3frac (problem 3.3.3)

Average Error: 9.7 → 0.5

Time: 25.3s

Precision: binary64

Cost: 15560

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{x - 1}\\ t_1 := \left(\frac{1}{x + 1} - \frac{2}{x}\right) + t_0\\ t_2 := \frac{1}{1 + x}\\ t_3 := t_2 \cdot t_2\\ t_4 := t_2 \cdot \left(t_2 \cdot t_3\right)\\ \mathbf{if}\;t_1 \leq -50:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-16}:\\ \;\;\;\;2 \cdot \left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t_4 \cdot t_4\right) \cdot \frac{\frac{\frac{1}{t_2}}{t_3}}{t_4} - \frac{2}{x}\right) + t_0\\ \end{array} \]

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 (+ x 1.0)) (/ 2.0 x)) t_0))
        (t_2 (/ 1.0 (+ 1.0 x)))
        (t_3 (* t_2 t_2))
        (t_4 (* t_2 (* t_2 t_3))))
   (if (<= t_1 -50.0)
     t_1
     (if (<= t_1 2e-16)
       (* 2.0 (+ (/ 1.0 (pow x 5.0)) (/ 1.0 (pow x 3.0))))
       (+ (- (* (* t_4 t_4) (/ (/ (/ 1.0 t_2) t_3) t_4)) (/ 2.0 x)) t_0)))))

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 / (x + 1.0)) - (2.0 / x)) + t_0;
	double t_2 = 1.0 / (1.0 + x);
	double t_3 = t_2 * t_2;
	double t_4 = t_2 * (t_2 * t_3);
	double tmp;
	if (t_1 <= -50.0) {
		tmp = t_1;
	} else if (t_1 <= 2e-16) {
		tmp = 2.0 * ((1.0 / pow(x, 5.0)) + (1.0 / pow(x, 3.0)));
	} else {
		tmp = (((t_4 * t_4) * (((1.0 / t_2) / t_3) / t_4)) - (2.0 / x)) + t_0;
	}
	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) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = 1.0d0 / (x - 1.0d0)
    t_1 = ((1.0d0 / (x + 1.0d0)) - (2.0d0 / x)) + t_0
    t_2 = 1.0d0 / (1.0d0 + x)
    t_3 = t_2 * t_2
    t_4 = t_2 * (t_2 * t_3)
    if (t_1 <= (-50.0d0)) then
        tmp = t_1
    else if (t_1 <= 2d-16) then
        tmp = 2.0d0 * ((1.0d0 / (x ** 5.0d0)) + (1.0d0 / (x ** 3.0d0)))
    else
        tmp = (((t_4 * t_4) * (((1.0d0 / t_2) / t_3) / t_4)) - (2.0d0 / x)) + t_0
    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 = ((1.0 / (x + 1.0)) - (2.0 / x)) + t_0;
	double t_2 = 1.0 / (1.0 + x);
	double t_3 = t_2 * t_2;
	double t_4 = t_2 * (t_2 * t_3);
	double tmp;
	if (t_1 <= -50.0) {
		tmp = t_1;
	} else if (t_1 <= 2e-16) {
		tmp = 2.0 * ((1.0 / Math.pow(x, 5.0)) + (1.0 / Math.pow(x, 3.0)));
	} else {
		tmp = (((t_4 * t_4) * (((1.0 / t_2) / t_3) / t_4)) - (2.0 / x)) + t_0;
	}
	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 = ((1.0 / (x + 1.0)) - (2.0 / x)) + t_0
	t_2 = 1.0 / (1.0 + x)
	t_3 = t_2 * t_2
	t_4 = t_2 * (t_2 * t_3)
	tmp = 0
	if t_1 <= -50.0:
		tmp = t_1
	elif t_1 <= 2e-16:
		tmp = 2.0 * ((1.0 / math.pow(x, 5.0)) + (1.0 / math.pow(x, 3.0)))
	else:
		tmp = (((t_4 * t_4) * (((1.0 / t_2) / t_3) / t_4)) - (2.0 / x)) + t_0
	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(x + 1.0)) - Float64(2.0 / x)) + t_0)
	t_2 = Float64(1.0 / Float64(1.0 + x))
	t_3 = Float64(t_2 * t_2)
	t_4 = Float64(t_2 * Float64(t_2 * t_3))
	tmp = 0.0
	if (t_1 <= -50.0)
		tmp = t_1;
	elseif (t_1 <= 2e-16)
		tmp = Float64(2.0 * Float64(Float64(1.0 / (x ^ 5.0)) + Float64(1.0 / (x ^ 3.0))));
	else
		tmp = Float64(Float64(Float64(Float64(t_4 * t_4) * Float64(Float64(Float64(1.0 / t_2) / t_3) / t_4)) - Float64(2.0 / x)) + t_0);
	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 = ((1.0 / (x + 1.0)) - (2.0 / x)) + t_0;
	t_2 = 1.0 / (1.0 + x);
	t_3 = t_2 * t_2;
	t_4 = t_2 * (t_2 * t_3);
	tmp = 0.0;
	if (t_1 <= -50.0)
		tmp = t_1;
	elseif (t_1 <= 2e-16)
		tmp = 2.0 * ((1.0 / (x ^ 5.0)) + (1.0 / (x ^ 3.0)));
	else
		tmp = (((t_4 * t_4) * (((1.0 / t_2) / t_3) / t_4)) - (2.0 / x)) + t_0;
	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[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -50.0], t$95$1, If[LessEqual[t$95$1, 2e-16], N[(2.0 * N[(N[(1.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$4 * t$95$4), $MachinePrecision] * N[(N[(N[(1.0 / t$95$2), $MachinePrecision] / t$95$3), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]]]]]

