3frac (problem 3.3.3)

Average Error: 9.8 → 0.2

Time: 11.4s

Precision: binary64

Cost: 8713

Math

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

↓

\[\begin{array}{l} t_0 := \left(\frac{1}{x + 1} + \frac{-2}{x}\right) + \frac{1}{x + -1}\\ t_1 := x \cdot \left(1 - x\right)\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-19} \lor \neg \left(t_0 \leq 5 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{t_1 + \left(x + 1\right) \cdot \left(x + -2\right)}{\left(x + 1\right) \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {x}^{-3}\\ \end{array} \]

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))))
        (t_1 (* x (- 1.0 x))))
   (if (or (<= t_0 -2e-19) (not (<= t_0 5e-30)))
     (/ (+ t_1 (* (+ x 1.0) (+ x -2.0))) (* (+ x 1.0) t_1))
     (* 2.0 (pow x -3.0)))))

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 t_1 = x * (1.0 - x);
	double tmp;
	if ((t_0 <= -2e-19) || !(t_0 <= 5e-30)) {
		tmp = (t_1 + ((x + 1.0) * (x + -2.0))) / ((x + 1.0) * t_1);
	} else {
		tmp = 2.0 * pow(x, -3.0);
	}
	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) :: tmp
    t_0 = ((1.0d0 / (x + 1.0d0)) + ((-2.0d0) / x)) + (1.0d0 / (x + (-1.0d0)))
    t_1 = x * (1.0d0 - x)
    if ((t_0 <= (-2d-19)) .or. (.not. (t_0 <= 5d-30))) then
        tmp = (t_1 + ((x + 1.0d0) * (x + (-2.0d0)))) / ((x + 1.0d0) * t_1)
    else
        tmp = 2.0d0 * (x ** (-3.0d0))
    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 t_1 = x * (1.0 - x);
	double tmp;
	if ((t_0 <= -2e-19) || !(t_0 <= 5e-30)) {
		tmp = (t_1 + ((x + 1.0) * (x + -2.0))) / ((x + 1.0) * t_1);
	} else {
		tmp = 2.0 * Math.pow(x, -3.0);
	}
	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))
	t_1 = x * (1.0 - x)
	tmp = 0
	if (t_0 <= -2e-19) or not (t_0 <= 5e-30):
		tmp = (t_1 + ((x + 1.0) * (x + -2.0))) / ((x + 1.0) * t_1)
	else:
		tmp = 2.0 * math.pow(x, -3.0)
	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)))
	t_1 = Float64(x * Float64(1.0 - x))
	tmp = 0.0
	if ((t_0 <= -2e-19) || !(t_0 <= 5e-30))
		tmp = Float64(Float64(t_1 + Float64(Float64(x + 1.0) * Float64(x + -2.0))) / Float64(Float64(x + 1.0) * t_1));
	else
		tmp = Float64(2.0 * (x ^ -3.0));
	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));
	t_1 = x * (1.0 - x);
	tmp = 0.0;
	if ((t_0 <= -2e-19) || ~((t_0 <= 5e-30)))
		tmp = (t_1 + ((x + 1.0) * (x + -2.0))) / ((x + 1.0) * t_1);
	else
		tmp = 2.0 * (x ^ -3.0);
	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]}, Block[{t$95$1 = N[(x * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e-19], N[Not[LessEqual[t$95$0, 5e-30]], $MachinePrecision]], N[(N[(t$95$1 + N[(N[(x + 1.0), $MachinePrecision] * N[(x + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision]]]]

Target

Original	9.8
Target	0.3
Herbie	0.2

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -2e-19 or 4.99999999999999972e-30 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 0.8

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

   2. Simplified 0.8

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

      [Start] 0.8	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 0.8	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 0.8	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 0.8	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 0.8	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 0.8	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 0.8	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 0.8	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 0.8	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 0.8	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Applied egg-rr 0.8

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

   4. Applied egg-rr 0.8

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

   5. Simplified 0.8

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

      [Start] 0.8	\[ \frac{1}{x + 1} + \left(-\frac{\frac{-2 + x}{x}}{x + -1}\right) \]
      sub-neg [<=] 0.8	\[ \color{blue}{\frac{1}{x + 1} - \frac{\frac{-2 + x}{x}}{x + -1}} \]
      associate-/l/ [=>] 0.8	\[ \frac{1}{x + 1} - \color{blue}{\frac{-2 + x}{\left(x + -1\right) \cdot x}} \]
      +-commutative [=>] 0.8	\[ \frac{1}{x + 1} - \frac{\color{blue}{x + -2}}{\left(x + -1\right) \cdot x} \]

