3frac (problem 3.3.3)

Average Error: 10.2 → 0.0

Time: 6.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \left(\frac{1}{1 + x} + \frac{-2}{x}\right) + \frac{1}{x + -1}\\ t_1 := \frac{x \cdot \left(\left(x + -1\right) + \left(1 + x\right)\right) + \mathsf{fma}\left(x, x, -1\right) \cdot -2}{x \cdot \mathsf{fma}\left(x, x, -1\right)}\\ \mathbf{if}\;t_0 \leq -0.5:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-23}:\\ \;\;\;\;2 \cdot {x}^{-3} + \left(\frac{2}{{x}^{5}} + \frac{2}{{x}^{7}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (+ 1.0 x)) (/ -2.0 x)) (/ 1.0 (+ x -1.0))))
        (t_1
         (/
          (+ (* x (+ (+ x -1.0) (+ 1.0 x))) (* (fma x x -1.0) -2.0))
          (* x (fma x x -1.0)))))
   (if (<= t_0 -0.5)
     t_1
     (if (<= t_0 5e-23)
       (+ (* 2.0 (pow x -3.0)) (+ (/ 2.0 (pow x 5.0)) (/ 2.0 (pow x 7.0))))
       t_1))))

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 / (1.0 + x)) + (-2.0 / x)) + (1.0 / (x + -1.0));
	double t_1 = ((x * ((x + -1.0) + (1.0 + x))) + (fma(x, x, -1.0) * -2.0)) / (x * fma(x, x, -1.0));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = t_1;
	} else if (t_0 <= 5e-23) {
		tmp = (2.0 * pow(x, -3.0)) + ((2.0 / pow(x, 5.0)) + (2.0 / pow(x, 7.0)));
	} else {
		tmp = t_1;
	}
	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(Float64(Float64(1.0 / Float64(1.0 + x)) + Float64(-2.0 / x)) + Float64(1.0 / Float64(x + -1.0)))
	t_1 = Float64(Float64(Float64(x * Float64(Float64(x + -1.0) + Float64(1.0 + x))) + Float64(fma(x, x, -1.0) * -2.0)) / Float64(x * fma(x, x, -1.0)))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = t_1;
	elseif (t_0 <= 5e-23)
		tmp = Float64(Float64(2.0 * (x ^ -3.0)) + Float64(Float64(2.0 / (x ^ 5.0)) + Float64(2.0 / (x ^ 7.0))));
	else
		tmp = t_1;
	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[(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]}, Block[{t$95$1 = N[(N[(N[(x * N[(N[(x + -1.0), $MachinePrecision] + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * x + -1.0), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] / N[(x * N[(x * x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], t$95$1, If[LessEqual[t$95$0, 5e-23], N[(N[(2.0 * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Target

Original	10.2
Target	0.3
Herbie	0.0

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -0.5 or 5.0000000000000002e-23 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 0.2

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

   2. Applied egg-rr 0.2

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

   3. Applied egg-rr 0.0

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

   if -0.5 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 5.0000000000000002e-23

   1. Initial program 20.6

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

   2. Taylor expanded in x around inf 0.7

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

   3. Simplified 0.7

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

   4. Applied egg-rr 0.1

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} + \frac{-2}{x}\right) + \frac{1}{x + -1} \leq -0.5:\\ \;\;\;\;\frac{x \cdot \left(\left(x + -1\right) + \left(1 + x\right)\right) + \mathsf{fma}\left(x, x, -1\right) \cdot -2}{x \cdot \mathsf{fma}\left(x, x, -1\right)}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} + \frac{-2}{x}\right) + \frac{1}{x + -1} \leq 5 \cdot 10^{-23}:\\ \;\;\;\;2 \cdot {x}^{-3} + \left(\frac{2}{{x}^{5}} + \frac{2}{{x}^{7}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(x + -1\right) + \left(1 + x\right)\right) + \mathsf{fma}\left(x, x, -1\right) \cdot -2}{x \cdot \mathsf{fma}\left(x, x, -1\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022209 
(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))))