3frac (problem 3.3.3)

Average Accuracy: 84.2% → 99.9%

Time: 12.9s

Precision: binary64

Cost: 704

Math

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

↓

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

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)) (+ x -1.0)))

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)) / (x + -1.0);
}

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)) / (x + (-1.0d0))
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)) / (x + -1.0);
}

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)) / (x + -1.0)

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(Float64(2.0 / x) / Float64(x + 1.0)) / Float64(x + -1.0))
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)) / (x + -1.0);
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[(N[(2.0 / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]

Target

Original	84.2%
Target	99.6%
Herbie	99.9%

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

Derivation

1. Initial program 84.2%

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

2. Simplified 84.2%

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

   [Start] 84.2	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
   associate-+l- [=>] 84.2	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   sub-neg [=>] 84.2	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
   neg-mul-1 [=>] 84.2	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   metadata-eval [<=] 84.2	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
   cancel-sign-sub-inv [<=] 84.2	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   +-commutative [=>] 84.2	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
   *-lft-identity [=>] 84.2	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   sub-neg [=>] 84.2	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
   metadata-eval [=>] 84.2	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

3. Applied egg-rr 59.0%

   \[\leadsto \frac{1}{1 + x} - \color{blue}{\frac{\left(x + -1\right) - \left(x \cdot 0.5\right) \cdot 1}{\left(x \cdot 0.5\right) \cdot \left(x + -1\right)}} \]
   Proof

   [Start] 84.2	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
   clear-num [=>] 84.2	\[ \frac{1}{1 + x} - \left(\color{blue}{\frac{1}{\frac{x}{2}}} - \frac{1}{x + -1}\right) \]
   frac-sub [=>] 59.0	\[ \frac{1}{1 + x} - \color{blue}{\frac{1 \cdot \left(x + -1\right) - \frac{x}{2} \cdot 1}{\frac{x}{2} \cdot \left(x + -1\right)}} \]
   *-un-lft-identity [<=] 59.0	\[ \frac{1}{1 + x} - \frac{\color{blue}{\left(x + -1\right)} - \frac{x}{2} \cdot 1}{\frac{x}{2} \cdot \left(x + -1\right)} \]
   div-inv [=>] 59.0	\[ \frac{1}{1 + x} - \frac{\left(x + -1\right) - \color{blue}{\left(x \cdot \frac{1}{2}\right)} \cdot 1}{\frac{x}{2} \cdot \left(x + -1\right)} \]
   metadata-eval [=>] 59.0	\[ \frac{1}{1 + x} - \frac{\left(x + -1\right) - \left(x \cdot \color{blue}{0.5}\right) \cdot 1}{\frac{x}{2} \cdot \left(x + -1\right)} \]
   div-inv [=>] 59.0	\[ \frac{1}{1 + x} - \frac{\left(x + -1\right) - \left(x \cdot 0.5\right) \cdot 1}{\color{blue}{\left(x \cdot \frac{1}{2}\right)} \cdot \left(x + -1\right)} \]
   metadata-eval [=>] 59.0	\[ \frac{1}{1 + x} - \frac{\left(x + -1\right) - \left(x \cdot 0.5\right) \cdot 1}{\left(x \cdot \color{blue}{0.5}\right) \cdot \left(x + -1\right)} \]

4. Simplified 84.2%

   \[\leadsto \frac{1}{1 + x} - \color{blue}{\frac{\frac{x + \left(-1 - x \cdot 0.5\right)}{x \cdot 0.5}}{x + -1}} \]
   Proof

   [Start] 59.0	\[ \frac{1}{1 + x} - \frac{\left(x + -1\right) - \left(x \cdot 0.5\right) \cdot 1}{\left(x \cdot 0.5\right) \cdot \left(x + -1\right)} \]
   associate-/r* [=>] 84.2	\[ \frac{1}{1 + x} - \color{blue}{\frac{\frac{\left(x + -1\right) - \left(x \cdot 0.5\right) \cdot 1}{x \cdot 0.5}}{x + -1}} \]
   *-rgt-identity [=>] 84.2	\[ \frac{1}{1 + x} - \frac{\frac{\left(x + -1\right) - \color{blue}{x \cdot 0.5}}{x \cdot 0.5}}{x + -1} \]
   associate--l+ [=>] 84.2	\[ \frac{1}{1 + x} - \frac{\frac{\color{blue}{x + \left(-1 - x \cdot 0.5\right)}}{x \cdot 0.5}}{x + -1} \]

