3frac (problem 3.3.3)

Average Error: 9.9 → 0.7

Time: 4.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{x - 1}\\ t_1 := \left(\frac{1}{1 + x} - \frac{2}{x}\right) + t_0\\ \mathbf{if}\;t_1 \leq -5700627.715827315:\\ \;\;\;\;t_0 + \left(e^{-\mathsf{log1p}\left(x\right)} - \frac{2}{x}\right)\\ \mathbf{elif}\;t_1 \leq 1.247041458181733 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{3}} + \frac{2}{{x}^{9}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, x + {x}^{3}, \frac{-2}{x}\right)\\ \end{array} \]

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 (+ (- (/ 1.0 (+ 1.0 x)) (/ 2.0 x)) t_0)))
   (if (<= t_1 -5700627.715827315)
     (+ t_0 (- (exp (- (log1p x))) (/ 2.0 x)))
     (if (<= t_1 1.247041458181733e-6)
       (+
        (/ 2.0 (pow x 5.0))
        (+ (/ 2.0 (pow x 7.0)) (+ (/ 2.0 (pow x 3.0)) (/ 2.0 (pow x 9.0)))))
       (fma -2.0 (+ x (pow x 3.0)) (/ -2.0 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 = ((1.0 / (1.0 + x)) - (2.0 / x)) + t_0;
	double tmp;
	if (t_1 <= -5700627.715827315) {
		tmp = t_0 + (exp(-log1p(x)) - (2.0 / x));
	} else if (t_1 <= 1.247041458181733e-6) {
		tmp = (2.0 / pow(x, 5.0)) + ((2.0 / pow(x, 7.0)) + ((2.0 / pow(x, 3.0)) + (2.0 / pow(x, 9.0))));
	} else {
		tmp = fma(-2.0, (x + pow(x, 3.0)), (-2.0 / 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(Float64(1.0 / Float64(1.0 + x)) - Float64(2.0 / x)) + t_0)
	tmp = 0.0
	if (t_1 <= -5700627.715827315)
		tmp = Float64(t_0 + Float64(exp(Float64(-log1p(x))) - Float64(2.0 / x)));
	elseif (t_1 <= 1.247041458181733e-6)
		tmp = Float64(Float64(2.0 / (x ^ 5.0)) + Float64(Float64(2.0 / (x ^ 7.0)) + Float64(Float64(2.0 / (x ^ 3.0)) + Float64(2.0 / (x ^ 9.0)))));
	else
		tmp = fma(-2.0, Float64(x + (x ^ 3.0)), Float64(-2.0 / 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[(1.0 / 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] + t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -5700627.715827315], N[(t$95$0 + N[(N[Exp[(-N[Log[1 + x], $MachinePrecision])], $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.247041458181733e-6], N[(N[(2.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 / N[Power[x, 9.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(x + N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-2.0 / x), $MachinePrecision]), $MachinePrecision]]]]]

Error

Bits error versus x

Target

Original	9.9
Target	0.3
Herbie	0.7

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -5700627.71582731511

   1. Initial program 0.0

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

   2. Applied add-exp-log_binary64 0.0

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

   3. Applied 1-exp_binary64 0.0

      \[\leadsto \left(\frac{\color{blue}{e^{0}}}{e^{\log \left(x + 1\right)}} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

   4. Applied div-exp_binary64 0.0

      \[\leadsto \left(\color{blue}{e^{0 - \log \left(x + 1\right)}} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

   5. Simplified 0.0

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

   if -5700627.71582731511 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 1.2470414581817e-6

   1. Initial program 19.8

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

   2. Taylor expanded in x around inf 1.2

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{5}} + \left(2 \cdot \frac{1}{{x}^{7}} + \left(2 \cdot \frac{1}{{x}^{9}} + 2 \cdot \frac{1}{{x}^{3}}\right)\right)} \]

   3. Simplified 1.2

      \[\leadsto \color{blue}{\frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{3}} + \frac{2}{{x}^{9}}\right)\right)} \]

   if 1.2470414581817e-6 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 0.0

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

   2. Taylor expanded in x around 0 0.5

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

   3. Simplified 0.5

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1} \leq -5700627.715827315:\\ \;\;\;\;\frac{1}{x - 1} + \left(e^{-\mathsf{log1p}\left(x\right)} - \frac{2}{x}\right)\\ \mathbf{elif}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1} \leq 1.247041458181733 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{3}} + \frac{2}{{x}^{9}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, x + {x}^{3}, \frac{-2}{x}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(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))))