3frac (problem 3.3.3)

Average Error: 85.8% → 99.1%

Time: 12.8s

Precision: binary64

Cost: 8712.00

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot \left(x + -1\right)\\ t_1 := \frac{1}{x + -1} + \left(\frac{1}{1 + x} + \frac{-2}{x}\right)\\ t_2 := x \cdot \left(1 + x\right)\\ \mathbf{if}\;t_1 \leq -0.004:\\ \;\;\;\;\frac{\frac{t_0 + \left(1 + x\right) \cdot \left(2 - x\right)}{1 + x}}{t_0}\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2 + \left(x + -1\right) \cdot \left(x + \left(1 + x\right) \cdot -2\right)}{\left(x + -1\right) \cdot t_2}\\ \end{array} \]

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 -1.0)))
        (t_1 (+ (/ 1.0 (+ x -1.0)) (+ (/ 1.0 (+ 1.0 x)) (/ -2.0 x))))
        (t_2 (* x (+ 1.0 x))))
   (if (<= t_1 -0.004)
     (/ (/ (+ t_0 (* (+ 1.0 x) (- 2.0 x))) (+ 1.0 x)) t_0)
     (if (<= t_1 0.0)
       (/ 2.0 (pow x 3.0))
       (/
        (+ t_2 (* (+ x -1.0) (+ x (* (+ 1.0 x) -2.0))))
        (* (+ x -1.0) t_2))))))

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 + -1.0);
	double t_1 = (1.0 / (x + -1.0)) + ((1.0 / (1.0 + x)) + (-2.0 / x));
	double t_2 = x * (1.0 + x);
	double tmp;
	if (t_1 <= -0.004) {
		tmp = ((t_0 + ((1.0 + x) * (2.0 - x))) / (1.0 + x)) / t_0;
	} else if (t_1 <= 0.0) {
		tmp = 2.0 / pow(x, 3.0);
	} else {
		tmp = (t_2 + ((x + -1.0) * (x + ((1.0 + x) * -2.0)))) / ((x + -1.0) * t_2);
	}
	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) :: tmp
    t_0 = x * (x + (-1.0d0))
    t_1 = (1.0d0 / (x + (-1.0d0))) + ((1.0d0 / (1.0d0 + x)) + ((-2.0d0) / x))
    t_2 = x * (1.0d0 + x)
    if (t_1 <= (-0.004d0)) then
        tmp = ((t_0 + ((1.0d0 + x) * (2.0d0 - x))) / (1.0d0 + x)) / t_0
    else if (t_1 <= 0.0d0) then
        tmp = 2.0d0 / (x ** 3.0d0)
    else
        tmp = (t_2 + ((x + (-1.0d0)) * (x + ((1.0d0 + x) * (-2.0d0))))) / ((x + (-1.0d0)) * t_2)
    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 + -1.0);
	double t_1 = (1.0 / (x + -1.0)) + ((1.0 / (1.0 + x)) + (-2.0 / x));
	double t_2 = x * (1.0 + x);
	double tmp;
	if (t_1 <= -0.004) {
		tmp = ((t_0 + ((1.0 + x) * (2.0 - x))) / (1.0 + x)) / t_0;
	} else if (t_1 <= 0.0) {
		tmp = 2.0 / Math.pow(x, 3.0);
	} else {
		tmp = (t_2 + ((x + -1.0) * (x + ((1.0 + x) * -2.0)))) / ((x + -1.0) * t_2);
	}
	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 + -1.0)
	t_1 = (1.0 / (x + -1.0)) + ((1.0 / (1.0 + x)) + (-2.0 / x))
	t_2 = x * (1.0 + x)
	tmp = 0
	if t_1 <= -0.004:
		tmp = ((t_0 + ((1.0 + x) * (2.0 - x))) / (1.0 + x)) / t_0
	elif t_1 <= 0.0:
		tmp = 2.0 / math.pow(x, 3.0)
	else:
		tmp = (t_2 + ((x + -1.0) * (x + ((1.0 + x) * -2.0)))) / ((x + -1.0) * t_2)
	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(x * Float64(x + -1.0))
	t_1 = Float64(Float64(1.0 / Float64(x + -1.0)) + Float64(Float64(1.0 / Float64(1.0 + x)) + Float64(-2.0 / x)))
	t_2 = Float64(x * Float64(1.0 + x))
	tmp = 0.0
	if (t_1 <= -0.004)
		tmp = Float64(Float64(Float64(t_0 + Float64(Float64(1.0 + x) * Float64(2.0 - x))) / Float64(1.0 + x)) / t_0);
	elseif (t_1 <= 0.0)
		tmp = Float64(2.0 / (x ^ 3.0));
	else
		tmp = Float64(Float64(t_2 + Float64(Float64(x + -1.0) * Float64(x + Float64(Float64(1.0 + x) * -2.0)))) / Float64(Float64(x + -1.0) * t_2));
	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 + -1.0);
	t_1 = (1.0 / (x + -1.0)) + ((1.0 / (1.0 + x)) + (-2.0 / x));
	t_2 = x * (1.0 + x);
	tmp = 0.0;
	if (t_1 <= -0.004)
		tmp = ((t_0 + ((1.0 + x) * (2.0 - x))) / (1.0 + x)) / t_0;
	elseif (t_1 <= 0.0)
		tmp = 2.0 / (x ^ 3.0);
	else
		tmp = (t_2 + ((x + -1.0) * (x + ((1.0 + x) * -2.0)))) / ((x + -1.0) * t_2);
	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[(x * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.004], N[(N[(N[(t$95$0 + N[(N[(1.0 + x), $MachinePrecision] * N[(2.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 + N[(N[(x + -1.0), $MachinePrecision] * N[(x + N[(N[(1.0 + x), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x + -1.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	85.8%
Target	99.6%
Herbie	99.1%

