Asymptote C

Average Error: 29.6 → 0.0

Time: 8.9s

Precision: binary64

Cost: 704

Math

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

↓

\[\frac{-3 + \frac{-1}{x}}{x + \frac{-1}{x}} \]

FPCore

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

↓

(FPCore (x) :precision binary64 (/ (+ -3.0 (/ -1.0 x)) (+ x (/ -1.0 x))))

C

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

↓

double code(double x) {
	return (-3.0 + (-1.0 / x)) / (x + (-1.0 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x / (x + 1.0d0)) - ((x + 1.0d0) / (x - 1.0d0))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((-3.0d0) + ((-1.0d0) / x)) / (x + ((-1.0d0) / x))
end function

Java

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

↓

public static double code(double x) {
	return (-3.0 + (-1.0 / x)) / (x + (-1.0 / x));
}

Python

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

↓

def code(x):
	return (-3.0 + (-1.0 / x)) / (x + (-1.0 / x))

Julia

function code(x)
	return Float64(Float64(x / Float64(x + 1.0)) - Float64(Float64(x + 1.0) / Float64(x - 1.0)))
end

↓

function code(x)
	return Float64(Float64(-3.0 + Float64(-1.0 / x)) / Float64(x + Float64(-1.0 / x)))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = (-3.0 + (-1.0 / x)) / (x + (-1.0 / x));
end

Wolfram

code[x_] := N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(N[(-3.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.6

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

2. Simplified 29.6

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

   [Start] 29.6	\[ \frac{x}{x + 1} - \frac{x + 1}{x - 1} \]
   sub-neg [=>] 29.6	\[ \color{blue}{\frac{x}{x + 1} + \left(-\frac{x + 1}{x - 1}\right)} \]
   +-commutative [=>] 29.6	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) + \frac{x}{x + 1}} \]
   remove-double-neg [<=] 29.6	\[ \left(-\frac{x + 1}{x - 1}\right) + \color{blue}{\left(-\left(-\frac{x}{x + 1}\right)\right)} \]
   sub-neg [<=] 29.6	\[ \color{blue}{\left(-\frac{x + 1}{x - 1}\right) - \left(-\frac{x}{x + 1}\right)} \]
   distribute-neg-frac [=>] 29.6	\[ \color{blue}{\frac{-\left(x + 1\right)}{x - 1}} - \left(-\frac{x}{x + 1}\right) \]
   neg-sub0 [=>] 29.6	\[ \frac{\color{blue}{0 - \left(x + 1\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
   +-commutative [=>] 29.6	\[ \frac{0 - \color{blue}{\left(1 + x\right)}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
   associate--r+ [=>] 29.6	\[ \frac{\color{blue}{\left(0 - 1\right) - x}}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
   metadata-eval [=>] 29.6	\[ \frac{\color{blue}{-1} - x}{x - 1} - \left(-\frac{x}{x + 1}\right) \]
   sub-neg [=>] 29.6	\[ \frac{-1 - x}{\color{blue}{x + \left(-1\right)}} - \left(-\frac{x}{x + 1}\right) \]
   metadata-eval [=>] 29.6	\[ \frac{-1 - x}{x + \color{blue}{-1}} - \left(-\frac{x}{x + 1}\right) \]
   /-rgt-identity [<=] 29.6	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{-\frac{x}{x + 1}}{1}} \]
   neg-mul-1 [=>] 29.6	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{-1 \cdot \frac{x}{x + 1}}}{1} \]
   metadata-eval [<=] 29.6	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\left(-1\right)} \cdot \frac{x}{x + 1}}{1} \]
   *-commutative [=>] 29.6	\[ \frac{-1 - x}{x + -1} - \frac{\color{blue}{\frac{x}{x + 1} \cdot \left(-1\right)}}{1} \]
   associate-/l* [=>] 29.6	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{\frac{x}{x + 1}}{\frac{1}{-1}}} \]
   metadata-eval [=>] 29.6	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\frac{1}{\color{blue}{-1}}} \]
   metadata-eval [=>] 29.6	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
   metadata-eval [<=] 29.6	\[ \frac{-1 - x}{x + -1} - \frac{\frac{x}{x + 1}}{\color{blue}{-1}} \]
   associate-/l/ [=>] 29.6	\[ \frac{-1 - x}{x + -1} - \color{blue}{\frac{x}{\left(-1\right) \cdot \left(x + 1\right)}} \]
   metadata-eval [=>] 29.6	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-1} \cdot \left(x + 1\right)} \]
   neg-mul-1 [<=] 29.6	\[ \frac{-1 - x}{x + -1} - \frac{x}{\color{blue}{-\left(x + 1\right)}} \]

