Hyperbolic arcsine

Average Error: 53.0 → 0.4

Time: 17.1s

Precision: binary64

Cost: 20488

Math

\[\log \left(x + \sqrt{x \cdot x + 1}\right) \]

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1.28:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\left(x + {x}^{3} \cdot -0.16666666666666666\right) + \left({x}^{5} \cdot 0.075 + -0.044642857142857144 \cdot {x}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + \log x\\ \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log (+ x (sqrt (+ (* x x) 1.0)))))

↓

(FPCore (x)
 :precision binary64
 (if (<= x -1.28)
   (log (/ -0.5 x))
   (if (<= x 1.25)
     (+
      (+ x (* (pow x 3.0) -0.16666666666666666))
      (+ (* (pow x 5.0) 0.075) (* -0.044642857142857144 (pow x 7.0))))
     (+ (log 2.0) (log x)))))

C

double code(double x) {
	return log((x + sqrt(((x * x) + 1.0))));
}

↓

double code(double x) {
	double tmp;
	if (x <= -1.28) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.25) {
		tmp = (x + (pow(x, 3.0) * -0.16666666666666666)) + ((pow(x, 5.0) * 0.075) + (-0.044642857142857144 * pow(x, 7.0)));
	} else {
		tmp = log(2.0) + log(x);
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.28d0)) then
        tmp = log(((-0.5d0) / x))
    else if (x <= 1.25d0) then
        tmp = (x + ((x ** 3.0d0) * (-0.16666666666666666d0))) + (((x ** 5.0d0) * 0.075d0) + ((-0.044642857142857144d0) * (x ** 7.0d0)))
    else
        tmp = log(2.0d0) + log(x)
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x) {
	double tmp;
	if (x <= -1.28) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 1.25) {
		tmp = (x + (Math.pow(x, 3.0) * -0.16666666666666666)) + ((Math.pow(x, 5.0) * 0.075) + (-0.044642857142857144 * Math.pow(x, 7.0)));
	} else {
		tmp = Math.log(2.0) + Math.log(x);
	}
	return tmp;
}

Python

def code(x):
	return math.log((x + math.sqrt(((x * x) + 1.0))))

↓

def code(x):
	tmp = 0
	if x <= -1.28:
		tmp = math.log((-0.5 / x))
	elif x <= 1.25:
		tmp = (x + (math.pow(x, 3.0) * -0.16666666666666666)) + ((math.pow(x, 5.0) * 0.075) + (-0.044642857142857144 * math.pow(x, 7.0)))
	else:
		tmp = math.log(2.0) + math.log(x)
	return tmp

Julia

function code(x)
	return log(Float64(x + sqrt(Float64(Float64(x * x) + 1.0))))
end

↓

function code(x)
	tmp = 0.0
	if (x <= -1.28)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.25)
		tmp = Float64(Float64(x + Float64((x ^ 3.0) * -0.16666666666666666)) + Float64(Float64((x ^ 5.0) * 0.075) + Float64(-0.044642857142857144 * (x ^ 7.0))));
	else
		tmp = Float64(log(2.0) + log(x));
	end
	return tmp
end

MATLAB

function tmp = code(x)
	tmp = log((x + sqrt(((x * x) + 1.0))));
end

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.28)
		tmp = log((-0.5 / x));
	elseif (x <= 1.25)
		tmp = (x + ((x ^ 3.0) * -0.16666666666666666)) + (((x ^ 5.0) * 0.075) + (-0.044642857142857144 * (x ^ 7.0)));
	else
		tmp = log(2.0) + log(x);
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_] := If[LessEqual[x, -1.28], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.25], N[(N[(x + N[(N[Power[x, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[x, 5.0], $MachinePrecision] * 0.075), $MachinePrecision] + N[(-0.044642857142857144 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[Log[x], $MachinePrecision]), $MachinePrecision]]]

Target

Original	53.0
Target	45.3
Herbie	0.4

\[\begin{array}{l} \mathbf{if}\;x < 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if x < -1.28000000000000003

   1. Initial program 62.9

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]

   2. Taylor expanded in x around -inf 0.6

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

   if -1.28000000000000003 < x < 1.25

   1. Initial program 58.4

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]

   2. Taylor expanded in x around 0 0.2

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(-0.044642857142857144 \cdot {x}^{7} + x\right)\right)} \]

