Prelude:atanh from fay-base-0.20.0.1

Average Accuracy: 100.0% → 100.0%

Time: 2.0s

Precision: binary64

Cost: 704

Math

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

↓

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

FPCore

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

↓

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

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))) + -1.0;
}

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

Python

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 100.0%

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

2. Applied egg-rr 100.0%

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

   [Start] 100.0	\[ \frac{x + 1}{1 - x} \]
   expm1-log1p-u [=>] 98.6	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x + 1}{1 - x}\right)\right)} \]
   expm1-udef [=>] 98.6	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{x + 1}{1 - x}\right)} - 1} \]
   log1p-udef [=>] 98.6	\[ e^{\color{blue}{\log \left(1 + \frac{x + 1}{1 - x}\right)}} - 1 \]
   add-exp-log [<=] 100.0	\[ \color{blue}{\left(1 + \frac{x + 1}{1 - x}\right)} - 1 \]

3. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	98.2%
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	98.8%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -2.05 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;-1 + \frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot 2\\ \end{array} \]

Alternative 3
Accuracy	100.0%
Cost	448

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

Alternative 4
Accuracy	97.6%
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	48.6%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023146 
(FPCore (x)
  :name "Prelude:atanh from fay-base-0.20.0.1"
  :precision binary64
  (/ (+ x 1.0) (- 1.0 x)))