Hyperbolic arcsine

Average Error: 53.4 → 0.2

Time: 10.1s

Precision: binary64

Cost: 20932

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -0.94:\\ \;\;\;\;\log \left(\frac{1}{\left(x \cdot -2 + 0.125 \cdot \frac{1}{{x}^{3}}\right) + \left(0.5 \cdot \frac{-1}{x} - 0.0625 \cdot \frac{1}{{x}^{5}}\right)}\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;x + \left({x}^{3} \cdot -0.16666666666666666 + {x}^{5} \cdot 0.075\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -0.94)
   (log
    (/
     1.0
     (+
      (+ (* x -2.0) (* 0.125 (/ 1.0 (pow x 3.0))))
      (- (* 0.5 (/ -1.0 x)) (* 0.0625 (/ 1.0 (pow x 5.0)))))))
   (if (<= x 1.05)
     (+ x (+ (* (pow x 3.0) -0.16666666666666666) (* (pow x 5.0) 0.075)))
     (log (+ x (+ x (/ 0.5 x)))))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -0.94) {
		tmp = log((1.0 / (((x * -2.0) + (0.125 * (1.0 / pow(x, 3.0)))) + ((0.5 * (-1.0 / x)) - (0.0625 * (1.0 / pow(x, 5.0)))))));
	} else if (x <= 1.05) {
		tmp = x + ((pow(x, 3.0) * -0.16666666666666666) + (pow(x, 5.0) * 0.075));
	} else {
		tmp = log((x + (x + (0.5 / 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 <= (-0.94d0)) then
        tmp = log((1.0d0 / (((x * (-2.0d0)) + (0.125d0 * (1.0d0 / (x ** 3.0d0)))) + ((0.5d0 * ((-1.0d0) / x)) - (0.0625d0 * (1.0d0 / (x ** 5.0d0)))))))
    else if (x <= 1.05d0) then
        tmp = x + (((x ** 3.0d0) * (-0.16666666666666666d0)) + ((x ** 5.0d0) * 0.075d0))
    else
        tmp = log((x + (x + (0.5d0 / 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 <= -0.94) {
		tmp = Math.log((1.0 / (((x * -2.0) + (0.125 * (1.0 / Math.pow(x, 3.0)))) + ((0.5 * (-1.0 / x)) - (0.0625 * (1.0 / Math.pow(x, 5.0)))))));
	} else if (x <= 1.05) {
		tmp = x + ((Math.pow(x, 3.0) * -0.16666666666666666) + (Math.pow(x, 5.0) * 0.075));
	} else {
		tmp = Math.log((x + (x + (0.5 / x))));
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -0.94:
		tmp = math.log((1.0 / (((x * -2.0) + (0.125 * (1.0 / math.pow(x, 3.0)))) + ((0.5 * (-1.0 / x)) - (0.0625 * (1.0 / math.pow(x, 5.0)))))))
	elif x <= 1.05:
		tmp = x + ((math.pow(x, 3.0) * -0.16666666666666666) + (math.pow(x, 5.0) * 0.075))
	else:
		tmp = math.log((x + (x + (0.5 / 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 <= -0.94)
		tmp = log(Float64(1.0 / Float64(Float64(Float64(x * -2.0) + Float64(0.125 * Float64(1.0 / (x ^ 3.0)))) + Float64(Float64(0.5 * Float64(-1.0 / x)) - Float64(0.0625 * Float64(1.0 / (x ^ 5.0)))))));
	elseif (x <= 1.05)
		tmp = Float64(x + Float64(Float64((x ^ 3.0) * -0.16666666666666666) + Float64((x ^ 5.0) * 0.075)));
	else
		tmp = log(Float64(x + Float64(x + Float64(0.5 / 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 <= -0.94)
		tmp = log((1.0 / (((x * -2.0) + (0.125 * (1.0 / (x ^ 3.0)))) + ((0.5 * (-1.0 / x)) - (0.0625 * (1.0 / (x ^ 5.0)))))));
	elseif (x <= 1.05)
		tmp = x + (((x ^ 3.0) * -0.16666666666666666) + ((x ^ 5.0) * 0.075));
	else
		tmp = log((x + (x + (0.5 / 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, -0.94], N[Log[N[(1.0 / N[(N[(N[(x * -2.0), $MachinePrecision] + N[(0.125 * N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] - N[(0.0625 * N[(1.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.05], N[(x + N[(N[(N[Power[x, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + N[(N[Power[x, 5.0], $MachinePrecision] * 0.075), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + N[(x + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Target

