3frac (problem 3.3.3)

Average Accuracy: 85.1% → 99.7%

Time: 15.2s

Precision: binary64

Cost: 1737

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot x - x\\ \mathbf{if}\;x \leq -12600000 \lor \neg \left(x \leq 100000000\right):\\ \;\;\;\;\frac{\frac{2}{x} + \frac{\frac{2}{x}}{x}}{x + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + \left(x + 1\right) \cdot \left(2 - x\right)}{t_0 \cdot \left(x + 1\right)}\\ \end{array} \]

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 (- (* x x) x)))
   (if (or (<= x -12600000.0) (not (<= x 100000000.0)))
     (/ (+ (/ 2.0 x) (/ (/ 2.0 x) x)) (+ x (* x x)))
     (/ (+ t_0 (* (+ x 1.0) (- 2.0 x))) (* t_0 (+ 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 = (x * x) - x;
	double tmp;
	if ((x <= -12600000.0) || !(x <= 100000000.0)) {
		tmp = ((2.0 / x) + ((2.0 / x) / x)) / (x + (x * x));
	} else {
		tmp = (t_0 + ((x + 1.0) * (2.0 - x))) / (t_0 * (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 = (x * x) - x
    if ((x <= (-12600000.0d0)) .or. (.not. (x <= 100000000.0d0))) then
        tmp = ((2.0d0 / x) + ((2.0d0 / x) / x)) / (x + (x * x))
    else
        tmp = (t_0 + ((x + 1.0d0) * (2.0d0 - x))) / (t_0 * (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 = (x * x) - x;
	double tmp;
	if ((x <= -12600000.0) || !(x <= 100000000.0)) {
		tmp = ((2.0 / x) + ((2.0 / x) / x)) / (x + (x * x));
	} else {
		tmp = (t_0 + ((x + 1.0) * (2.0 - x))) / (t_0 * (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 = (x * x) - x
	tmp = 0
	if (x <= -12600000.0) or not (x <= 100000000.0):
		tmp = ((2.0 / x) + ((2.0 / x) / x)) / (x + (x * x))
	else:
		tmp = (t_0 + ((x + 1.0) * (2.0 - x))) / (t_0 * (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(x * x) - x)
	tmp = 0.0
	if ((x <= -12600000.0) || !(x <= 100000000.0))
		tmp = Float64(Float64(Float64(2.0 / x) + Float64(Float64(2.0 / x) / x)) / Float64(x + Float64(x * x)));
	else
		tmp = Float64(Float64(t_0 + Float64(Float64(x + 1.0) * Float64(2.0 - x))) / Float64(t_0 * Float64(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 = (x * x) - x;
	tmp = 0.0;
	if ((x <= -12600000.0) || ~((x <= 100000000.0)))
		tmp = ((2.0 / x) + ((2.0 / x) / x)) / (x + (x * x));
	else
		tmp = (t_0 + ((x + 1.0) * (2.0 - x))) / (t_0 * (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[(x * x), $MachinePrecision] - x), $MachinePrecision]}, If[Or[LessEqual[x, -12600000.0], N[Not[LessEqual[x, 100000000.0]], $MachinePrecision]], N[(N[(N[(2.0 / x), $MachinePrecision] + N[(N[(2.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(x + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 + N[(N[(x + 1.0), $MachinePrecision] * N[(2.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	85.1%
Target	99.5%
Herbie	99.7%

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

Derivation

1. Split input into 2 regimes

2. if x < -1.26e7 or 1e8 < x

   1. Initial program 71.2%

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

   2. Simplified 71.2%

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

      [Start] 71.2	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 71.2	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 71.2	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 71.2	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 71.2	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 71.2	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 71.2	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 71.2	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 71.2	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 71.2	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Applied egg-rr 71.2%