3. Applied egg-rr 29.3

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

4. Simplified 29.6

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

   [Start] 29.3	\[ \frac{\left(x + 1\right) \cdot \frac{-1 - x}{x} - \left(1 - x\right) \cdot 1}{\left(1 - x\right) \cdot \frac{-1 - x}{x}} \]
   *-rgt-identity [=>] 29.3	\[ \frac{\left(x + 1\right) \cdot \frac{-1 - x}{x} - \color{blue}{\left(1 - x\right)}}{\left(1 - x\right) \cdot \frac{-1 - x}{x}} \]
   fma-neg [=>] 29.6	\[ \frac{\color{blue}{\mathsf{fma}\left(x + 1, \frac{-1 - x}{x}, -\left(1 - x\right)\right)}}{\left(1 - x\right) \cdot \frac{-1 - x}{x}} \]
   +-commutative [=>] 29.6	\[ \frac{\mathsf{fma}\left(\color{blue}{1 + x}, \frac{-1 - x}{x}, -\left(1 - x\right)\right)}{\left(1 - x\right) \cdot \frac{-1 - x}{x}} \]
   div-sub [=>] 29.6	\[ \frac{\mathsf{fma}\left(1 + x, \color{blue}{\frac{-1}{x} - \frac{x}{x}}, -\left(1 - x\right)\right)}{\left(1 - x\right) \cdot \frac{-1 - x}{x}} \]
   *-inverses [=>] 29.6	\[ \frac{\mathsf{fma}\left(1 + x, \frac{-1}{x} - \color{blue}{1}, -\left(1 - x\right)\right)}{\left(1 - x\right) \cdot \frac{-1 - x}{x}} \]
   sub-neg [=>] 29.6	\[ \frac{\mathsf{fma}\left(1 + x, \color{blue}{\frac{-1}{x} + \left(-1\right)}, -\left(1 - x\right)\right)}{\left(1 - x\right) \cdot \frac{-1 - x}{x}} \]
   metadata-eval [=>] 29.6	\[ \frac{\mathsf{fma}\left(1 + x, \frac{-1}{x} + \color{blue}{-1}, -\left(1 - x\right)\right)}{\left(1 - x\right) \cdot \frac{-1 - x}{x}} \]
   sub-neg [=>] 29.6	\[ \frac{\mathsf{fma}\left(1 + x, \frac{-1}{x} + -1, -\color{blue}{\left(1 + \left(-x\right)\right)}\right)}{\left(1 - x\right) \cdot \frac{-1 - x}{x}} \]
   distribute-neg-in [=>] 29.6	\[ \frac{\mathsf{fma}\left(1 + x, \frac{-1}{x} + -1, \color{blue}{\left(-1\right) + \left(-\left(-x\right)\right)}\right)}{\left(1 - x\right) \cdot \frac{-1 - x}{x}} \]
   metadata-eval [=>] 29.6	\[ \frac{\mathsf{fma}\left(1 + x, \frac{-1}{x} + -1, \color{blue}{-1} + \left(-\left(-x\right)\right)\right)}{\left(1 - x\right) \cdot \frac{-1 - x}{x}} \]
   mul-1-neg [<=] 29.6	\[ \frac{\mathsf{fma}\left(1 + x, \frac{-1}{x} + -1, -1 + \color{blue}{-1 \cdot \left(-x\right)}\right)}{\left(1 - x\right) \cdot \frac{-1 - x}{x}} \]
   mul-1-neg [<=] 29.6	\[ \frac{\mathsf{fma}\left(1 + x, \frac{-1}{x} + -1, -1 + -1 \cdot \color{blue}{\left(-1 \cdot x\right)}\right)}{\left(1 - x\right) \cdot \frac{-1 - x}{x}} \]
   associate-*r* [=>] 29.6	\[ \frac{\mathsf{fma}\left(1 + x, \frac{-1}{x} + -1, -1 + \color{blue}{\left(-1 \cdot -1\right) \cdot x}\right)}{\left(1 - x\right) \cdot \frac{-1 - x}{x}} \]
   metadata-eval [=>] 29.6	\[ \frac{\mathsf{fma}\left(1 + x, \frac{-1}{x} + -1, -1 + \color{blue}{1} \cdot x\right)}{\left(1 - x\right) \cdot \frac{-1 - x}{x}} \]
   *-lft-identity [=>] 29.6	\[ \frac{\mathsf{fma}\left(1 + x, \frac{-1}{x} + -1, -1 + \color{blue}{x}\right)}{\left(1 - x\right) \cdot \frac{-1 - x}{x}} \]
   +-commutative [=>] 29.6	\[ \frac{\mathsf{fma}\left(1 + x, \frac{-1}{x} + -1, \color{blue}{x + -1}\right)}{\left(1 - x\right) \cdot \frac{-1 - x}{x}} \]
   div-sub [=>] 29.6	\[ \frac{\mathsf{fma}\left(1 + x, \frac{-1}{x} + -1, x + -1\right)}{\left(1 - x\right) \cdot \color{blue}{\left(\frac{-1}{x} - \frac{x}{x}\right)}} \]
   *-inverses [=>] 29.6	\[ \frac{\mathsf{fma}\left(1 + x, \frac{-1}{x} + -1, x + -1\right)}{\left(1 - x\right) \cdot \left(\frac{-1}{x} - \color{blue}{1}\right)} \]
   sub-neg [=>] 29.6	\[ \frac{\mathsf{fma}\left(1 + x, \frac{-1}{x} + -1, x + -1\right)}{\left(1 - x\right) \cdot \color{blue}{\left(\frac{-1}{x} + \left(-1\right)\right)}} \]
   metadata-eval [=>] 29.6	\[ \frac{\mathsf{fma}\left(1 + x, \frac{-1}{x} + -1, x + -1\right)}{\left(1 - x\right) \cdot \left(\frac{-1}{x} + \color{blue}{-1}\right)} \]