Target

Original	9.7
Target	0.3
Herbie	0.5

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -50

   1. Initial program 0.0

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

   if -50 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 2e-16

   1. Initial program 19.6

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

   2. Taylor expanded in x around inf 0.8

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{3}}} \]

   3. Simplified 0.8

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

      [Start] 0.8	\[ 2 \cdot \frac{1}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{3}} \]
      rational.json-simplify-1 [=>] 0.8	\[ \color{blue}{2 \cdot \frac{1}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{5}}} \]
      rational.json-simplify-2 [=>] 0.8	\[ 2 \cdot \frac{1}{{x}^{3}} + \color{blue}{\frac{1}{{x}^{5}} \cdot 2} \]
      rational.json-simplify-47 [=>] 0.8	\[ \color{blue}{2 \cdot \left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{3}}\right)} \]

   if 2e-16 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 0.3

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

   2. Applied egg-rr 0.3

      \[\leadsto \left(\color{blue}{\left(\left(\frac{1}{1 + x} \cdot \left(\frac{1}{1 + x} \cdot \left(\frac{1}{1 + x} \cdot \frac{1}{1 + x}\right)\right)\right) \cdot \left(\frac{1}{1 + x} \cdot \left(\frac{1}{1 + x} \cdot \left(\frac{1}{1 + x} \cdot \frac{1}{1 + x}\right)\right)\right)\right) \cdot \frac{\frac{\frac{1}{\frac{1}{1 + x}}}{\frac{1}{1 + x} \cdot \frac{1}{1 + x}}}{\frac{1}{1 + x} \cdot \left(\frac{1}{1 + x} \cdot \left(\frac{1}{1 + x} \cdot \frac{1}{1 + x}\right)\right)}} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \leq -50:\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{elif}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;2 \cdot \left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{1}{1 + x} \cdot \left(\frac{1}{1 + x} \cdot \left(\frac{1}{1 + x} \cdot \frac{1}{1 + x}\right)\right)\right) \cdot \left(\frac{1}{1 + x} \cdot \left(\frac{1}{1 + x} \cdot \left(\frac{1}{1 + x} \cdot \frac{1}{1 + x}\right)\right)\right)\right) \cdot \frac{\frac{\frac{1}{\frac{1}{1 + x}}}{\frac{1}{1 + x} \cdot \frac{1}{1 + x}}}{\frac{1}{1 + x} \cdot \left(\frac{1}{1 + x} \cdot \left(\frac{1}{1 + x} \cdot \frac{1}{1 + x}\right)\right)} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	8712

\[\begin{array}{l} t_0 := \frac{1}{1 + x}\\ t_1 := \frac{1}{x - 1}\\ t_2 := \left(\frac{1}{x + 1} - \frac{2}{x}\right) + t_1\\ t_3 := t_0 \cdot t_0\\ t_4 := t_0 \cdot \left(t_0 \cdot t_3\right)\\ \mathbf{if}\;t_2 \leq -50:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t_4 \cdot t_4\right) \cdot \frac{\frac{\frac{1}{t_0}}{t_3}}{t_4} - \frac{2}{x}\right) + t_1\\ \end{array} \]

Alternative 2
Error	9.7
Cost	960

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

Alternative 3
Error	10.6
Cost	448

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

Alternative 4
Error	30.5
Cost	192

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

Alternative 5
Error	61.9
Cost	64

\[-1 \]

Alternative 6
Error	61.9
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023077 
(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))))