3frac (problem 3.3.3)

Average Accuracy: 84.4% → 98.9%

Time: 14.1s

Precision: binary64

Cost: 8712

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{1 + x}\\ t_1 := \left(t_0 - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{if}\;t_1 \leq -0.2:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{x - -2 \cdot \left(1 - x\right)}{x \cdot x - x}\\ \end{array} \]

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 (+ 1.0 x))) (t_1 (+ (- t_0 (/ 2.0 x)) (/ 1.0 (+ x -1.0)))))
   (if (<= t_1 -0.2)
     t_1
     (if (<= t_1 2e-28)
       (/ 2.0 (pow x 3.0))
       (+ t_0 (/ (- x (* -2.0 (- 1.0 x))) (- (* x x) x)))))))

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 / (1.0 + x);
	double t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if (t_1 <= -0.2) {
		tmp = t_1;
	} else if (t_1 <= 2e-28) {
		tmp = 2.0 / pow(x, 3.0);
	} else {
		tmp = t_0 + ((x - (-2.0 * (1.0 - x))) / ((x * x) - x));
	}
	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 / (1.0d0 + x)
    t_1 = (t_0 - (2.0d0 / x)) + (1.0d0 / (x + (-1.0d0)))
    if (t_1 <= (-0.2d0)) then
        tmp = t_1
    else if (t_1 <= 2d-28) then
        tmp = 2.0d0 / (x ** 3.0d0)
    else
        tmp = t_0 + ((x - ((-2.0d0) * (1.0d0 - x))) / ((x * x) - x))
    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 / (1.0 + x);
	double t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0));
	double tmp;
	if (t_1 <= -0.2) {
		tmp = t_1;
	} else if (t_1 <= 2e-28) {
		tmp = 2.0 / Math.pow(x, 3.0);
	} else {
		tmp = t_0 + ((x - (-2.0 * (1.0 - x))) / ((x * x) - x));
	}
	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 / (1.0 + x)
	t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0))
	tmp = 0
	if t_1 <= -0.2:
		tmp = t_1
	elif t_1 <= 2e-28:
		tmp = 2.0 / math.pow(x, 3.0)
	else:
		tmp = t_0 + ((x - (-2.0 * (1.0 - x))) / ((x * x) - x))
	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(1.0 / Float64(1.0 + x))
	t_1 = Float64(Float64(t_0 - Float64(2.0 / x)) + Float64(1.0 / Float64(x + -1.0)))
	tmp = 0.0
	if (t_1 <= -0.2)
		tmp = t_1;
	elseif (t_1 <= 2e-28)
		tmp = Float64(2.0 / (x ^ 3.0));
	else
		tmp = Float64(t_0 + Float64(Float64(x - Float64(-2.0 * Float64(1.0 - x))) / Float64(Float64(x * x) - x)));
	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 / (1.0 + x);
	t_1 = (t_0 - (2.0 / x)) + (1.0 / (x + -1.0));
	tmp = 0.0;
	if (t_1 <= -0.2)
		tmp = t_1;
	elseif (t_1 <= 2e-28)
		tmp = 2.0 / (x ^ 3.0);
	else
		tmp = t_0 + ((x - (-2.0 * (1.0 - x))) / ((x * x) - x));
	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[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.2], t$95$1, If[LessEqual[t$95$1, 2e-28], N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(N[(x - N[(-2.0 * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	84.4%
Target	99.5%
Herbie	98.9%

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -0.20000000000000001

   1. Initial program 100.0%

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

   if -0.20000000000000001 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 1.99999999999999994e-28

   1. Initial program 69.0%

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

   2. Simplified 69.0%

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

      [Start] 69.0	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 69.0	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 69.0	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 69.0	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 69.0	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 69.0	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 69.0	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 69.0	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 69.0	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 69.0	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Taylor expanded in x around inf 98.7%

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

   if 1.99999999999999994e-28 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 98.4%

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

   2. Simplified 98.4%

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

      [Start] 98.4	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
      associate-+l- [=>] 98.4	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 98.4	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
      neg-mul-1 [=>] 98.4	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      metadata-eval [<=] 98.4	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      cancel-sign-sub-inv [<=] 98.4	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      +-commutative [=>] 98.4	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
      *-lft-identity [=>] 98.4	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
      sub-neg [=>] 98.4	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
      metadata-eval [=>] 98.4	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

