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 99.7%	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 99.7%	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 99.7%	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 99.7%	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 99.7%	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 99.7%	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 99.7%	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 99.7%	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 99.7%	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Applied egg-rr 99.8%

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

      [Start] 99.7%	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      flip-+ [=>] 99.8%	\[ \frac{1}{\color{blue}{\frac{1 \cdot 1 - x \cdot x}{1 - x}}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      sub-neg [=>] 99.8%	\[ \frac{1}{\frac{1 \cdot 1 - x \cdot x}{\color{blue}{1 + \left(-x\right)}}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      metadata-eval [<=] 99.8%	\[ \frac{1}{\frac{1 \cdot 1 - x \cdot x}{\color{blue}{\left(--1\right)} + \left(-x\right)}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      distribute-neg-in [<=] 99.8%	\[ \frac{1}{\frac{1 \cdot 1 - x \cdot x}{\color{blue}{-\left(-1 + x\right)}}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      +-commutative [<=] 99.8%	\[ \frac{1}{\frac{1 \cdot 1 - x \cdot x}{-\color{blue}{\left(x + -1\right)}}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      associate-/r/ [=>] 99.8%	\[ \color{blue}{\frac{1}{1 \cdot 1 - x \cdot x} \cdot \left(-\left(x + -1\right)\right)} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      metadata-eval [=>] 99.8%	\[ \frac{1}{\color{blue}{1} - x \cdot x} \cdot \left(-\left(x + -1\right)\right) - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      +-commutative [=>] 99.8%	\[ \frac{1}{1 - x \cdot x} \cdot \left(-\color{blue}{\left(-1 + x\right)}\right) - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      distribute-neg-in [=>] 99.8%	\[ \frac{1}{1 - x \cdot x} \cdot \color{blue}{\left(\left(--1\right) + \left(-x\right)\right)} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      metadata-eval [=>] 99.8%	\[ \frac{1}{1 - x \cdot x} \cdot \left(\color{blue}{1} + \left(-x\right)\right) - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      sub-neg [<=] 99.8%	\[ \frac{1}{1 - x \cdot x} \cdot \color{blue}{\left(1 - x\right)} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

   4. Applied egg-rr 99.8%

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

      [Start] 99.8%	\[ \frac{1}{1 - x \cdot x} \cdot \left(1 - x\right) - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      flip-- [=>] 99.7%	\[ \frac{1}{1 - x \cdot x} \cdot \color{blue}{\frac{1 \cdot 1 - x \cdot x}{1 + x}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      metadata-eval [=>] 99.7%	\[ \frac{1}{1 - x \cdot x} \cdot \frac{\color{blue}{1} - x \cdot x}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      +-commutative [<=] 99.7%	\[ \frac{1}{1 - x \cdot x} \cdot \frac{1 - x \cdot x}{\color{blue}{x + 1}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      frac-times [=>] 99.8%	\[ \color{blue}{\frac{1 \cdot \left(1 - x \cdot x\right)}{\left(1 - x \cdot x\right) \cdot \left(x + 1\right)}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      *-un-lft-identity [<=] 99.8%	\[ \frac{\color{blue}{1 - x \cdot x}}{\left(1 - x \cdot x\right) \cdot \left(x + 1\right)} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      +-commutative [=>] 99.8%	\[ \frac{1 - x \cdot x}{\left(1 - x \cdot x\right) \cdot \color{blue}{\left(1 + x\right)}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

