3frac (problem 3.3.3)

Average Error: 10.1 → 1.0

Time: 19.7s

Precision: binary64

Cost: 15560

Math

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

↓

\[\begin{array}{l} t_0 := \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{if}\;t_0 \leq -2000000000:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{elif}\;t_0 \leq 10^{-20}:\\ \;\;\;\;2 \cdot \left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{x} + \frac{x + x}{-1 + x \cdot x}}{1}\\ \end{array} \]

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 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0)))))
   (if (<= t_0 -2000000000.0)
     (/ -2.0 x)
     (if (<= t_0 1e-20)
       (* 2.0 (+ (/ 1.0 (pow x 5.0)) (/ 1.0 (pow x 3.0))))
       (/ (+ (/ -2.0 x) (/ (+ x x) (+ -1.0 (* x x)))) 1.0)))))

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 <= -2000000000.0) {
		tmp = -2.0 / x;
	} else if (t_0 <= 1e-20) {
		tmp = 2.0 * ((1.0 / pow(x, 5.0)) + (1.0 / pow(x, 3.0)));
	} else {
		tmp = ((-2.0 / x) + ((x + x) / (-1.0 + (x * x)))) / 1.0;
	}
	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 <= (-2000000000.0d0)) then
        tmp = (-2.0d0) / x
    else if (t_0 <= 1d-20) then
        tmp = 2.0d0 * ((1.0d0 / (x ** 5.0d0)) + (1.0d0 / (x ** 3.0d0)))
    else
        tmp = (((-2.0d0) / x) + ((x + x) / ((-1.0d0) + (x * x)))) / 1.0d0
    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 <= -2000000000.0) {
		tmp = -2.0 / x;
	} else if (t_0 <= 1e-20) {
		tmp = 2.0 * ((1.0 / Math.pow(x, 5.0)) + (1.0 / Math.pow(x, 3.0)));
	} else {
		tmp = ((-2.0 / x) + ((x + x) / (-1.0 + (x * x)))) / 1.0;
	}
	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 <= -2000000000.0:
		tmp = -2.0 / x
	elif t_0 <= 1e-20:
		tmp = 2.0 * ((1.0 / math.pow(x, 5.0)) + (1.0 / math.pow(x, 3.0)))
	else:
		tmp = ((-2.0 / x) + ((x + x) / (-1.0 + (x * x)))) / 1.0
	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 <= -2000000000.0)
		tmp = Float64(-2.0 / x);
	elseif (t_0 <= 1e-20)
		tmp = Float64(2.0 * Float64(Float64(1.0 / (x ^ 5.0)) + Float64(1.0 / (x ^ 3.0))));
	else
		tmp = Float64(Float64(Float64(-2.0 / x) + Float64(Float64(x + x) / Float64(-1.0 + Float64(x * x)))) / 1.0);
	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 <= -2000000000.0)
		tmp = -2.0 / x;
	elseif (t_0 <= 1e-20)
		tmp = 2.0 * ((1.0 / (x ^ 5.0)) + (1.0 / (x ^ 3.0)));
	else
		tmp = ((-2.0 / x) + ((x + x) / (-1.0 + (x * x)))) / 1.0;
	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, -2000000000.0], N[(-2.0 / x), $MachinePrecision], If[LessEqual[t$95$0, 1e-20], 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[(-2.0 / x), $MachinePrecision] + N[(N[(x + x), $MachinePrecision] / N[(-1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]]]]

Target

Original	10.1
Target	0.3
Herbie	1.0

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -2e9

   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

      \[\leadsto \color{blue}{\frac{-2}{x}} \]

   if -2e9 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 9.99999999999999945e-21

   1. Initial program 19.9

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

   2. Taylor expanded in x around inf 1.7

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

   3. Simplified 1.7

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

      [Start] 1.7	\[ 2 \cdot \frac{1}{{x}^{5}} + 2 \cdot \frac{1}{{x}^{3}} \]
      rational.json-simplify-1 [=>] 1.7	\[ \color{blue}{2 \cdot \frac{1}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{5}}} \]
      rational.json-simplify-2 [=>] 1.7	\[ 2 \cdot \frac{1}{{x}^{3}} + \color{blue}{\frac{1}{{x}^{5}} \cdot 2} \]
      rational.json-simplify-51 [=>] 1.7	\[ \color{blue}{2 \cdot \left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{3}}\right)} \]

   if 9.99999999999999945e-21 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 0.6

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

   2. Applied egg-rr 0.9

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

   3. Applied egg-rr 0.6

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

   4. Applied egg-rr 0.6

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

   5. Simplified 0.6

      \[\leadsto \frac{\color{blue}{\frac{-2}{x} + \frac{x + x}{-1 + x \cdot x}}}{1} \]
      Proof

      [Start] 0.6	\[ \frac{\left(\frac{x + x}{-1 + x \cdot x} + \frac{-2}{x}\right) - 0}{1} \]
      rational.json-simplify-5 [=>] 0.6	\[ \frac{\color{blue}{\frac{x + x}{-1 + x \cdot x} + \frac{-2}{x}}}{1} \]
      rational.json-simplify-1 [=>] 0.6	\[ \frac{\color{blue}{\frac{-2}{x} + \frac{x + x}{-1 + x \cdot x}}}{1} \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \leq -2000000000:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{elif}\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \leq 10^{-20}:\\ \;\;\;\;2 \cdot \left(\frac{1}{{x}^{5}} + \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{x} + \frac{x + x}{-1 + x \cdot x}}{1}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.1
Cost	8712

\[\begin{array}{l} t_0 := \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{if}\;t_0 \leq -2000000000:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{elif}\;t_0 \leq 10^{-20}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{x} + \frac{x + x}{-1 + x \cdot x}}{1}\\ \end{array} \]

Alternative 2
Error	10.2
Cost	1152

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

Alternative 3
Error	10.1
Cost	960

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

Alternative 4
Error	11.1
Cost	448

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

Alternative 5
Error	30.8
Cost	192

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

Reproduce

herbie shell --seed 2023073 
(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))))