   3. Simplified 0.2

      \[\leadsto \color{blue}{\left(x + {x}^{3} \cdot -0.16666666666666666\right) + \left({x}^{5} \cdot 0.075 + -0.044642857142857144 \cdot {x}^{7}\right)} \]
      Proof

      [Start] 0.2	\[ -0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(-0.044642857142857144 \cdot {x}^{7} + x\right)\right) \]
      rational.json-simplify-1 [=>] 0.2	\[ -0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \color{blue}{\left(x + -0.044642857142857144 \cdot {x}^{7}\right)}\right) \]
      rational.json-simplify-41 [=>] 0.2	\[ -0.16666666666666666 \cdot {x}^{3} + \color{blue}{\left(x + \left(-0.044642857142857144 \cdot {x}^{7} + 0.075 \cdot {x}^{5}\right)\right)} \]
      rational.json-simplify-41 [<=] 0.2	\[ \color{blue}{\left(-0.044642857142857144 \cdot {x}^{7} + 0.075 \cdot {x}^{5}\right) + \left(-0.16666666666666666 \cdot {x}^{3} + x\right)} \]
      rational.json-simplify-1 [<=] 0.2	\[ \color{blue}{\left(-0.16666666666666666 \cdot {x}^{3} + x\right) + \left(-0.044642857142857144 \cdot {x}^{7} + 0.075 \cdot {x}^{5}\right)} \]
      rational.json-simplify-1 [=>] 0.2	\[ \left(-0.16666666666666666 \cdot {x}^{3} + x\right) + \color{blue}{\left(0.075 \cdot {x}^{5} + -0.044642857142857144 \cdot {x}^{7}\right)} \]
      rational.json-simplify-1 [=>] 0.2	\[ \color{blue}{\left(x + -0.16666666666666666 \cdot {x}^{3}\right)} + \left(0.075 \cdot {x}^{5} + -0.044642857142857144 \cdot {x}^{7}\right) \]
      rational.json-simplify-2 [=>] 0.2	\[ \left(x + \color{blue}{{x}^{3} \cdot -0.16666666666666666}\right) + \left(0.075 \cdot {x}^{5} + -0.044642857142857144 \cdot {x}^{7}\right) \]
      rational.json-simplify-2 [=>] 0.2	\[ \left(x + {x}^{3} \cdot -0.16666666666666666\right) + \left(\color{blue}{{x}^{5} \cdot 0.075} + -0.044642857142857144 \cdot {x}^{7}\right) \]

   if 1.25 < x

   1. Initial program 31.9

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]

   2. Taylor expanded in x around inf 0.5

      \[\leadsto \log \left(x + \color{blue}{x}\right) \]

   3. Taylor expanded in x around 0 0.6

      \[\leadsto \color{blue}{\log 2 + \log x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.28:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\left(x + {x}^{3} \cdot -0.16666666666666666\right) + \left({x}^{5} \cdot 0.075 + -0.044642857142857144 \cdot {x}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + \log x\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	13768

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;{x}^{5} \cdot 0.075 + \left(x + {x}^{3} \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + \log x\\ \end{array} \]

Alternative 2
Error	0.5
Cost	13256

\[\begin{array}{l} \mathbf{if}\;x \leq -1.28:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3} + x\\ \mathbf{else}:\\ \;\;\;\;\log 2 + \log x\\ \end{array} \]

Alternative 3
Error	0.4
Cost	7240

\[\begin{array}{l} \mathbf{if}\;x \leq -1.28:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.96:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3} + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right)\\ \end{array} \]

Alternative 4
Error	0.4
Cost	7048

\[\begin{array}{l} \mathbf{if}\;x \leq -1.28:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3} + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternative 5
Error	0.6
Cost	6856

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternative 6
Error	26.2
Cost	6724

\[\begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 1\right)\\ \end{array} \]

Alternative 7
Error	15.6
Cost	6724

\[\begin{array}{l} \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternative 8
Error	30.3
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023077 
(FPCore (x)
  :name "Hyperbolic arcsine"
  :precision binary64

  :herbie-target
  (if (< x 0.0) (log (/ -1.0 (- x (sqrt (+ (* x x) 1.0))))) (log (+ x (sqrt (+ (* x x) 1.0)))))

  (log (+ x (sqrt (+ (* x x) 1.0)))))