3frac (problem 3.3.3)

Average Error: 10.0 → 0.3

Time: 3.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{x - 1}\\ t_1 := \left(\frac{1}{1 + x} - \frac{2}{x}\right) + t_0\\ t_2 := t_0 + \left(e^{-\mathsf{log1p}\left(x\right)} - \frac{2}{x}\right)\\ \mathbf{if}\;t_1 \leq -19144.86656850329:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 1.3439767451283585 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left({x}^{-5}, 2, 2 \cdot \left({x}^{-3} + {x}^{-7}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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))
        (t_2 (+ t_0 (- (exp (- (log1p x))) (/ 2.0 x)))))
   (if (<= t_1 -19144.86656850329)
     t_2
     (if (<= t_1 1.3439767451283585e-8)
       (fma (pow x -5.0) 2.0 (* 2.0 (+ (pow x -3.0) (pow x -7.0))))
       t_2))))

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 t_2 = t_0 + (exp(-log1p(x)) - (2.0 / x));
	double tmp;
	if (t_1 <= -19144.86656850329) {
		tmp = t_2;
	} else if (t_1 <= 1.3439767451283585e-8) {
		tmp = fma(pow(x, -5.0), 2.0, (2.0 * (pow(x, -3.0) + pow(x, -7.0))));
	} else {
		tmp = t_2;
	}
	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)
	t_2 = Float64(t_0 + Float64(exp(Float64(-log1p(x))) - Float64(2.0 / x)))
	tmp = 0.0
	if (t_1 <= -19144.86656850329)
		tmp = t_2;
	elseif (t_1 <= 1.3439767451283585e-8)
		tmp = fma((x ^ -5.0), 2.0, Float64(2.0 * Float64((x ^ -3.0) + (x ^ -7.0))));
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = 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, -19144.86656850329], t$95$2, If[LessEqual[t$95$1, 1.3439767451283585e-8], N[(N[Power[x, -5.0], $MachinePrecision] * 2.0 + N[(2.0 * N[(N[Power[x, -3.0], $MachinePrecision] + N[Power[x, -7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Error

Bits error versus x

Target

Original	10.0
Target	0.3
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -19144.86656850329 or 1.34397674513e-8 < (+.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. Applied egg-rr 0.0

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

   if -19144.86656850329 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 1.34397674513e-8

   1. Initial program 19.7

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

   2. Taylor expanded in x around inf 0.9

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

   3. Simplified 0.9

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

   4. Applied egg-rr 0.5

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

   5. Applied egg-rr 0.5

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{-5}, 2, 2 \cdot \left({x}^{-3} + {x}^{-7}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1} \leq -19144.86656850329:\\ \;\;\;\;\frac{1}{x - 1} + \left(e^{-\mathsf{log1p}\left(x\right)} - \frac{2}{x}\right)\\ \mathbf{elif}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1} \leq 1.3439767451283585 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left({x}^{-5}, 2, 2 \cdot \left({x}^{-3} + {x}^{-7}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x - 1} + \left(e^{-\mathsf{log1p}\left(x\right)} - \frac{2}{x}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022134 
(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))))