5. Taylor expanded in x around 0 84.2%

   \[\leadsto \frac{1}{1 + x} - \frac{\color{blue}{1 - 2 \cdot \frac{1}{x}}}{x + -1} \]

6. Simplified 84.2%

   \[\leadsto \frac{1}{1 + x} - \frac{\color{blue}{1 - \frac{2}{x}}}{x + -1} \]
   Proof

   [Start] 84.2	\[ \frac{1}{1 + x} - \frac{1 - 2 \cdot \frac{1}{x}}{x + -1} \]
   associate-*r/ [=>] 84.2	\[ \frac{1}{1 + x} - \frac{1 - \color{blue}{\frac{2 \cdot 1}{x}}}{x + -1} \]
   metadata-eval [=>] 84.2	\[ \frac{1}{1 + x} - \frac{1 - \frac{\color{blue}{2}}{x}}{x + -1} \]

7. Applied egg-rr 84.2%

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

   [Start] 84.2	\[ \frac{1}{1 + x} - \frac{1 - \frac{2}{x}}{x + -1} \]
   frac-sub [=>] 84.2	\[ \color{blue}{\frac{1 \cdot \left(x + -1\right) - \left(1 + x\right) \cdot \left(1 - \frac{2}{x}\right)}{\left(1 + x\right) \cdot \left(x + -1\right)}} \]
   associate-/r* [=>] 84.2	\[ \color{blue}{\frac{\frac{1 \cdot \left(x + -1\right) - \left(1 + x\right) \cdot \left(1 - \frac{2}{x}\right)}{1 + x}}{x + -1}} \]
   /-rgt-identity [<=] 84.2	\[ \frac{\frac{1 \cdot \left(x + -1\right) - \color{blue}{\frac{1 + x}{1}} \cdot \left(1 - \frac{2}{x}\right)}{1 + x}}{x + -1} \]
   *-un-lft-identity [<=] 84.2	\[ \frac{\frac{\color{blue}{\left(x + -1\right)} - \frac{1 + x}{1} \cdot \left(1 - \frac{2}{x}\right)}{1 + x}}{x + -1} \]
   /-rgt-identity [=>] 84.2	\[ \frac{\frac{\left(x + -1\right) - \color{blue}{\left(1 + x\right)} \cdot \left(1 - \frac{2}{x}\right)}{1 + x}}{x + -1} \]
   sub-neg [=>] 84.2	\[ \frac{\frac{\left(x + -1\right) - \left(1 + x\right) \cdot \color{blue}{\left(1 + \left(-\frac{2}{x}\right)\right)}}{1 + x}}{x + -1} \]
   distribute-neg-frac [=>] 84.2	\[ \frac{\frac{\left(x + -1\right) - \left(1 + x\right) \cdot \left(1 + \color{blue}{\frac{-2}{x}}\right)}{1 + x}}{x + -1} \]
   metadata-eval [=>] 84.2	\[ \frac{\frac{\left(x + -1\right) - \left(1 + x\right) \cdot \left(1 + \frac{\color{blue}{-2}}{x}\right)}{1 + x}}{x + -1} \]

8. Taylor expanded in x around 0 99.9%

   \[\leadsto \frac{\frac{\color{blue}{\frac{2}{x}}}{1 + x}}{x + -1} \]

9. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	98.6%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.85\right):\\ \;\;\;\;\frac{\frac{2}{x \cdot x}}{x + -1}\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \end{array} \]

Alternative 2
Accuracy	99.9%
Cost	704

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

Alternative 3
Accuracy	75.8%
Cost	585

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

Alternative 4
Accuracy	82.8%
Cost	448

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

Alternative 5
Accuracy	51.8%
Cost	192

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

Alternative 6
Accuracy	3.3%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023153 
(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))))