Asymptote C

Average Error: 30.0 → 0.2

Time: 7.7s

Precision: binary64

Cost: 832

Math

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

↓

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

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (+ (/ 1.0 (- 1.0 x)) (/ -2.0 (+ 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 (1.0 / (1.0 - x)) + (-2.0 / (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 = (1.0d0 / (1.0d0 - x)) + ((-2.0d0) / (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 (1.0 / (1.0 - x)) + (-2.0 / (x + (-1.0 / x)));
}

Python

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

↓

def code(x):
	return (1.0 / (1.0 - x)) + (-2.0 / (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(1.0 / Float64(1.0 - x)) + Float64(-2.0 / 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 = (1.0 / (1.0 - x)) + (-2.0 / (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[(1.0 / N[(1.0 - x), $MachinePrecision]), $MachinePrecision] + N[(-2.0 / N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.0

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

2. Simplified 30.0

   \[\leadsto \color{blue}{\frac{x}{x + 1} + \frac{-1 - x}{x + -1}} \]
   Proof
   (+.f64 (/.f64 x (+.f64 x 1)) (/.f64 (-.f64 -1 x) (+.f64 x -1))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 x (+.f64 x 1)) (/.f64 (-.f64 (Rewrite<= metadata-eval (neg.f64 1)) x) (+.f64 x -1))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 x (+.f64 x 1)) (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (neg.f64 1) (neg.f64 x))) (+.f64 x -1))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 x (+.f64 x 1)) (/.f64 (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 1 x))) (+.f64 x -1))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 x (+.f64 x 1)) (/.f64 (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 x 1))) (+.f64 x -1))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 x (+.f64 x 1)) (/.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (+.f64 x 1))) (+.f64 x -1))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 x (+.f64 x 1)) (/.f64 (*.f64 (Rewrite<= metadata-eval (/.f64 1 -1)) (+.f64 x 1)) (+.f64 x -1))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 x (+.f64 x 1)) (/.f64 (*.f64 (/.f64 1 (Rewrite<= metadata-eval (neg.f64 1))) (+.f64 x 1)) (+.f64 x -1))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 x (+.f64 x 1)) (/.f64 (Rewrite<= associate-/r/_binary64 (/.f64 1 (/.f64 (neg.f64 1) (+.f64 x 1)))) (+.f64 x -1))): 19 points increase in error, 9 points decrease in error
   (+.f64 (/.f64 x (+.f64 x 1)) (/.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 1 (+.f64 x 1)) (neg.f64 1))) (+.f64 x -1))): 9 points increase in error, 19 points decrease in error
   (+.f64 (/.f64 x (+.f64 x 1)) (/.f64 (/.f64 (Rewrite=> *-lft-identity_binary64 (+.f64 x 1)) (neg.f64 1)) (+.f64 x -1))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 x (+.f64 x 1)) (/.f64 (/.f64 (+.f64 x 1) (neg.f64 1)) (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 x (+.f64 x 1)) (/.f64 (/.f64 (+.f64 x 1) (neg.f64 1)) (Rewrite<= sub-neg_binary64 (-.f64 x 1)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 x (+.f64 x 1)) (Rewrite<= associate-/r*_binary64 (/.f64 (+.f64 x 1) (*.f64 (neg.f64 1) (-.f64 x 1))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 x (+.f64 x 1)) (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (+.f64 x 1))) (*.f64 (neg.f64 1) (-.f64 x 1)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 x (+.f64 x 1)) (Rewrite=> times-frac_binary64 (*.f64 (/.f64 1 (neg.f64 1)) (/.f64 (+.f64 x 1) (-.f64 x 1))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 x (+.f64 x 1)) (*.f64 (/.f64 1 (Rewrite=> metadata-eval -1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 x (+.f64 x 1)) (*.f64 (Rewrite=> metadata-eval -1) (/.f64 (+.f64 x 1) (-.f64 x 1)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 x (+.f64 x 1)) (Rewrite<= neg-mul-1_binary64 (neg.f64 (/.f64 (+.f64 x 1) (-.f64 x 1))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 x (+.f64 x 1)) (/.f64 (+.f64 x 1) (-.f64 x 1)))): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 26.2

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

4. Applied egg-rr 25.7

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

5. Taylor expanded in x around 0 0.2

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

6. Taylor expanded in x around 0 0.2

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

7. Simplified 0.2

   \[\leadsto \frac{1}{1 - x} - \frac{2}{\color{blue}{x + \frac{-1}{x}}} \]
   Proof
   (+.f64 x (/.f64 -1 x)): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 (Rewrite<= metadata-eval (neg.f64 1)) x)): 0 points increase in error, 0 points decrease in error
   (+.f64 x (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 1 x)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= sub-neg_binary64 (-.f64 x (/.f64 1 x))): 0 points increase in error, 0 points decrease in error

8. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.8
Cost	840

\[\begin{array}{l} t_0 := \frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{if}\;x \leq -137220053.93023926:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.05591673386039308:\\ \;\;\;\;\frac{1}{1 - x} + x \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.9
Cost	712

\[\begin{array}{l} t_0 := \frac{-3 + \frac{-1}{x}}{x}\\ \mathbf{if}\;x \leq -137220053.93023926:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 0.05591673386039308:\\ \;\;\;\;1 + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	1.1
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -137220053.93023926:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 0.05591673386039308:\\ \;\;\;\;1 + x \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]

Alternative 4
Error	1.4
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -137220053.93023926:\\ \;\;\;\;\frac{-3}{x}\\ \mathbf{elif}\;x \leq 0.05591673386039308:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{-3}{x}\\ \end{array} \]

Alternative 5
Error	32.2
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022300 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1.0)) (/ (+ x 1.0) (- x 1.0))))