3frac (problem 3.3.3)

Average Accuracy: 85.0% → 99.9%

Time: 19.0s

Precision: binary64

Cost: 704

Math

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

↓

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

FPCore

(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))

↓

(FPCore (x) :precision binary64 (/ (/ -2.0 x) (* (+ x -1.0) (- -1.0 x))))

C

double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

↓

double code(double x) {
	return (-2.0 / x) / ((x + -1.0) * (-1.0 - x));
}

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
    code = ((-2.0d0) / x) / ((x + (-1.0d0)) * ((-1.0d0) - x))
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) {
	return (-2.0 / x) / ((x + -1.0) * (-1.0 - x));
}

Python

def code(x):
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0))

↓

def code(x):
	return (-2.0 / x) / ((x + -1.0) * (-1.0 - x))

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)
	return Float64(Float64(-2.0 / x) / Float64(Float64(x + -1.0) * Float64(-1.0 - x)))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
end

↓

function tmp = code(x)
	tmp = (-2.0 / x) / ((x + -1.0) * (-1.0 - x));
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_] := N[(N[(-2.0 / x), $MachinePrecision] / N[(N[(x + -1.0), $MachinePrecision] * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	85.0%
Target	99.7%
Herbie	99.9%

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

Derivation

1. Initial program 85.0%

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

2. Simplified 85.0%

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

   [Start] 85.0	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
   associate-+l- [=>] 85.0	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   sub-neg [=>] 85.0	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
   neg-mul-1 [=>] 85.0	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   metadata-eval [<=] 85.0	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
   cancel-sign-sub-inv [<=] 85.0	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   +-commutative [=>] 85.0	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
   *-lft-identity [=>] 85.0	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   sub-neg [=>] 85.0	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
   metadata-eval [=>] 85.0	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

3. Applied egg-rr 84.9%

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

   [Start] 85.0	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
   *-un-lft-identity [=>] 85.0	\[ \frac{1}{1 + x} - \color{blue}{1 \cdot \left(\frac{2}{x} - \frac{1}{x + -1}\right)} \]
   *-commutative [=>] 85.0	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x + -1}\right) \cdot 1} \]
   frac-sub [=>] 59.4	\[ \frac{1}{1 + x} - \color{blue}{\frac{2 \cdot \left(x + -1\right) - x \cdot 1}{x \cdot \left(x + -1\right)}} \cdot 1 \]
   associate-/r* [=>] 84.9	\[ \frac{1}{1 + x} - \color{blue}{\frac{\frac{2 \cdot \left(x + -1\right) - x \cdot 1}{x}}{x + -1}} \cdot 1 \]
   associate-/r/ [<=] 84.9	\[ \frac{1}{1 + x} - \color{blue}{\frac{\frac{2 \cdot \left(x + -1\right) - x \cdot 1}{x}}{\frac{x + -1}{1}}} \]
   clear-num [=>] 84.9	\[ \frac{1}{1 + x} - \frac{\frac{2 \cdot \left(x + -1\right) - x \cdot 1}{x}}{\color{blue}{\frac{1}{\frac{1}{x + -1}}}} \]
   associate-/r/ [=>] 84.9	\[ \frac{1}{1 + x} - \color{blue}{\frac{\frac{2 \cdot \left(x + -1\right) - x \cdot 1}{x}}{1} \cdot \frac{1}{x + -1}} \]
   +-commutative [=>] 84.9	\[ \frac{1}{1 + x} - \frac{\frac{2 \cdot \color{blue}{\left(-1 + x\right)} - x \cdot 1}{x}}{1} \cdot \frac{1}{x + -1} \]
   distribute-lft-in [=>] 84.9	\[ \frac{1}{1 + x} - \frac{\frac{\color{blue}{\left(2 \cdot -1 + 2 \cdot x\right)} - x \cdot 1}{x}}{1} \cdot \frac{1}{x + -1} \]
   metadata-eval [=>] 84.9	\[ \frac{1}{1 + x} - \frac{\frac{\left(\color{blue}{-2} + 2 \cdot x\right) - x \cdot 1}{x}}{1} \cdot \frac{1}{x + -1} \]
   metadata-eval [<=] 84.9	\[ \frac{1}{1 + x} - \frac{\frac{\left(\color{blue}{\left(-2\right)} + 2 \cdot x\right) - x \cdot 1}{x}}{1} \cdot \frac{1}{x + -1} \]
   *-rgt-identity [=>] 84.9	\[ \frac{1}{1 + x} - \frac{\frac{\left(\left(-2\right) + 2 \cdot x\right) - \color{blue}{x}}{x}}{1} \cdot \frac{1}{x + -1} \]
   associate--l+ [=>] 84.9	\[ \frac{1}{1 + x} - \frac{\frac{\color{blue}{\left(-2\right) + \left(2 \cdot x - x\right)}}{x}}{1} \cdot \frac{1}{x + -1} \]
   metadata-eval [=>] 84.9	\[ \frac{1}{1 + x} - \frac{\frac{\color{blue}{-2} + \left(2 \cdot x - x\right)}{x}}{1} \cdot \frac{1}{x + -1} \]