   5. Applied egg-rr 99.9%

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

      [Start] 99.8%	\[ \frac{1 - x \cdot x}{\left(1 - x \cdot x\right) \cdot \left(1 + x\right)} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      clear-num [=>] 99.7%	\[ \color{blue}{\frac{1}{\frac{\left(1 - x \cdot x\right) \cdot \left(1 + x\right)}{1 - x \cdot x}}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      frac-sub [=>] 99.6%	\[ \frac{1}{\frac{\left(1 - x \cdot x\right) \cdot \left(1 + x\right)}{1 - x \cdot x}} - \color{blue}{\frac{2 \cdot \left(x + -1\right) - x \cdot 1}{x \cdot \left(x + -1\right)}} \]
      frac-sub [=>] 99.9%	\[ \color{blue}{\frac{1 \cdot \left(x \cdot \left(x + -1\right)\right) - \frac{\left(1 - x \cdot x\right) \cdot \left(1 + x\right)}{1 - x \cdot x} \cdot \left(2 \cdot \left(x + -1\right) - x \cdot 1\right)}{\frac{\left(1 - x \cdot x\right) \cdot \left(1 + x\right)}{1 - x \cdot x} \cdot \left(x \cdot \left(x + -1\right)\right)}} \]

   if -9.99999999999999955e-7 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 1.99999999999999994e-28

   1. Initial program 69.0%

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

   2. Simplified 69.0%

      \[\leadsto \color{blue}{\frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right)} \]
      Step-by-step derivation

      [Start] 69.0%	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 69.0%	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 69.0%	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 69.0%	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 69.0%	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 69.0%	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 69.0%	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 69.0%	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 69.0%	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 69.0%	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Taylor expanded in x around inf 99.8%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{2 \cdot {x}^{-3}} \]
      Step-by-step derivation

      [Start] 99.8%	\[ \frac{2}{{x}^{3}} \]
      div-inv [=>] 99.8%	\[ \color{blue}{2 \cdot \frac{1}{{x}^{3}}} \]
      pow-flip [=>] 100.0%	\[ 2 \cdot \color{blue}{{x}^{\left(-3\right)}} \]
      metadata-eval [=>] 100.0%	\[ 2 \cdot {x}^{\color{blue}{-3}} \]

   if 1.99999999999999994e-28 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 99.6%

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

   2. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right)} \]
      Step-by-step derivation

      [Start] 99.6%	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 99.6%	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 99.6%	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 99.6%	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 99.6%	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 99.6%	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 99.6%	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 99.6%	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 99.6%	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 99.6%	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Applied egg-rr 99.6%

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

      [Start] 99.6%	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      flip-+ [=>] 99.6%	\[ \frac{1}{\color{blue}{\frac{1 \cdot 1 - x \cdot x}{1 - x}}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      sub-neg [=>] 99.6%	\[ \frac{1}{\frac{1 \cdot 1 - x \cdot x}{\color{blue}{1 + \left(-x\right)}}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      metadata-eval [<=] 99.6%	\[ \frac{1}{\frac{1 \cdot 1 - x \cdot x}{\color{blue}{\left(--1\right)} + \left(-x\right)}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      distribute-neg-in [<=] 99.6%	\[ \frac{1}{\frac{1 \cdot 1 - x \cdot x}{\color{blue}{-\left(-1 + x\right)}}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      +-commutative [<=] 99.6%	\[ \frac{1}{\frac{1 \cdot 1 - x \cdot x}{-\color{blue}{\left(x + -1\right)}}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      associate-/r/ [=>] 99.6%	\[ \color{blue}{\frac{1}{1 \cdot 1 - x \cdot x} \cdot \left(-\left(x + -1\right)\right)} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      metadata-eval [=>] 99.6%	\[ \frac{1}{\color{blue}{1} - x \cdot x} \cdot \left(-\left(x + -1\right)\right) - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      +-commutative [=>] 99.6%	\[ \frac{1}{1 - x \cdot x} \cdot \left(-\color{blue}{\left(-1 + x\right)}\right) - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      distribute-neg-in [=>] 99.6%	\[ \frac{1}{1 - x \cdot x} \cdot \color{blue}{\left(\left(--1\right) + \left(-x\right)\right)} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      metadata-eval [=>] 99.6%	\[ \frac{1}{1 - x \cdot x} \cdot \left(\color{blue}{1} + \left(-x\right)\right) - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      sub-neg [<=] 99.6%	\[ \frac{1}{1 - x \cdot x} \cdot \color{blue}{\left(1 - x\right)} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