Original	53.4
Target	45.9
Herbie	0.2

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

Derivation

1. Split input into 3 regimes

2. if x < -0.93999999999999995

   1. Initial program 62.9

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

   2. Applied egg-rr 0.1

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

   3. Simplified 0.1

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

      [Start] 0.1	\[ \log \left(\left(x \cdot \left(x - x\right) - 1\right) \cdot \frac{1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      *-commutative [=>] 0.1	\[ \log \left(\left(\color{blue}{\left(x - x\right) \cdot x} - 1\right) \cdot \frac{1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      +-inverses [=>] 0.1	\[ \log \left(\left(\color{blue}{0} \cdot x - 1\right) \cdot \frac{1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      mul0-lft [=>] 0.1	\[ \log \left(\left(\color{blue}{0} - 1\right) \cdot \frac{1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      metadata-eval [=>] 0.1	\[ \log \left(\color{blue}{-1} \cdot \frac{1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      associate-*r/ [=>] 0.1	\[ \log \color{blue}{\left(\frac{-1 \cdot 1}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]
      metadata-eval [=>] 0.1	\[ \log \left(\frac{\color{blue}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      metadata-eval [<=] 0.1	\[ \log \left(\frac{\color{blue}{\frac{1}{-1}}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      associate-/r* [<=] 0.1	\[ \log \color{blue}{\left(\frac{1}{-1 \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)} \]
      neg-mul-1 [<=] 0.1	\[ \log \left(\frac{1}{\color{blue}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right) \]
      neg-sub0 [=>] 0.1	\[ \log \left(\frac{1}{\color{blue}{0 - \left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right) \]
      associate--r- [=>] 0.1	\[ \log \left(\frac{1}{\color{blue}{\left(0 - x\right) + \mathsf{hypot}\left(1, x\right)}}\right) \]
      neg-sub0 [<=] 0.1	\[ \log \left(\frac{1}{\color{blue}{\left(-x\right)} + \mathsf{hypot}\left(1, x\right)}\right) \]
      mul-1-neg [<=] 0.1	\[ \log \left(\frac{1}{\color{blue}{-1 \cdot x} + \mathsf{hypot}\left(1, x\right)}\right) \]
      +-commutative [<=] 0.1	\[ \log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) + -1 \cdot x}}\right) \]
      mul-1-neg [=>] 0.1	\[ \log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) + \color{blue}{\left(-x\right)}}\right) \]
      sub-neg [<=] 0.1	\[ \log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) - x}}\right) \]

   4. Taylor expanded in x around -inf 0.2

      \[\leadsto \log \left(\frac{1}{\color{blue}{\left(-2 \cdot x + 0.125 \cdot \frac{1}{{x}^{3}}\right) - \left(0.5 \cdot \frac{1}{x} + 0.0625 \cdot \frac{1}{{x}^{5}}\right)}}\right) \]

   if -0.93999999999999995 < x < 1.05000000000000004

   1. Initial program 58.7

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

   2. Applied egg-rr 58.7

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

   3. Simplified 58.7

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

      [Start] 58.7	\[ \log \left(\left(x \cdot \left(x - x\right) - 1\right) \cdot \frac{1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      *-commutative [=>] 58.7	\[ \log \left(\left(\color{blue}{\left(x - x\right) \cdot x} - 1\right) \cdot \frac{1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      +-inverses [=>] 58.7	\[ \log \left(\left(\color{blue}{0} \cdot x - 1\right) \cdot \frac{1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      mul0-lft [=>] 58.7	\[ \log \left(\left(\color{blue}{0} - 1\right) \cdot \frac{1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      metadata-eval [=>] 58.7	\[ \log \left(\color{blue}{-1} \cdot \frac{1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      associate-*r/ [=>] 58.7	\[ \log \color{blue}{\left(\frac{-1 \cdot 1}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]
      metadata-eval [=>] 58.7	\[ \log \left(\frac{\color{blue}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      metadata-eval [<=] 58.7	\[ \log \left(\frac{\color{blue}{\frac{1}{-1}}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      associate-/r* [<=] 58.7	\[ \log \color{blue}{\left(\frac{1}{-1 \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)} \]
      neg-mul-1 [<=] 58.7	\[ \log \left(\frac{1}{\color{blue}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right) \]
      neg-sub0 [=>] 58.7	\[ \log \left(\frac{1}{\color{blue}{0 - \left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right) \]
      associate--r- [=>] 58.7	\[ \log \left(\frac{1}{\color{blue}{\left(0 - x\right) + \mathsf{hypot}\left(1, x\right)}}\right) \]
      neg-sub0 [<=] 58.7	\[ \log \left(\frac{1}{\color{blue}{\left(-x\right)} + \mathsf{hypot}\left(1, x\right)}\right) \]
      mul-1-neg [<=] 58.7	\[ \log \left(\frac{1}{\color{blue}{-1 \cdot x} + \mathsf{hypot}\left(1, x\right)}\right) \]
      +-commutative [<=] 58.7	\[ \log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) + -1 \cdot x}}\right) \]
      mul-1-neg [=>] 58.7	\[ \log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) + \color{blue}{\left(-x\right)}}\right) \]
      sub-neg [<=] 58.7	\[ \log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) - x}}\right) \]

