Hyperbolic arcsine

Average Error: 53.2 → 0.4

Time: 4.8s

Precision: binary64

Cost: 7108

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -0.96:\\ \;\;\;\;\log \left(\frac{1}{x \cdot -2 + \frac{-0.5}{x}}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -0.96)
   (log (/ 1.0 (+ (* x -2.0) (/ -0.5 x))))
   (if (<= x 1.25) (+ x (* -0.16666666666666666 (pow x 3.0))) (log (+ x x)))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -0.96) {
		tmp = log((1.0 / ((x * -2.0) + (-0.5 / x))));
	} else if (x <= 1.25) {
		tmp = x + (-0.16666666666666666 * pow(x, 3.0));
	} else {
		tmp = log((x + 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 <= (-0.96d0)) then
        tmp = log((1.0d0 / ((x * (-2.0d0)) + ((-0.5d0) / x))))
    else if (x <= 1.25d0) then
        tmp = x + ((-0.16666666666666666d0) * (x ** 3.0d0))
    else
        tmp = log((x + 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 <= -0.96) {
		tmp = Math.log((1.0 / ((x * -2.0) + (-0.5 / x))));
	} else if (x <= 1.25) {
		tmp = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	} else {
		tmp = Math.log((x + x));
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -0.96:
		tmp = math.log((1.0 / ((x * -2.0) + (-0.5 / x))))
	elif x <= 1.25:
		tmp = x + (-0.16666666666666666 * math.pow(x, 3.0))
	else:
		tmp = math.log((x + 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 <= -0.96)
		tmp = log(Float64(1.0 / Float64(Float64(x * -2.0) + Float64(-0.5 / x))));
	elseif (x <= 1.25)
		tmp = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)));
	else
		tmp = log(Float64(x + 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 <= -0.96)
		tmp = log((1.0 / ((x * -2.0) + (-0.5 / x))));
	elseif (x <= 1.25)
		tmp = x + (-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = log((x + 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, -0.96], N[Log[N[(1.0 / N[(N[(x * -2.0), $MachinePrecision] + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.25], N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]]]

Target

Original	53.2
Target	45.1
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 < -0.95999999999999996

   1. Initial program 62.8

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

   2. Simplified 62.8

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
      Proof

      [Start] 62.8	\[ \log \left(x + \sqrt{x \cdot x + 1}\right) \]
      +-commutative [=>] 62.8	\[ \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]
      hypot-1-def [=>] 62.8	\[ \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Applied egg-rr 62.8

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

   4. Simplified 0.2

      \[\leadsto \log \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)} \]
      Proof

      [Start] 62.8	\[ \log \left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{1 + x \cdot x}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      div-sub [<=] 62.1	\[ \log \color{blue}{\left(\frac{x \cdot x - \left(1 + x \cdot x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]
      +-commutative [=>] 62.1	\[ \log \left(\frac{x \cdot x - \color{blue}{\left(x \cdot x + 1\right)}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      associate--r+ [=>] 30.7	\[ \log \left(\frac{\color{blue}{\left(x \cdot x - x \cdot x\right) - 1}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      +-inverses [=>] 0.2	\[ \log \left(\frac{\color{blue}{0} - 1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      metadata-eval [=>] 0.2	\[ \log \left(\frac{\color{blue}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      metadata-eval [<=] 0.2	\[ \log \left(\frac{\color{blue}{\frac{1}{-1}}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      associate-/r* [<=] 0.2	\[ \log \color{blue}{\left(\frac{1}{-1 \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)} \]
      neg-mul-1 [<=] 0.2	\[ \log \left(\frac{1}{\color{blue}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right) \]
      neg-sub0 [=>] 0.2	\[ \log \left(\frac{1}{\color{blue}{0 - \left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right) \]
      associate--r- [=>] 0.2	\[ \log \left(\frac{1}{\color{blue}{\left(0 - x\right) + \mathsf{hypot}\left(1, x\right)}}\right) \]
      neg-sub0 [<=] 0.2	\[ \log \left(\frac{1}{\color{blue}{\left(-x\right)} + \mathsf{hypot}\left(1, x\right)}\right) \]
      +-commutative [<=] 0.2	\[ \log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) + \left(-x\right)}}\right) \]
      sub-neg [<=] 0.2	\[ \log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) - x}}\right) \]

   5. Taylor expanded in x around -inf 0.5

      \[\leadsto \log \left(\frac{1}{\color{blue}{-2 \cdot x - 0.5 \cdot \frac{1}{x}}}\right) \]

   6. Simplified 0.5

      \[\leadsto \log \left(\frac{1}{\color{blue}{x \cdot -2 - \frac{0.5}{x}}}\right) \]
      Proof

      [Start] 0.5	\[ \log \left(\frac{1}{-2 \cdot x - 0.5 \cdot \frac{1}{x}}\right) \]
      *-commutative [=>] 0.5	\[ \log \left(\frac{1}{\color{blue}{x \cdot -2} - 0.5 \cdot \frac{1}{x}}\right) \]
      associate-*r/ [=>] 0.5	\[ \log \left(\frac{1}{x \cdot -2 - \color{blue}{\frac{0.5 \cdot 1}{x}}}\right) \]
      metadata-eval [=>] 0.5	\[ \log \left(\frac{1}{x \cdot -2 - \frac{\color{blue}{0.5}}{x}}\right) \]

   if -0.95999999999999996 < x < 1.25

   1. Initial program 58.6

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

   2. Simplified 58.6

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
      Proof

      [Start] 58.6	\[ \log \left(x + \sqrt{x \cdot x + 1}\right) \]
      +-commutative [=>] 58.6	\[ \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]
      hypot-1-def [=>] 58.6	\[ \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Taylor expanded in x around 0 0.2

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {x}^{3} + x} \]

   if 1.25 < x

   1. Initial program 33.2

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

   2. Simplified 0.0

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
      Proof

      [Start] 33.2	\[ \log \left(x + \sqrt{x \cdot x + 1}\right) \]
      +-commutative [=>] 33.2	\[ \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]
      hypot-1-def [=>] 0.0	\[ \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Taylor expanded in x around inf 0.5

      \[\leadsto \log \color{blue}{\left(2 \cdot x\right)} \]

   4. Simplified 0.5

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

      [Start] 0.5	\[ \log \left(2 \cdot x\right) \]
      count-2 [<=] 0.5	\[ \log \color{blue}{\left(x + x\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.96:\\ \;\;\;\;\log \left(\frac{1}{x \cdot -2 + \frac{-0.5}{x}}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	7048

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

Alternative 2
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 3
Error	15.7
Cost	6724

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

Alternative 4
Error	31.0
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023039 
(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)))))