Hyperbolic arc-(co)tangent

Average Accuracy: 8.5% → 99.5%

Time: 6.3s

Precision: binary64

Cost: 832

Math

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

↓

\[0.5 \cdot \left(2 \cdot x + 0.6666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \]

FPCore

(FPCore (x) :precision binary64 (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))

↓

(FPCore (x)
 :precision binary64
 (* 0.5 (+ (* 2.0 x) (* 0.6666666666666666 (* x (* x x))))))

C

double code(double x) {
	return (1.0 / 2.0) * log(((1.0 + x) / (1.0 - x)));
}

↓

double code(double x) {
	return 0.5 * ((2.0 * x) + (0.6666666666666666 * (x * (x * x))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / 2.0d0) * log(((1.0d0 + x) / (1.0d0 - x)))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 * ((2.0d0 * x) + (0.6666666666666666d0 * (x * (x * x))))
end function

Java

public static double code(double x) {
	return (1.0 / 2.0) * Math.log(((1.0 + x) / (1.0 - x)));
}

↓

public static double code(double x) {
	return 0.5 * ((2.0 * x) + (0.6666666666666666 * (x * (x * x))));
}

Python

def code(x):
	return (1.0 / 2.0) * math.log(((1.0 + x) / (1.0 - x)))

↓

def code(x):
	return 0.5 * ((2.0 * x) + (0.6666666666666666 * (x * (x * x))))

Julia

function code(x)
	return Float64(Float64(1.0 / 2.0) * log(Float64(Float64(1.0 + x) / Float64(1.0 - x))))
end

↓

function code(x)
	return Float64(0.5 * Float64(Float64(2.0 * x) + Float64(0.6666666666666666 * Float64(x * Float64(x * x)))))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / 2.0) * log(((1.0 + x) / (1.0 - x)));
end

↓

function tmp = code(x)
	tmp = 0.5 * ((2.0 * x) + (0.6666666666666666 * (x * (x * x))));
end

Wolfram

code[x_] := N[(N[(1.0 / 2.0), $MachinePrecision] * N[Log[N[(N[(1.0 + x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(0.5 * N[(N[(2.0 * x), $MachinePrecision] + N[(0.6666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.5%

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

2. Simplified 8.5%

   \[\leadsto \color{blue}{0.5 \cdot \log \left(\frac{1 + x}{1 - x}\right)} \]
   Proof

   [Start] 8.5	\[ \frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]
   metadata-eval [=>] 8.5	\[ \color{blue}{0.5} \cdot \log \left(\frac{1 + x}{1 - x}\right) \]

3. Taylor expanded in x around 0 99.5%

   \[\leadsto 0.5 \cdot \color{blue}{\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right)} \]

4. Applied egg-rr 99.5%

   \[\leadsto 0.5 \cdot \left(2 \cdot x + 0.6666666666666666 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}\right) \]
   Proof

   [Start] 99.5	\[ 0.5 \cdot \left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) \]
   unpow3 [=>] 99.5	\[ 0.5 \cdot \left(2 \cdot x + 0.6666666666666666 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}\right) \]

5. Final simplification 99.5%

   \[\leadsto 0.5 \cdot \left(2 \cdot x + 0.6666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \]

Alternatives

Alternative 1
Accuracy	99.0%
Cost	320

\[0.5 \cdot \left(2 \cdot x\right) \]

Reproduce

herbie shell --seed 2023146 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  :precision binary64
  (* (/ 1.0 2.0) (log (/ (+ 1.0 x) (- 1.0 x)))))