   4. Applied egg-rr 58.7

      \[\leadsto \color{blue}{0 + \left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\right)} \]

   5. Simplified 58.7

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

      [Start] 58.7	\[ 0 + \left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\right) \]
      +-lft-identity [=>] 58.7	\[ \color{blue}{-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)} \]

   6. Taylor expanded in x around 0 0.1

      \[\leadsto -\color{blue}{\left(0.16666666666666666 \cdot {x}^{3} + \left(-1 \cdot x + -0.075 \cdot {x}^{5}\right)\right)} \]

   7. Simplified 0.1

      \[\leadsto -\color{blue}{\left(\mathsf{fma}\left(0.16666666666666666, {x}^{3}, -0.075 \cdot {x}^{5}\right) - x\right)} \]
      Proof

      [Start] 0.1	\[ -\left(0.16666666666666666 \cdot {x}^{3} + \left(-1 \cdot x + -0.075 \cdot {x}^{5}\right)\right) \]
      +-commutative [=>] 0.1	\[ -\left(0.16666666666666666 \cdot {x}^{3} + \color{blue}{\left(-0.075 \cdot {x}^{5} + -1 \cdot x\right)}\right) \]
      mul-1-neg [=>] 0.1	\[ -\left(0.16666666666666666 \cdot {x}^{3} + \left(-0.075 \cdot {x}^{5} + \color{blue}{\left(-x\right)}\right)\right) \]
      unsub-neg [=>] 0.1	\[ -\left(0.16666666666666666 \cdot {x}^{3} + \color{blue}{\left(-0.075 \cdot {x}^{5} - x\right)}\right) \]
      associate-+r- [=>] 0.1	\[ -\color{blue}{\left(\left(0.16666666666666666 \cdot {x}^{3} + -0.075 \cdot {x}^{5}\right) - x\right)} \]
      fma-def [=>] 0.1	\[ -\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, {x}^{3}, -0.075 \cdot {x}^{5}\right)} - x\right) \]

   8. Taylor expanded in x around 0 0.1

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

   if 1.05000000000000004 < x

   1. Initial program 33.0

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

   2. Taylor expanded in x around inf 0.4

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

   3. Simplified 0.4

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

      [Start] 0.4	\[ \log \left(x + \left(0.5 \cdot \frac{1}{x} + x\right)\right) \]
      +-commutative [=>] 0.4	\[ \log \left(x + \color{blue}{\left(x + 0.5 \cdot \frac{1}{x}\right)}\right) \]
      associate-*r/ [=>] 0.4	\[ \log \left(x + \left(x + \color{blue}{\frac{0.5 \cdot 1}{x}}\right)\right) \]
      metadata-eval [=>] 0.4	\[ \log \left(x + \left(x + \frac{\color{blue}{0.5}}{x}\right)\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.94:\\ \;\;\;\;\log \left(\frac{1}{\left(x \cdot -2 + 0.125 \cdot \frac{1}{{x}^{3}}\right) + \left(0.5 \cdot \frac{-1}{x} - 0.0625 \cdot \frac{1}{{x}^{5}}\right)}\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;x + \left({x}^{3} \cdot -0.16666666666666666 + {x}^{5} \cdot 0.075\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	13768

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0065:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;x + \left({x}^{3} \cdot -0.16666666666666666 + {x}^{5} \cdot 0.075\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array} \]

Alternative 2
Error	0.2
Cost	13252

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0065:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x + {x}^{5} \cdot 0.075\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array} \]

Alternative 3
Error	0.2
Cost	7560

\[\begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;-\log \left(x \cdot -2 + \frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x + {x}^{5} \cdot 0.075\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array} \]

Alternative 4
Error	0.3
Cost	7112

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

Alternative 5
Error	0.3
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -0.95:\\ \;\;\;\;-\log \left(x \cdot -2 + \frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.96:\\ \;\;\;\;x + -0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array} \]

Alternative 6
Error	0.4
Cost	6856

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

Alternative 7
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 8
Error	30.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023045 
(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)))))