   4. Simplified 99.6%

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

      [Start] 99.6%	\[ \frac{1}{1 - x \cdot x} \cdot \left(1 - x\right) - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      associate-*l/ [=>] 99.6%	\[ \color{blue}{\frac{1 \cdot \left(1 - x\right)}{1 - x \cdot x}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      *-lft-identity [=>] 99.6%	\[ \frac{\color{blue}{1 - x}}{1 - x \cdot x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]

   5. Applied egg-rr 100.0%

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

      [Start] 99.6%	\[ \frac{1 - x}{1 - x \cdot x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      clear-num [=>] 99.6%	\[ \color{blue}{\frac{1}{\frac{1 - x \cdot x}{1 - x}}} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      frac-sub [=>] 99.6%	\[ \frac{1}{\frac{1 - x \cdot x}{1 - x}} - \color{blue}{\frac{2 \cdot \left(x + -1\right) - x \cdot 1}{x \cdot \left(x + -1\right)}} \]
      frac-sub [=>] 99.9%	\[ \color{blue}{\frac{1 \cdot \left(x \cdot \left(x + -1\right)\right) - \frac{1 - x \cdot x}{1 - x} \cdot \left(2 \cdot \left(x + -1\right) - x \cdot 1\right)}{\frac{1 - x \cdot x}{1 - x} \cdot \left(x \cdot \left(x + -1\right)\right)}} \]
      *-un-lft-identity [<=] 99.9%	\[ \frac{\color{blue}{x \cdot \left(x + -1\right)} - \frac{1 - x \cdot x}{1 - x} \cdot \left(2 \cdot \left(x + -1\right) - x \cdot 1\right)}{\frac{1 - x \cdot x}{1 - x} \cdot \left(x \cdot \left(x + -1\right)\right)} \]
      metadata-eval [<=] 99.9%	\[ \frac{x \cdot \left(x + -1\right) - \frac{\color{blue}{1 \cdot 1} - x \cdot x}{1 - x} \cdot \left(2 \cdot \left(x + -1\right) - x \cdot 1\right)}{\frac{1 - x \cdot x}{1 - x} \cdot \left(x \cdot \left(x + -1\right)\right)} \]
      flip-+ [<=] 99.9%	\[ \frac{x \cdot \left(x + -1\right) - \color{blue}{\left(1 + x\right)} \cdot \left(2 \cdot \left(x + -1\right) - x \cdot 1\right)}{\frac{1 - x \cdot x}{1 - x} \cdot \left(x \cdot \left(x + -1\right)\right)} \]
      fma-neg [=>] 99.9%	\[ \frac{x \cdot \left(x + -1\right) - \left(1 + x\right) \cdot \color{blue}{\mathsf{fma}\left(2, x + -1, -x \cdot 1\right)}}{\frac{1 - x \cdot x}{1 - x} \cdot \left(x \cdot \left(x + -1\right)\right)} \]
      *-rgt-identity [=>] 99.9%	\[ \frac{x \cdot \left(x + -1\right) - \left(1 + x\right) \cdot \mathsf{fma}\left(2, x + -1, -\color{blue}{x}\right)}{\frac{1 - x \cdot x}{1 - x} \cdot \left(x \cdot \left(x + -1\right)\right)} \]
      metadata-eval [<=] 99.9%	\[ \frac{x \cdot \left(x + -1\right) - \left(1 + x\right) \cdot \mathsf{fma}\left(2, x + -1, -x\right)}{\frac{\color{blue}{1 \cdot 1} - x \cdot x}{1 - x} \cdot \left(x \cdot \left(x + -1\right)\right)} \]
      flip-+ [<=] 100.0%	\[ \frac{x \cdot \left(x + -1\right) - \left(1 + x\right) \cdot \mathsf{fma}\left(2, x + -1, -x\right)}{\color{blue}{\left(1 + x\right)} \cdot \left(x \cdot \left(x + -1\right)\right)} \]