5. Taylor expanded in x around 0 0.0

   \[\leadsto \frac{\color{blue}{-\left(3 + \frac{1}{x}\right)}}{\left(1 - x\right) \cdot \left(\frac{-1}{x} + -1\right)} \]

6. Simplified 0.0

   \[\leadsto \frac{\color{blue}{-3 - \frac{1}{x}}}{\left(1 - x\right) \cdot \left(\frac{-1}{x} + -1\right)} \]
   Proof

   [Start] 0.0	\[ \frac{-\left(3 + \frac{1}{x}\right)}{\left(1 - x\right) \cdot \left(\frac{-1}{x} + -1\right)} \]
   distribute-neg-in [=>] 0.0	\[ \frac{\color{blue}{\left(-3\right) + \left(-\frac{1}{x}\right)}}{\left(1 - x\right) \cdot \left(\frac{-1}{x} + -1\right)} \]
   metadata-eval [=>] 0.0	\[ \frac{\color{blue}{-3} + \left(-\frac{1}{x}\right)}{\left(1 - x\right) \cdot \left(\frac{-1}{x} + -1\right)} \]
   unsub-neg [=>] 0.0	\[ \frac{\color{blue}{-3 - \frac{1}{x}}}{\left(1 - x\right) \cdot \left(\frac{-1}{x} + -1\right)} \]

7. Taylor expanded in x around 0 0.0

   \[\leadsto \frac{-3 - \frac{1}{x}}{\color{blue}{x - \frac{1}{x}}} \]

8. Final simplification 0.0

   \[\leadsto \frac{-3 + \frac{-1}{x}}{x + \frac{-1}{x}} \]

Alternatives

Alternative 1
Error	0.6
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-1 - x\right) \cdot \left(-1 - x\right)\\ \end{array} \]

Alternative 2
Error	0.6
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-1}{x \cdot x} + \frac{-3}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \left(-1 - x\right) \cdot \left(-1 - x\right)\\ \end{array} \]

Alternative 3
Error	0.7
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot 3\\ \end{array} \]

Alternative 4
Error	0.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]

Alternative 5
Error	1.3
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]

Alternative 6
Error	1.3
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + x\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]

Alternative 7
Error	31.8
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023187 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))