4. Simplified 84.9%

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

   [Start] 84.9	\[ \frac{1}{1 + x} - \frac{\frac{-2 + \left(2 \cdot x - x\right)}{x}}{1} \cdot \frac{1}{x + -1} \]
   *-commutative [=>] 84.9	\[ \frac{1}{1 + x} - \color{blue}{\frac{1}{x + -1} \cdot \frac{\frac{-2 + \left(2 \cdot x - x\right)}{x}}{1}} \]
   /-rgt-identity [=>] 84.9	\[ \frac{1}{1 + x} - \frac{1}{x + -1} \cdot \color{blue}{\frac{-2 + \left(2 \cdot x - x\right)}{x}} \]
   *-commutative [=>] 84.9	\[ \frac{1}{1 + x} - \frac{1}{x + -1} \cdot \frac{-2 + \left(\color{blue}{x \cdot 2} - x\right)}{x} \]

5. Applied egg-rr 85.0%

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

   [Start] 84.9	\[ \frac{1}{1 + x} - \frac{1}{x + -1} \cdot \frac{-2 + \left(x \cdot 2 - x\right)}{x} \]
   frac-2neg [=>] 84.9	\[ \color{blue}{\frac{-1}{-\left(1 + x\right)}} - \frac{1}{x + -1} \cdot \frac{-2 + \left(x \cdot 2 - x\right)}{x} \]
   metadata-eval [=>] 84.9	\[ \frac{\color{blue}{-1}}{-\left(1 + x\right)} - \frac{1}{x + -1} \cdot \frac{-2 + \left(x \cdot 2 - x\right)}{x} \]
   associate-*l/ [=>] 84.9	\[ \frac{-1}{-\left(1 + x\right)} - \color{blue}{\frac{1 \cdot \frac{-2 + \left(x \cdot 2 - x\right)}{x}}{x + -1}} \]
   *-un-lft-identity [<=] 84.9	\[ \frac{-1}{-\left(1 + x\right)} - \frac{\color{blue}{\frac{-2 + \left(x \cdot 2 - x\right)}{x}}}{x + -1} \]
   frac-sub [=>] 84.9	\[ \color{blue}{\frac{-1 \cdot \left(x + -1\right) - \left(-\left(1 + x\right)\right) \cdot \frac{-2 + \left(x \cdot 2 - x\right)}{x}}{\left(-\left(1 + x\right)\right) \cdot \left(x + -1\right)}} \]

6. Simplified 77.1%

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

   [Start] 85.0	\[ \frac{\left(1 - x\right) - \left(-1 - x\right) \cdot \frac{x + -2}{x}}{\left(-1 - x\right) \cdot \left(x + -1\right)} \]
   associate--l- [=>] 77.1	\[ \frac{\color{blue}{1 - \left(x + \left(-1 - x\right) \cdot \frac{x + -2}{x}\right)}}{\left(-1 - x\right) \cdot \left(x + -1\right)} \]
   *-commutative [=>] 77.1	\[ \frac{1 - \left(x + \color{blue}{\frac{x + -2}{x} \cdot \left(-1 - x\right)}\right)}{\left(-1 - x\right) \cdot \left(x + -1\right)} \]
   +-commutative [=>] 77.1	\[ \frac{1 - \left(x + \frac{\color{blue}{-2 + x}}{x} \cdot \left(-1 - x\right)\right)}{\left(-1 - x\right) \cdot \left(x + -1\right)} \]
   *-commutative [=>] 77.1	\[ \frac{1 - \left(x + \frac{-2 + x}{x} \cdot \left(-1 - x\right)\right)}{\color{blue}{\left(x + -1\right) \cdot \left(-1 - x\right)}} \]

7. Taylor expanded in x around 0 99.9%

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

8. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	83.9%
Cost	969

\[\begin{array}{l} \mathbf{if}\;x \leq -0.54 \lor \neg \left(x \leq 2.65 \cdot 10^{+51}\right):\\ \;\;\;\;\frac{1}{x + 1} + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{1}{x + -1} + \frac{-2}{x}\right)\\ \end{array} \]

Alternative 2
Accuracy	84.0%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -0.65 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{1}{x + 1} + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot x + \frac{-2}{x}\\ \end{array} \]

Alternative 3
Accuracy	51.6%
Cost	192

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

Alternative 4
Accuracy	3.3%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023138 
(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))))