3frac (problem 3.3.3)

Average Accuracy: 84.6% → 99.9%

Time: 12.0s

Precision: binary64

Cost: 576

Math

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

↓

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

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) (- 1.0 (* 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) {
	return (-2.0 / x) / (1.0 - (x * 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) / (1.0d0 - (x * 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) / (1.0 - (x * 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) / (1.0 - (x * 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(1.0 - Float64(x * 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) / (1.0 - (x * 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[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	84.6%
Target	99.6%
Herbie	99.9%

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

Derivation

1. Initial program 84.6%

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

2. Simplified 84.6%

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

   [Start] 84.6	\[ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
   associate-+l- [=>] 84.6	\[ \color{blue}{\frac{1}{x + 1} - \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   sub-neg [=>] 84.6	\[ \color{blue}{\frac{1}{x + 1} + \left(-\left(\frac{2}{x} - \frac{1}{x - 1}\right)\right)} \]
   neg-mul-1 [=>] 84.6	\[ \frac{1}{x + 1} + \color{blue}{-1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   metadata-eval [<=] 84.6	\[ \frac{1}{x + 1} + \color{blue}{\left(-1\right)} \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
   cancel-sign-sub-inv [<=] 84.6	\[ \color{blue}{\frac{1}{x + 1} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   +-commutative [=>] 84.6	\[ \frac{1}{\color{blue}{1 + x}} - 1 \cdot \left(\frac{2}{x} - \frac{1}{x - 1}\right) \]
   *-lft-identity [=>] 84.6	\[ \frac{1}{1 + x} - \color{blue}{\left(\frac{2}{x} - \frac{1}{x - 1}\right)} \]
   sub-neg [=>] 84.6	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{\color{blue}{x + \left(-1\right)}}\right) \]
   metadata-eval [=>] 84.6	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + \color{blue}{-1}}\right) \]

3. Applied egg-rr 84.5%

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

   [Start] 84.6	\[ \frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right) \]
   frac-sub [=>] 59.6	\[ \frac{1}{1 + x} - \color{blue}{\frac{2 \cdot \left(x + -1\right) - x \cdot 1}{x \cdot \left(x + -1\right)}} \]
   associate-/r* [=>] 84.5	\[ \frac{1}{1 + x} - \color{blue}{\frac{\frac{2 \cdot \left(x + -1\right) - x \cdot 1}{x}}{x + -1}} \]
   +-commutative [=>] 84.5	\[ \frac{1}{1 + x} - \frac{\frac{2 \cdot \color{blue}{\left(-1 + x\right)} - x \cdot 1}{x}}{x + -1} \]
   distribute-lft-in [=>] 84.5	\[ \frac{1}{1 + x} - \frac{\frac{\color{blue}{\left(2 \cdot -1 + 2 \cdot x\right)} - x \cdot 1}{x}}{x + -1} \]
   metadata-eval [=>] 84.5	\[ \frac{1}{1 + x} - \frac{\frac{\left(\color{blue}{-2} + 2 \cdot x\right) - x \cdot 1}{x}}{x + -1} \]
   metadata-eval [<=] 84.5	\[ \frac{1}{1 + x} - \frac{\frac{\left(\color{blue}{\left(-2\right)} + 2 \cdot x\right) - x \cdot 1}{x}}{x + -1} \]
   *-rgt-identity [=>] 84.5	\[ \frac{1}{1 + x} - \frac{\frac{\left(\left(-2\right) + 2 \cdot x\right) - \color{blue}{x}}{x}}{x + -1} \]
   associate--l+ [=>] 84.5	\[ \frac{1}{1 + x} - \frac{\frac{\color{blue}{\left(-2\right) + \left(2 \cdot x - x\right)}}{x}}{x + -1} \]
   metadata-eval [=>] 84.5	\[ \frac{1}{1 + x} - \frac{\frac{\color{blue}{-2} + \left(2 \cdot x - x\right)}{x}}{x + -1} \]

4. Applied egg-rr 77.3%

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

   [Start] 84.5	\[ \frac{1}{1 + x} - \frac{\frac{-2 + \left(2 \cdot x - x\right)}{x}}{x + -1} \]
   frac-2neg [=>] 84.5	\[ \color{blue}{\frac{-1}{-\left(1 + x\right)}} - \frac{\frac{-2 + \left(2 \cdot x - x\right)}{x}}{x + -1} \]
   metadata-eval [=>] 84.5	\[ \frac{\color{blue}{-1}}{-\left(1 + x\right)} - \frac{\frac{-2 + \left(2 \cdot x - x\right)}{x}}{x + -1} \]
   frac-sub [=>] 84.5	\[ \color{blue}{\frac{-1 \cdot \left(x + -1\right) - \left(-\left(1 + x\right)\right) \cdot \frac{-2 + \left(2 \cdot x - x\right)}{x}}{\left(-\left(1 + x\right)\right) \cdot \left(x + -1\right)}} \]

5. Taylor expanded in x around 0 99.9%

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

6. Taylor expanded in x around 0 99.9%

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

7. Simplified 99.9%

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

   [Start] 99.9	\[ \frac{\frac{-2}{x}}{1 + -1 \cdot {x}^{2}} \]
   mul-1-neg [=>] 99.9	\[ \frac{\frac{-2}{x}}{1 + \color{blue}{\left(-{x}^{2}\right)}} \]
   unpow2 [=>] 99.9	\[ \frac{\frac{-2}{x}}{1 + \left(-\color{blue}{x \cdot x}\right)} \]
   sub-neg [<=] 99.9	\[ \frac{\frac{-2}{x}}{\color{blue}{1 - x \cdot x}} \]

8. Final simplification 99.9%

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

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	76.0%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.38\right):\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x}\\ \end{array} \]

Alternative 3
Accuracy	83.1%
Cost	448

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

Alternative 4
Accuracy	52.1%
Cost	192

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

Alternative 5
Accuracy	3.3%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023140 
(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))))