   3. Applied egg-rr 98.4%

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

      [Start] 98.4	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
      frac-2neg [=>] 98.4	\[ \frac{1}{1 + x} - \left(\color{blue}{\frac{-2}{-x}} - \frac{1}{x + -1}\right) \]
      frac-2neg [=>] 98.4	\[ \frac{1}{1 + x} - \left(\frac{-2}{-x} - \color{blue}{\frac{-1}{-\left(x + -1\right)}}\right) \]
      metadata-eval [=>] 98.4	\[ \frac{1}{1 + x} - \left(\frac{-2}{-x} - \frac{\color{blue}{-1}}{-\left(x + -1\right)}\right) \]
      frac-sub [=>] 98.4	\[ \frac{1}{1 + x} - \color{blue}{\frac{\left(-2\right) \cdot \left(-\left(x + -1\right)\right) - \left(-x\right) \cdot -1}{\left(-x\right) \cdot \left(-\left(x + -1\right)\right)}} \]
      metadata-eval [=>] 98.4	\[ \frac{1}{1 + x} - \frac{\color{blue}{-2} \cdot \left(-\left(x + -1\right)\right) - \left(-x\right) \cdot -1}{\left(-x\right) \cdot \left(-\left(x + -1\right)\right)} \]
      +-commutative [=>] 98.4	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(-\color{blue}{\left(-1 + x\right)}\right) - \left(-x\right) \cdot -1}{\left(-x\right) \cdot \left(-\left(x + -1\right)\right)} \]
      distribute-neg-in [=>] 98.4	\[ \frac{1}{1 + x} - \frac{-2 \cdot \color{blue}{\left(\left(--1\right) + \left(-x\right)\right)} - \left(-x\right) \cdot -1}{\left(-x\right) \cdot \left(-\left(x + -1\right)\right)} \]
      metadata-eval [=>] 98.4	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(\color{blue}{1} + \left(-x\right)\right) - \left(-x\right) \cdot -1}{\left(-x\right) \cdot \left(-\left(x + -1\right)\right)} \]
      sub-neg [<=] 98.4	\[ \frac{1}{1 + x} - \frac{-2 \cdot \color{blue}{\left(1 - x\right)} - \left(-x\right) \cdot -1}{\left(-x\right) \cdot \left(-\left(x + -1\right)\right)} \]
      +-commutative [=>] 98.4	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - \left(-x\right) \cdot -1}{\left(-x\right) \cdot \left(-\color{blue}{\left(-1 + x\right)}\right)} \]
      distribute-neg-in [=>] 98.4	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - \left(-x\right) \cdot -1}{\left(-x\right) \cdot \color{blue}{\left(\left(--1\right) + \left(-x\right)\right)}} \]
      metadata-eval [=>] 98.4	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - \left(-x\right) \cdot -1}{\left(-x\right) \cdot \left(\color{blue}{1} + \left(-x\right)\right)} \]
      sub-neg [<=] 98.4	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - \left(-x\right) \cdot -1}{\left(-x\right) \cdot \color{blue}{\left(1 - x\right)}} \]

   4. Simplified 98.3%

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

      [Start] 98.4	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - \left(-x\right) \cdot -1}{\left(-x\right) \cdot \left(1 - x\right)} \]
      cancel-sign-sub [=>] 98.4	\[ \frac{1}{1 + x} - \frac{\color{blue}{-2 \cdot \left(1 - x\right) + x \cdot -1}}{\left(-x\right) \cdot \left(1 - x\right)} \]
      *-commutative [<=] 98.4	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) + \color{blue}{-1 \cdot x}}{\left(-x\right) \cdot \left(1 - x\right)} \]
      neg-mul-1 [<=] 98.4	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) + \color{blue}{\left(-x\right)}}{\left(-x\right) \cdot \left(1 - x\right)} \]
      unsub-neg [=>] 98.4	\[ \frac{1}{1 + x} - \frac{\color{blue}{-2 \cdot \left(1 - x\right) - x}}{\left(-x\right) \cdot \left(1 - x\right)} \]
      sub-neg [=>] 98.4	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - x}{\left(-x\right) \cdot \color{blue}{\left(1 + \left(-x\right)\right)}} \]
      +-commutative [=>] 98.4	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - x}{\left(-x\right) \cdot \color{blue}{\left(\left(-x\right) + 1\right)}} \]
      distribute-lft-in [=>] 98.3	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - x}{\color{blue}{\left(-x\right) \cdot \left(-x\right) + \left(-x\right) \cdot 1}} \]
      sqr-neg [=>] 98.3	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - x}{\color{blue}{x \cdot x} + \left(-x\right) \cdot 1} \]
      unpow2 [<=] 98.3	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - x}{\color{blue}{{x}^{2}} + \left(-x\right) \cdot 1} \]
      *-rgt-identity [=>] 98.3	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - x}{{x}^{2} + \color{blue}{\left(-x\right)}} \]
      sub-neg [<=] 98.3	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - x}{\color{blue}{{x}^{2} - x}} \]
      unpow2 [=>] 98.3	\[ \frac{1}{1 + x} - \frac{-2 \cdot \left(1 - x\right) - x}{\color{blue}{x \cdot x} - x} \]

3. Recombined 3 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq -0.2:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 2 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + x} + \frac{x - -2 \cdot \left(1 - x\right)}{x \cdot x - x}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	84.4%
Cost	960

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

Alternative 2
Accuracy	84.4%
Cost	960

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

Alternative 3
Accuracy	76.2%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.52\right):\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x}\\ \end{array} \]

Alternative 4
Accuracy	76.2%
Cost	585

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

Alternative 5
Accuracy	83.1%
Cost	448

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

Alternative 6
Accuracy	52.1%
Cost	192

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

Alternative 7
Accuracy	3.3%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023147 
(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))))