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

   4. Simplified 71.2%

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

      [Start] 71.2	\[ \frac{1}{1 + x} - \frac{\frac{-2 + \left(2 \cdot x - x\right)}{x}}{1} \cdot \frac{1}{x + -1} \]
      *-commutative [=>] 71.2	\[ \frac{1}{1 + x} - \color{blue}{\frac{1}{x + -1} \cdot \frac{\frac{-2 + \left(2 \cdot x - x\right)}{x}}{1}} \]
      /-rgt-identity [=>] 71.2	\[ \frac{1}{1 + x} - \frac{1}{x + -1} \cdot \color{blue}{\frac{-2 + \left(2 \cdot x - x\right)}{x}} \]
      +-commutative [=>] 71.2	\[ \frac{1}{1 + x} - \frac{1}{x + -1} \cdot \frac{\color{blue}{\left(2 \cdot x - x\right) + -2}}{x} \]
      associate-+l- [=>] 71.2	\[ \frac{1}{1 + x} - \frac{1}{x + -1} \cdot \frac{\color{blue}{2 \cdot x - \left(x - -2\right)}}{x} \]
      *-commutative [=>] 71.2	\[ \frac{1}{1 + x} - \frac{1}{x + -1} \cdot \frac{\color{blue}{x \cdot 2} - \left(x - -2\right)}{x} \]

   5. Applied egg-rr 71.1%

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

   6. Simplified 71.1%

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

      [Start] 71.1	\[ \frac{x - \left(1 + x\right) \cdot \frac{x + -2}{x + -1}}{\left(1 + x\right) \cdot x} \]
      +-commutative [=>] 71.1	\[ \frac{x - \color{blue}{\left(x + 1\right)} \cdot \frac{x + -2}{x + -1}}{\left(1 + x\right) \cdot x} \]
      +-commutative [=>] 71.1	\[ \frac{x - \left(x + 1\right) \cdot \frac{\color{blue}{-2 + x}}{x + -1}}{\left(1 + x\right) \cdot x} \]
      *-commutative [=>] 71.1	\[ \frac{x - \left(x + 1\right) \cdot \frac{-2 + x}{x + -1}}{\color{blue}{x \cdot \left(1 + x\right)}} \]
      distribute-rgt-in [=>] 71.1	\[ \frac{x - \left(x + 1\right) \cdot \frac{-2 + x}{x + -1}}{\color{blue}{1 \cdot x + x \cdot x}} \]
      metadata-eval [<=] 71.1	\[ \frac{x - \left(x + 1\right) \cdot \frac{-2 + x}{x + -1}}{\color{blue}{\left(-1 \cdot -1\right)} \cdot x + x \cdot x} \]
      associate-*r* [<=] 71.1	\[ \frac{x - \left(x + 1\right) \cdot \frac{-2 + x}{x + -1}}{\color{blue}{-1 \cdot \left(-1 \cdot x\right)} + x \cdot x} \]
      neg-mul-1 [<=] 71.1	\[ \frac{x - \left(x + 1\right) \cdot \frac{-2 + x}{x + -1}}{-1 \cdot \color{blue}{\left(-x\right)} + x \cdot x} \]
      neg-mul-1 [<=] 71.1	\[ \frac{x - \left(x + 1\right) \cdot \frac{-2 + x}{x + -1}}{\color{blue}{\left(-\left(-x\right)\right)} + x \cdot x} \]
      remove-double-neg [=>] 71.1	\[ \frac{x - \left(x + 1\right) \cdot \frac{-2 + x}{x + -1}}{\color{blue}{x} + x \cdot x} \]

   7. Taylor expanded in x around inf 99.8%

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

   8. Simplified 99.8%

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

      [Start] 99.8	\[ \frac{2 \cdot \frac{1}{{x}^{2}} + 2 \cdot \frac{1}{x}}{x + x \cdot x} \]
      associate-*r/ [=>] 99.8	\[ \frac{\color{blue}{\frac{2 \cdot 1}{{x}^{2}}} + 2 \cdot \frac{1}{x}}{x + x \cdot x} \]
      metadata-eval [=>] 99.8	\[ \frac{\frac{\color{blue}{2}}{{x}^{2}} + 2 \cdot \frac{1}{x}}{x + x \cdot x} \]
      unpow2 [=>] 99.8	\[ \frac{\frac{2}{\color{blue}{x \cdot x}} + 2 \cdot \frac{1}{x}}{x + x \cdot x} \]
      associate-/r* [=>] 99.8	\[ \frac{\color{blue}{\frac{\frac{2}{x}}{x}} + 2 \cdot \frac{1}{x}}{x + x \cdot x} \]
      associate-*r/ [=>] 99.8	\[ \frac{\frac{\frac{2}{x}}{x} + \color{blue}{\frac{2 \cdot 1}{x}}}{x + x \cdot x} \]
      metadata-eval [=>] 99.8	\[ \frac{\frac{\frac{2}{x}}{x} + \frac{\color{blue}{2}}{x}}{x + x \cdot x} \]