   6. Simplified 100.0%

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

      [Start] 100.0%	\[ \frac{x \cdot \left(x + -1\right) - \left(1 + x\right) \cdot \mathsf{fma}\left(2, x + -1, -x\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]
      +-commutative [=>] 100.0%	\[ \frac{x \cdot \color{blue}{\left(-1 + x\right)} - \left(1 + x\right) \cdot \mathsf{fma}\left(2, x + -1, -x\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]
      +-commutative [=>] 100.0%	\[ \frac{x \cdot \left(-1 + x\right) - \left(1 + x\right) \cdot \mathsf{fma}\left(2, \color{blue}{-1 + x}, -x\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]
      associate-*r* [=>] 100.0%	\[ \frac{x \cdot \left(-1 + x\right) - \left(1 + x\right) \cdot \mathsf{fma}\left(2, -1 + x, -x\right)}{\color{blue}{\left(\left(1 + x\right) \cdot x\right) \cdot \left(x + -1\right)}} \]
      +-commutative [=>] 100.0%	\[ \frac{x \cdot \left(-1 + x\right) - \left(1 + x\right) \cdot \mathsf{fma}\left(2, -1 + x, -x\right)}{\left(\color{blue}{\left(x + 1\right)} \cdot x\right) \cdot \left(x + -1\right)} \]
      distribute-lft1-in [<=] 100.0%	\[ \frac{x \cdot \left(-1 + x\right) - \left(1 + x\right) \cdot \mathsf{fma}\left(2, -1 + x, -x\right)}{\color{blue}{\left(x \cdot x + x\right)} \cdot \left(x + -1\right)} \]
      fma-udef [<=] 100.0%	\[ \frac{x \cdot \left(-1 + x\right) - \left(1 + x\right) \cdot \mathsf{fma}\left(2, -1 + x, -x\right)}{\color{blue}{\mathsf{fma}\left(x, x, x\right)} \cdot \left(x + -1\right)} \]
      +-commutative [=>] 100.0%	\[ \frac{x \cdot \left(-1 + x\right) - \left(1 + x\right) \cdot \mathsf{fma}\left(2, -1 + x, -x\right)}{\mathsf{fma}\left(x, x, x\right) \cdot \color{blue}{\left(-1 + x\right)}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq -1 \cdot 10^{-6}:\\ \;\;\;\;\frac{x \cdot \left(x + -1\right) - \left(-1 - x\right) \cdot \left(x - 2 \cdot \left(x + -1\right)\right)}{\left(x \cdot \left(x + -1\right)\right) \cdot \left(1 + x\right)}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 2 \cdot 10^{-28}:\\ \;\;\;\;2 \cdot {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x + -1\right) + \mathsf{fma}\left(2, x + -1, -x\right) \cdot \left(-1 - x\right)}{\left(x + -1\right) \cdot \mathsf{fma}\left(x, x, x\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.7%
Cost	16392

\[\begin{array}{l} t_0 := x \cdot \left(x + -1\right)\\ t_1 := \left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_0 - \left(-1 - x\right) \cdot \left(x - 2 \cdot \left(x + -1\right)\right)}{t_0 \cdot \left(1 + x\right)}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-28}:\\ \;\;\;\;2 \cdot {x}^{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + \mathsf{fma}\left(2, x + -1, -x\right) \cdot \left(-1 - x\right)}{\left(x + -1\right) \cdot \mathsf{fma}\left(x, x, x\right)}\\ \end{array} \]

Alternative 2
Accuracy	99.7%
Cost	8713

\[\begin{array}{l} t_0 := \left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ t_1 := x \cdot \left(x + -1\right)\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-6} \lor \neg \left(t_0 \leq 2 \cdot 10^{-28}\right):\\ \;\;\;\;\frac{t_1 - \left(-1 - x\right) \cdot \left(x - 2 \cdot \left(x + -1\right)\right)}{t_1 \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {x}^{-3}\\ \end{array} \]

Alternative 3
Accuracy	84.2%
Cost	960

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

Alternative 4
Accuracy	75.4%
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 5
Accuracy	83.0%
Cost	448

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

Alternative 6
Accuracy	51.4%
Cost	192

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

Reproduce

herbie shell --seed 2023167 
(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))))