Prelude:atanh from fay-base-0.20.0.1

Average Accuracy: 100.0% → 100.0%

Time: 4.2s

Precision: binary64

Cost: 576

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Fortran

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

↓

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

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 100.0%

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

2. Simplified 100.0%

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

   [Start] 100.0	\[ \frac{x + 1}{1 - x} \]
   metadata-eval [<=] 100.0	\[ \frac{x + \color{blue}{\left(--1\right)}}{1 - x} \]
   sub-neg [<=] 100.0	\[ \frac{\color{blue}{x - -1}}{1 - x} \]

3. Applied egg-rr 100.0%

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

   No proof available- proof too large to flatten.

4. Applied egg-rr 100.0%

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

   No proof available- proof too large to flatten.

5. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	98.4%
Cost	585

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

Alternative 2
Accuracy	98.9%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -2.1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;-1 + \frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot 2\\ \end{array} \]

Alternative 3
Accuracy	100.0%
Cost	448

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

Alternative 4
Accuracy	97.9%
Cost	328

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

Alternative 5
Accuracy	49.4%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023133 
(FPCore (x)
  :name "Prelude:atanh from fay-base-0.20.0.1"
  :precision binary64
  (/ (+ x 1.0) (- 1.0 x)))