3frac (problem 3.3.3)

Average Error: 10.2 → 0.7

Time: 16.6s

Precision: binary64

Cost: 22280

Math

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

↓

\[\begin{array}{l} t_0 := \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{if}\;t_0 \leq -2000:\\ \;\;\;\;\left(\frac{1}{1 + x} + \frac{1}{x + -1}\right) - \frac{2}{x}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-31}:\\ \;\;\;\;\left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{3}}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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)) (/ 2.0 x)) (/ 1.0 (- x 1.0)))))
   (if (<= t_0 -2000.0)
     (- (+ (/ 1.0 (+ 1.0 x)) (/ 1.0 (+ x -1.0))) (/ 2.0 x))
     (if (<= t_0 5e-31)
       (*
        (+ (/ 1.0 (pow x 7.0)) (+ (/ 1.0 (pow x 5.0)) (/ 1.0 (pow x 3.0))))
        2.0)
       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)) - (2.0 / x)) + (1.0 / (x - 1.0));
	double tmp;
	if (t_0 <= -2000.0) {
		tmp = ((1.0 / (1.0 + x)) + (1.0 / (x + -1.0))) - (2.0 / x);
	} else if (t_0 <= 5e-31) {
		tmp = ((1.0 / pow(x, 7.0)) + ((1.0 / pow(x, 5.0)) + (1.0 / pow(x, 3.0)))) * 2.0;
	} else {
		tmp = 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) :: tmp
    t_0 = ((1.0d0 / (x + 1.0d0)) - (2.0d0 / x)) + (1.0d0 / (x - 1.0d0))
    if (t_0 <= (-2000.0d0)) then
        tmp = ((1.0d0 / (1.0d0 + x)) + (1.0d0 / (x + (-1.0d0)))) - (2.0d0 / x)
    else if (t_0 <= 5d-31) then
        tmp = ((1.0d0 / (x ** 7.0d0)) + ((1.0d0 / (x ** 5.0d0)) + (1.0d0 / (x ** 3.0d0)))) * 2.0d0
    else
        tmp = 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)) - (2.0 / x)) + (1.0 / (x - 1.0));
	double tmp;
	if (t_0 <= -2000.0) {
		tmp = ((1.0 / (1.0 + x)) + (1.0 / (x + -1.0))) - (2.0 / x);
	} else if (t_0 <= 5e-31) {
		tmp = ((1.0 / Math.pow(x, 7.0)) + ((1.0 / Math.pow(x, 5.0)) + (1.0 / Math.pow(x, 3.0)))) * 2.0;
	} else {
		tmp = 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)) - (2.0 / x)) + (1.0 / (x - 1.0))
	tmp = 0
	if t_0 <= -2000.0:
		tmp = ((1.0 / (1.0 + x)) + (1.0 / (x + -1.0))) - (2.0 / x)
	elif t_0 <= 5e-31:
		tmp = ((1.0 / math.pow(x, 7.0)) + ((1.0 / math.pow(x, 5.0)) + (1.0 / math.pow(x, 3.0)))) * 2.0
	else:
		tmp = 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(Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x - 1.0)))
	tmp = 0.0
	if (t_0 <= -2000.0)
		tmp = Float64(Float64(Float64(1.0 / Float64(1.0 + x)) + Float64(1.0 / Float64(x + -1.0))) - Float64(2.0 / x));
	elseif (t_0 <= 5e-31)
		tmp = Float64(Float64(Float64(1.0 / (x ^ 7.0)) + Float64(Float64(1.0 / (x ^ 5.0)) + Float64(1.0 / (x ^ 3.0)))) * 2.0);
	else
		tmp = 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)) - (2.0 / x)) + (1.0 / (x - 1.0));
	tmp = 0.0;
	if (t_0 <= -2000.0)
		tmp = ((1.0 / (1.0 + x)) + (1.0 / (x + -1.0))) - (2.0 / x);
	elseif (t_0 <= 5e-31)
		tmp = ((1.0 / (x ^ 7.0)) + ((1.0 / (x ^ 5.0)) + (1.0 / (x ^ 3.0)))) * 2.0;
	else
		tmp = 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[(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]}, If[LessEqual[t$95$0, -2000.0], N[(N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-31], N[(N[(N[(1.0 / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], t$95$0]]]

Target

Original	10.2
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))) < -2e3

   1. Initial program 0.0

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

   2. Simplified 0.0

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

      [Start] 0.0	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.0	\[ \color{blue}{\frac{1}{x - 1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-108 [=>] 0.0	\[ \color{blue}{\left(\frac{1}{x + 1} + \frac{1}{x - 1}\right) - \frac{2}{x}} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.0	\[ \left(\frac{1}{\color{blue}{1 + x}} + \frac{1}{x - 1}\right) - \frac{2}{x} \]
      rational_best_oopsla_all_46_json_45_simplify-45 [=>] 0.0	\[ \left(\frac{1}{1 + x} + \frac{1}{\color{blue}{x + -1}}\right) - \frac{2}{x} \]

   if -2e3 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 5e-31

   1. Initial program 19.7

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

   2. Applied egg-rr 23.0

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

   3. Taylor expanded in x around inf 0.9

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

   4. Simplified 0.9

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

      [Start] 0.9	\[ 2 \cdot \frac{1}{{x}^{5}} + \left(2 \cdot \frac{1}{{x}^{7}} + 2 \cdot \frac{1}{{x}^{3}}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.9	\[ 2 \cdot \frac{1}{{x}^{5}} + \color{blue}{\left(2 \cdot \frac{1}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{7}}\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.9	\[ 2 \cdot \frac{1}{{x}^{5}} + \left(\color{blue}{\frac{1}{{x}^{3}} \cdot 2} + 2 \cdot \frac{1}{{x}^{7}}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-23 [=>] 0.9	\[ 2 \cdot \frac{1}{{x}^{5}} + \color{blue}{2 \cdot \left(\frac{1}{{x}^{3}} + \frac{1}{{x}^{7}}\right)} \]

   5. Applied egg-rr 0.9

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

   if 5e-31 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 1.1

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \leq -2000:\\ \;\;\;\;\left(\frac{1}{1 + x} + \frac{1}{x + -1}\right) - \frac{2}{x}\\ \mathbf{elif}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \leq 5 \cdot 10^{-31}:\\ \;\;\;\;\left(\frac{1}{{x}^{7}} + \left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{3}}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.7
Cost	15560

\[\begin{array}{l} t_0 := \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{if}\;t_0 \leq -2000:\\ \;\;\;\;\left(\frac{1}{1 + x} + \frac{1}{x + -1}\right) - \frac{2}{x}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-31}:\\ \;\;\;\;2 \cdot \left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.6
Cost	8712

\[\begin{array}{l} t_0 := \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-31}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	10.7
Cost	964

\[\begin{array}{l} \mathbf{if}\;x \leq -0.65:\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x}\\ \mathbf{elif}\;x \leq 0.64:\\ \;\;\;\;-2 \cdot x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + \frac{1}{x - 1}\\ \end{array} \]

Alternative 4
Error	10.2
Cost	960

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

Alternative 5
Error	10.8
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{2}{x} - \frac{2}{x}\\ \mathbf{elif}\;x \leq 0.64:\\ \;\;\;\;-2 \cdot x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + \frac{1}{x - 1}\\ \end{array} \]

Alternative 6
Error	10.9
Cost	712

\[\begin{array}{l} t_0 := \frac{2}{x} - \frac{2}{x}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;-2 \cdot x - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	31.4
Cost	192

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

Reproduce

herbie shell --seed 2023090 
(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))))