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -0.0040000000000000001

   1. Initial program 100.0

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

   2. Simplified 100.0

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

      [Start] 100.0	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 100.0	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 100.0	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 100.0	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 100.0	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 100.0	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 100.0	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 100.0	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 100.0	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 100.0	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Applied egg-rr 100.0

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

   4. Simplified 100.0

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

      [Start] 100.0	\[ \frac{1}{1 + x} - \frac{\frac{-2 + \left(2 \cdot x - x\right)}{x}}{x + -1} \]
      associate-/l/ [=>] 100.0	\[ \frac{1}{1 + x} - \color{blue}{\frac{-2 + \left(2 \cdot x - x\right)}{\left(x + -1\right) \cdot x}} \]
      +-commutative [=>] 100.0	\[ \frac{1}{1 + x} - \frac{\color{blue}{\left(2 \cdot x - x\right) + -2}}{\left(x + -1\right) \cdot x} \]
      associate-+l- [=>] 100.0	\[ \frac{1}{1 + x} - \frac{\color{blue}{2 \cdot x - \left(x - -2\right)}}{\left(x + -1\right) \cdot x} \]
      *-commutative [=>] 100.0	\[ \frac{1}{1 + x} - \frac{\color{blue}{x \cdot 2} - \left(x - -2\right)}{\left(x + -1\right) \cdot x} \]
      *-commutative [<=] 100.0	\[ \frac{1}{1 + x} - \frac{x \cdot 2 - \left(x - -2\right)}{\color{blue}{x \cdot \left(x + -1\right)}} \]

   5. Applied egg-rr 99.9

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

   if -0.0040000000000000001 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 0.0

   1. Initial program 71.6

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

   2. Taylor expanded in x around inf 98.8

      \[\leadsto \color{blue}{\frac{2}{{x}^{3}}} \]

   if 0.0 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 97.8

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

   2. Applied egg-rr 98.7

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

4. Final simplification 99.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{x + -1} + \left(\frac{1}{1 + x} + \frac{-2}{x}\right) \leq -0.004:\\ \;\;\;\;\frac{\frac{x \cdot \left(x + -1\right) + \left(1 + x\right) \cdot \left(2 - x\right)}{1 + x}}{x \cdot \left(x + -1\right)}\\ \mathbf{elif}\;\frac{1}{x + -1} + \left(\frac{1}{1 + x} + \frac{-2}{x}\right) \leq 0:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 + x\right) + \left(x + -1\right) \cdot \left(x + \left(1 + x\right) \cdot -2\right)}{\left(x + -1\right) \cdot \left(x \cdot \left(1 + x\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	87.5%
Cost	3784.00

\[\begin{array}{l} t_0 := x \cdot \left(x + -1\right)\\ t_1 := \frac{1}{x + -1} + \left(\frac{1}{1 + x} + \frac{-2}{x}\right)\\ t_2 := x \cdot \left(1 + x\right)\\ \mathbf{if}\;t_1 \leq -0.004:\\ \;\;\;\;\frac{\frac{t_0 + \left(1 + x\right) \cdot \left(2 - x\right)}{1 + x}}{t_0}\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{1 + x \cdot \left(x + \left(-1 - x\right)\right)}{\left(1 + x\right) \cdot \left(1 + x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2 + \left(x + -1\right) \cdot \left(x + \left(1 + x\right) \cdot -2\right)}{\left(x + -1\right) \cdot t_2}\\ \end{array} \]

Alternative 2
Error	87.5%
Cost	3529.00

\[\begin{array}{l} t_0 := \frac{1}{x + -1} + \left(\frac{1}{1 + x} + \frac{-2}{x}\right)\\ t_1 := x \cdot \left(x + -1\right)\\ \mathbf{if}\;t_0 \leq -0.004 \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;\frac{\frac{t_1 + \left(1 + x\right) \cdot \left(2 - x\right)}{1 + x}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x \cdot \left(x + \left(-1 - x\right)\right)}{\left(1 + x\right) \cdot \left(1 + x \cdot x\right)}\\ \end{array} \]

Alternative 3
Error	87.2%
Cost	3272.00

\[\begin{array}{l} t_0 := \frac{1}{x + -1}\\ t_1 := \frac{1}{1 + x}\\ t_2 := t_0 + \left(t_1 + \frac{-2}{x}\right)\\ \mathbf{if}\;t_2 \leq -0.004:\\ \;\;\;\;t_1 - \left(x + -2\right) \cdot \left(\frac{1}{x} \cdot t_0\right)\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\frac{1 + x \cdot \left(x + \left(-1 - x\right)\right)}{\left(1 + x\right) \cdot \left(1 + x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{t_2}}\\ \end{array} \]

Alternative 4
Error	85.8%
Cost	960.00

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

Alternative 5
Error	85.7%
Cost	960.00

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

Alternative 6
Error	85.8%
Cost	960.00

\[\frac{1}{1 + x} + \frac{-1 - \frac{-2}{x}}{x + -1} \]

Alternative 7
Error	76.8%
Cost	585.00

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

Alternative 8
Error	84.4%
Cost	448.00

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

Alternative 9
Error	52.8%
Cost	192.00

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

Alternative 10
Error	3.3%
Cost	64.00

\[-1 \]

Reproduce

herbie shell --seed 2023097 
(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))))