   6. Applied egg-rr 0.3

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

   7. Simplified 0.3

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

      [Start] 0.3	\[ \frac{\left(x + -1\right) \cdot \left(-x\right) - \left(x + 1\right) \cdot \left(2 - x\right)}{\left(x + 1\right) \cdot \left(\left(x + -1\right) \cdot \left(-x\right)\right)} \]
      *-commutative [=>] 0.3	\[ \frac{\color{blue}{\left(-x\right) \cdot \left(x + -1\right)} - \left(x + 1\right) \cdot \left(2 - x\right)}{\left(x + 1\right) \cdot \left(\left(x + -1\right) \cdot \left(-x\right)\right)} \]
      mul-1-neg [<=] 0.3	\[ \frac{\color{blue}{\left(-1 \cdot x\right)} \cdot \left(x + -1\right) - \left(x + 1\right) \cdot \left(2 - x\right)}{\left(x + 1\right) \cdot \left(\left(x + -1\right) \cdot \left(-x\right)\right)} \]
      *-commutative [<=] 0.3	\[ \frac{\color{blue}{\left(x \cdot -1\right)} \cdot \left(x + -1\right) - \left(x + 1\right) \cdot \left(2 - x\right)}{\left(x + 1\right) \cdot \left(\left(x + -1\right) \cdot \left(-x\right)\right)} \]
      associate-*l* [=>] 0.3	\[ \frac{\color{blue}{x \cdot \left(-1 \cdot \left(x + -1\right)\right)} - \left(x + 1\right) \cdot \left(2 - x\right)}{\left(x + 1\right) \cdot \left(\left(x + -1\right) \cdot \left(-x\right)\right)} \]
      +-commutative [=>] 0.3	\[ \frac{x \cdot \left(-1 \cdot \color{blue}{\left(-1 + x\right)}\right) - \left(x + 1\right) \cdot \left(2 - x\right)}{\left(x + 1\right) \cdot \left(\left(x + -1\right) \cdot \left(-x\right)\right)} \]
      distribute-lft-in [=>] 0.3	\[ \frac{x \cdot \color{blue}{\left(-1 \cdot -1 + -1 \cdot x\right)} - \left(x + 1\right) \cdot \left(2 - x\right)}{\left(x + 1\right) \cdot \left(\left(x + -1\right) \cdot \left(-x\right)\right)} \]
      metadata-eval [=>] 0.3	\[ \frac{x \cdot \left(\color{blue}{1} + -1 \cdot x\right) - \left(x + 1\right) \cdot \left(2 - x\right)}{\left(x + 1\right) \cdot \left(\left(x + -1\right) \cdot \left(-x\right)\right)} \]
      mul-1-neg [=>] 0.3	\[ \frac{x \cdot \left(1 + \color{blue}{\left(-x\right)}\right) - \left(x + 1\right) \cdot \left(2 - x\right)}{\left(x + 1\right) \cdot \left(\left(x + -1\right) \cdot \left(-x\right)\right)} \]
      sub-neg [<=] 0.3	\[ \frac{x \cdot \color{blue}{\left(1 - x\right)} - \left(x + 1\right) \cdot \left(2 - x\right)}{\left(x + 1\right) \cdot \left(\left(x + -1\right) \cdot \left(-x\right)\right)} \]
      *-commutative [=>] 0.3	\[ \frac{x \cdot \left(1 - x\right) - \left(x + 1\right) \cdot \left(2 - x\right)}{\left(x + 1\right) \cdot \color{blue}{\left(\left(-x\right) \cdot \left(x + -1\right)\right)}} \]
      mul-1-neg [<=] 0.3	\[ \frac{x \cdot \left(1 - x\right) - \left(x + 1\right) \cdot \left(2 - x\right)}{\left(x + 1\right) \cdot \left(\color{blue}{\left(-1 \cdot x\right)} \cdot \left(x + -1\right)\right)} \]
      *-commutative [<=] 0.3	\[ \frac{x \cdot \left(1 - x\right) - \left(x + 1\right) \cdot \left(2 - x\right)}{\left(x + 1\right) \cdot \left(\color{blue}{\left(x \cdot -1\right)} \cdot \left(x + -1\right)\right)} \]
      associate-*l* [=>] 0.3	\[ \frac{x \cdot \left(1 - x\right) - \left(x + 1\right) \cdot \left(2 - x\right)}{\left(x + 1\right) \cdot \color{blue}{\left(x \cdot \left(-1 \cdot \left(x + -1\right)\right)\right)}} \]
      +-commutative [=>] 0.3	\[ \frac{x \cdot \left(1 - x\right) - \left(x + 1\right) \cdot \left(2 - x\right)}{\left(x + 1\right) \cdot \left(x \cdot \left(-1 \cdot \color{blue}{\left(-1 + x\right)}\right)\right)} \]
      distribute-lft-in [=>] 0.3	\[ \frac{x \cdot \left(1 - x\right) - \left(x + 1\right) \cdot \left(2 - x\right)}{\left(x + 1\right) \cdot \left(x \cdot \color{blue}{\left(-1 \cdot -1 + -1 \cdot x\right)}\right)} \]
      metadata-eval [=>] 0.3	\[ \frac{x \cdot \left(1 - x\right) - \left(x + 1\right) \cdot \left(2 - x\right)}{\left(x + 1\right) \cdot \left(x \cdot \left(\color{blue}{1} + -1 \cdot x\right)\right)} \]
      mul-1-neg [=>] 0.3	\[ \frac{x \cdot \left(1 - x\right) - \left(x + 1\right) \cdot \left(2 - x\right)}{\left(x + 1\right) \cdot \left(x \cdot \left(1 + \color{blue}{\left(-x\right)}\right)\right)} \]
      sub-neg [<=] 0.3	\[ \frac{x \cdot \left(1 - x\right) - \left(x + 1\right) \cdot \left(2 - x\right)}{\left(x + 1\right) \cdot \left(x \cdot \color{blue}{\left(1 - x\right)}\right)} \]