   if -1.26e7 < x < 1e8

   1. Initial program 99.0%

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

   2. Simplified 99.0%

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

      [Start] 99.0	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 99.0	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 99.0	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 99.0	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 99.0	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 99.0	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 99.0	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 99.0	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 99.0	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 99.0	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Applied egg-rr 99.0%

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

   4. Simplified 99.0%

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

      [Start] 99.0	\[ \frac{1}{1 + x} - \frac{\frac{-2 + \left(2 \cdot x - x\right)}{x}}{1} \cdot \frac{1}{x + -1} \]
      *-commutative [=>] 99.0	\[ \frac{1}{1 + x} - \color{blue}{\frac{1}{x + -1} \cdot \frac{\frac{-2 + \left(2 \cdot x - x\right)}{x}}{1}} \]
      /-rgt-identity [=>] 99.0	\[ \frac{1}{1 + x} - \frac{1}{x + -1} \cdot \color{blue}{\frac{-2 + \left(2 \cdot x - x\right)}{x}} \]
      +-commutative [=>] 99.0	\[ \frac{1}{1 + x} - \frac{1}{x + -1} \cdot \frac{\color{blue}{\left(2 \cdot x - x\right) + -2}}{x} \]
      associate-+l- [=>] 99.0	\[ \frac{1}{1 + x} - \frac{1}{x + -1} \cdot \frac{\color{blue}{2 \cdot x - \left(x - -2\right)}}{x} \]
      *-commutative [=>] 99.0	\[ \frac{1}{1 + x} - \frac{1}{x + -1} \cdot \frac{\color{blue}{x \cdot 2} - \left(x - -2\right)}{x} \]

   5. Applied egg-rr 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -12600000 \lor \neg \left(x \leq 100000000\right):\\ \;\;\;\;\frac{\frac{2}{x} + \frac{\frac{2}{x}}{x}}{x + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x - x\right) + \left(x + 1\right) \cdot \left(2 - x\right)}{\left(x \cdot x - x\right) \cdot \left(x + 1\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.5%
Cost	3400

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

Alternative 2
Accuracy	99.2%
Cost	3017

\[\begin{array}{l} t_0 := \left(\frac{1}{x + 1} + \frac{-2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-15} \lor \neg \left(t_0 \leq 10^{-23}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x}}{x + x \cdot x}\\ \end{array} \]

Alternative 3
Accuracy	99.2%
Cost	3016

\[\begin{array}{l} t_0 := \frac{1}{x + 1}\\ t_1 := \left(t_0 + \frac{-2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-15}:\\ \;\;\;\;t_0 + \frac{2 - x}{x \cdot \left(x + -1\right)}\\ \mathbf{elif}\;t_1 \leq 10^{-23}:\\ \;\;\;\;\frac{\frac{2}{x}}{x + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	99.5%
Cost	3016

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

Alternative 5
Accuracy	99.2%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;x \leq -250000 \lor \neg \left(x \leq 410000\right):\\ \;\;\;\;\frac{\frac{2}{x}}{x + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot -2}{1 - x \cdot x} + \frac{-2}{x}\\ \end{array} \]

Alternative 6
Accuracy	98.5%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -0.85 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{\frac{2}{x}}{x + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \end{array} \]

Alternative 7
Accuracy	83.7%
Cost	448

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

Alternative 8
Accuracy	50.8%
Cost	192

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

Alternative 9
Accuracy	3.3%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023122 
(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))))