3frac (problem 3.3.3)

Average Accuracy: 85.0% → 99.9%

Time: 13.6s

Precision: binary64

Cost: 832

Math

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

↓

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

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (* (/ -2.0 (- -1.0 x)) (/ 1.0 (* x (+ -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 / (-1.0 - x)) * (1.0 / (x * (-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) / ((-1.0d0) - x)) * (1.0d0 / (x * ((-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 / (-1.0 - x)) * (1.0 / (x * (-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 / (-1.0 - x)) * (1.0 / (x * (-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 / Float64(-1.0 - x)) * Float64(1.0 / Float64(x * 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 / (-1.0 - x)) * (1.0 / (x * (-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 / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x * N[(-1.0 + x), $MachinePrecision]), $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 85.0%

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

   [Start] 85.0	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
   frac-sub [=>] 58.9	\[ \frac{1}{1 + x} - \color{blue}{\frac{2 \cdot \left(x + -1\right) - x \cdot 1}{x \cdot \left(x + -1\right)}} \]
   associate-/r* [=>] 85.0	\[ \frac{1}{1 + x} - \color{blue}{\frac{\frac{2 \cdot \left(x + -1\right) - x \cdot 1}{x}}{x + -1}} \]
   +-commutative [=>] 85.0	\[ \frac{1}{1 + x} - \frac{\frac{2 \cdot \color{blue}{\left(-1 + x\right)} - x \cdot 1}{x}}{x + -1} \]
   distribute-lft-in [=>] 85.0	\[ \frac{1}{1 + x} - \frac{\frac{\color{blue}{\left(2 \cdot -1 + 2 \cdot x\right)} - x \cdot 1}{x}}{x + -1} \]
   metadata-eval [=>] 85.0	\[ \frac{1}{1 + x} - \frac{\frac{\left(\color{blue}{-2} + 2 \cdot x\right) - x \cdot 1}{x}}{x + -1} \]
   metadata-eval [<=] 85.0	\[ \frac{1}{1 + x} - \frac{\frac{\left(\color{blue}{\left(-2\right)} + 2 \cdot x\right) - x \cdot 1}{x}}{x + -1} \]
   *-rgt-identity [=>] 85.0	\[ \frac{1}{1 + x} - \frac{\frac{\left(\left(-2\right) + 2 \cdot x\right) - \color{blue}{x}}{x}}{x + -1} \]
   associate--l+ [=>] 85.0	\[ \frac{1}{1 + x} - \frac{\frac{\color{blue}{\left(-2\right) + \left(2 \cdot x - x\right)}}{x}}{x + -1} \]
   metadata-eval [=>] 85.0	\[ \frac{1}{1 + x} - \frac{\frac{\color{blue}{-2} + \left(2 \cdot x - x\right)}{x}}{x + -1} \]

4. Simplified 58.9%

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

   [Start] 85.0	\[ \frac{1}{1 + x} - \frac{\frac{-2 + \left(2 \cdot x - x\right)}{x}}{x + -1} \]
   associate-/l/ [=>] 58.9	\[ \frac{1}{1 + x} - \color{blue}{\frac{-2 + \left(2 \cdot x - x\right)}{\left(x + -1\right) \cdot x}} \]
   +-commutative [=>] 58.9	\[ \frac{1}{1 + x} - \frac{\color{blue}{\left(2 \cdot x - x\right) + -2}}{\left(x + -1\right) \cdot x} \]
   associate-+l- [=>] 58.9	\[ \frac{1}{1 + x} - \frac{\color{blue}{2 \cdot x - \left(x - -2\right)}}{\left(x + -1\right) \cdot x} \]
   *-commutative [=>] 58.9	\[ \frac{1}{1 + x} - \frac{\color{blue}{x \cdot 2} - \left(x - -2\right)}{\left(x + -1\right) \cdot x} \]
   *-commutative [=>] 58.9	\[ \frac{1}{1 + x} - \frac{x \cdot 2 - \left(x - -2\right)}{\color{blue}{x \cdot \left(x + -1\right)}} \]

5. Applied egg-rr 85.0%

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

   [Start] 58.9	\[ \frac{1}{1 + x} - \frac{x \cdot 2 - \left(x - -2\right)}{x \cdot \left(x + -1\right)} \]
   frac-2neg [=>] 58.9	\[ \color{blue}{\frac{-1}{-\left(1 + x\right)}} - \frac{x \cdot 2 - \left(x - -2\right)}{x \cdot \left(x + -1\right)} \]
   metadata-eval [=>] 58.9	\[ \frac{\color{blue}{-1}}{-\left(1 + x\right)} - \frac{x \cdot 2 - \left(x - -2\right)}{x \cdot \left(x + -1\right)} \]
   associate-/r* [=>] 85.0	\[ \frac{-1}{-\left(1 + x\right)} - \color{blue}{\frac{\frac{x \cdot 2 - \left(x - -2\right)}{x}}{x + -1}} \]
   frac-sub [=>] 85.0	\[ \color{blue}{\frac{-1 \cdot \left(x + -1\right) - \left(-\left(1 + x\right)\right) \cdot \frac{x \cdot 2 - \left(x - -2\right)}{x}}{\left(-\left(1 + x\right)\right) \cdot \left(x + -1\right)}} \]

6. Simplified 85.0%

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

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

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. Applied egg-rr 99.9%

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

   [Start] 99.9	\[ \frac{\frac{-2}{x}}{\left(x + -1\right) \cdot \left(-1 - x\right)} \]
   div-inv [=>] 99.9	\[ \frac{\color{blue}{-2 \cdot \frac{1}{x}}}{\left(x + -1\right) \cdot \left(-1 - x\right)} \]
   *-commutative [=>] 99.9	\[ \frac{-2 \cdot \frac{1}{x}}{\color{blue}{\left(-1 - x\right) \cdot \left(x + -1\right)}} \]
   times-frac [=>] 99.8	\[ \color{blue}{\frac{-2}{-1 - x} \cdot \frac{\frac{1}{x}}{x + -1}} \]
   associate-/l/ [=>] 99.9	\[ \frac{-2}{-1 - x} \cdot \color{blue}{\frac{1}{\left(x + -1\right) \cdot x}} \]

9. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.0%
Cost	777

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

Alternative 2
Accuracy	99.9%
Cost	704

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

Alternative 3
Accuracy	76.6%
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 4
Accuracy	77.0%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot x}\\ \end{array} \]

Alternative 5
Accuracy	99.7%
Cost	576

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

Alternative 6
Accuracy	83.6%
Cost	448

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

Alternative 7
Accuracy	51.4%
Cost	192

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

Reproduce

herbie shell --seed 2023126 
(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))))