   if -2e-19 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 4.99999999999999972e-30

   1. Initial program 19.1

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

   2. Simplified 19.1

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

      [Start] 19.1	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 19.1	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 19.1	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 19.1	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 19.1	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 19.1	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 19.1	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 19.1	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 19.1	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 19.1	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Taylor expanded in x around inf 0.5

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

   4. Applied egg-rr 0.0

      \[\leadsto \color{blue}{{x}^{-3} \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{x + 1} + \frac{-2}{x}\right) + \frac{1}{x + -1} \leq -2 \cdot 10^{-19} \lor \neg \left(\left(\frac{1}{x + 1} + \frac{-2}{x}\right) + \frac{1}{x + -1} \leq 5 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{x \cdot \left(1 - x\right) + \left(x + 1\right) \cdot \left(x + -2\right)}{\left(x + 1\right) \cdot \left(x \cdot \left(1 - x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {x}^{-3}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	6848

\[\frac{\frac{2}{x}}{\mathsf{fma}\left(x, x, -1\right)} \]

Alternative 2
Error	0.3
Cost	6784

\[\frac{2}{{x}^{3} - x} \]

Alternative 3
Error	0.5
Cost	1869

\[\begin{array}{l} t_0 := x \cdot \left(1 - x\right)\\ \mathbf{if}\;x \leq -4 \cdot 10^{+155}:\\ \;\;\;\;1 + \left(-1 + \frac{-2}{x}\right)\\ \mathbf{elif}\;x \leq -135000000 \lor \neg \left(x \leq 135000000\right):\\ \;\;\;\;\frac{2}{\left(x + 1\right) \cdot \left(x \cdot x\right) + x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + \left(x + 1\right) \cdot \left(x + -2\right)}{\left(x + 1\right) \cdot t_0}\\ \end{array} \]

Alternative 4
Error	0.8
Cost	1357

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+155}:\\ \;\;\;\;1 + \left(-1 + \frac{-2}{x}\right)\\ \mathbf{elif}\;x \leq -340000 \lor \neg \left(x \leq 350000\right):\\ \;\;\;\;\frac{2}{\left(x + 1\right) \cdot \left(x \cdot x\right) + x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot -2}{1 - x \cdot x} + \frac{-2}{x}\\ \end{array} \]

Alternative 5
Error	9.8
Cost	1096

\[\begin{array}{l} \mathbf{if}\;x \leq -118000000:\\ \;\;\;\;\frac{1}{x + 1} + \frac{-1}{x}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+34}:\\ \;\;\;\;\frac{x \cdot -2}{1 - x \cdot x} + \frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-1 + \frac{-2}{x}\right)\\ \end{array} \]

Alternative 6
Error	9.8
Cost	960

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

Alternative 7
Error	10.5
Cost	448

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

Alternative 8
Error	30.9
Cost	192

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

Alternative 9
Error	61.9
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023045 
(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))))