Hyperbolic arcsine

Average Error: 53.2 → 0.2

Time: 11.2s

Precision: binary64

Cost: 20548

Math

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

↓

\[\begin{array}{l} t_0 := 0.5 \cdot \frac{1}{x}\\ \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\log \left(0.125 \cdot \frac{1}{{x}^{3}} - \left(t_0 + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;x + \left(\left(-0.044642857142857144 \cdot {x}^{7} + 0.075 \cdot {x}^{5}\right) - {x}^{3} \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 \cdot x + t_0\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 0.5 (/ 1.0 x))))
   (if (<= x -1.05)
     (log
      (- (* 0.125 (/ 1.0 (pow x 3.0))) (+ t_0 (* 0.0625 (/ 1.0 (pow x 5.0))))))
     (if (<= x 1.05)
       (+
        x
        (-
         (+ (* -0.044642857142857144 (pow x 7.0)) (* 0.075 (pow x 5.0)))
         (* (pow x 3.0) 0.16666666666666666)))
       (log (+ (* 2.0 x) t_0))))))

C

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

↓

double code(double x) {
	double t_0 = 0.5 * (1.0 / x);
	double tmp;
	if (x <= -1.05) {
		tmp = log(((0.125 * (1.0 / pow(x, 3.0))) - (t_0 + (0.0625 * (1.0 / pow(x, 5.0))))));
	} else if (x <= 1.05) {
		tmp = x + (((-0.044642857142857144 * pow(x, 7.0)) + (0.075 * pow(x, 5.0))) - (pow(x, 3.0) * 0.16666666666666666));
	} else {
		tmp = log(((2.0 * x) + t_0));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (1.0d0 / x)
    if (x <= (-1.05d0)) then
        tmp = log(((0.125d0 * (1.0d0 / (x ** 3.0d0))) - (t_0 + (0.0625d0 * (1.0d0 / (x ** 5.0d0))))))
    else if (x <= 1.05d0) then
        tmp = x + ((((-0.044642857142857144d0) * (x ** 7.0d0)) + (0.075d0 * (x ** 5.0d0))) - ((x ** 3.0d0) * 0.16666666666666666d0))
    else
        tmp = log(((2.0d0 * x) + t_0))
    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 t_0 = 0.5 * (1.0 / x);
	double tmp;
	if (x <= -1.05) {
		tmp = Math.log(((0.125 * (1.0 / Math.pow(x, 3.0))) - (t_0 + (0.0625 * (1.0 / Math.pow(x, 5.0))))));
	} else if (x <= 1.05) {
		tmp = x + (((-0.044642857142857144 * Math.pow(x, 7.0)) + (0.075 * Math.pow(x, 5.0))) - (Math.pow(x, 3.0) * 0.16666666666666666));
	} else {
		tmp = Math.log(((2.0 * x) + t_0));
	}
	return tmp;
}

Python

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

↓

def code(x):
	t_0 = 0.5 * (1.0 / x)
	tmp = 0
	if x <= -1.05:
		tmp = math.log(((0.125 * (1.0 / math.pow(x, 3.0))) - (t_0 + (0.0625 * (1.0 / math.pow(x, 5.0))))))
	elif x <= 1.05:
		tmp = x + (((-0.044642857142857144 * math.pow(x, 7.0)) + (0.075 * math.pow(x, 5.0))) - (math.pow(x, 3.0) * 0.16666666666666666))
	else:
		tmp = math.log(((2.0 * x) + t_0))
	return tmp

Julia

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

↓

function code(x)
	t_0 = Float64(0.5 * Float64(1.0 / x))
	tmp = 0.0
	if (x <= -1.05)
		tmp = log(Float64(Float64(0.125 * Float64(1.0 / (x ^ 3.0))) - Float64(t_0 + Float64(0.0625 * Float64(1.0 / (x ^ 5.0))))));
	elseif (x <= 1.05)
		tmp = Float64(x + Float64(Float64(Float64(-0.044642857142857144 * (x ^ 7.0)) + Float64(0.075 * (x ^ 5.0))) - Float64((x ^ 3.0) * 0.16666666666666666)));
	else
		tmp = log(Float64(Float64(2.0 * x) + t_0));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x)
	t_0 = 0.5 * (1.0 / x);
	tmp = 0.0;
	if (x <= -1.05)
		tmp = log(((0.125 * (1.0 / (x ^ 3.0))) - (t_0 + (0.0625 * (1.0 / (x ^ 5.0))))));
	elseif (x <= 1.05)
		tmp = x + (((-0.044642857142857144 * (x ^ 7.0)) + (0.075 * (x ^ 5.0))) - ((x ^ 3.0) * 0.16666666666666666));
	else
		tmp = log(((2.0 * x) + t_0));
	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_] := Block[{t$95$0 = N[(0.5 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.05], N[Log[N[(N[(0.125 * N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 + N[(0.0625 * N[(1.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.05], N[(x + N[(N[(N[(-0.044642857142857144 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] + N[(0.075 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[x, 3.0], $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(2.0 * x), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]]]]

Target

Original	53.2
Target	45.4
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 < -1.05000000000000004

   1. Initial program 63.0

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

   2. Taylor expanded in x around -inf 0.2

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

   3. Simplified 0.2

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

      [Start] 0.2	\[ \log \left(0.125 \cdot \frac{1}{{x}^{3}} - \left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-35 [<=] 0.2	\[ \log \left(0.125 \cdot \frac{1}{{x}^{3}} - \color{blue}{\left(0.5 \cdot \frac{1}{x} + 0.0625 \cdot \frac{1}{{x}^{5}}\right)}\right) \]

   if -1.05000000000000004 < x < 1.05000000000000004

   1. Initial program 58.5

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

   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

      \[\leadsto \color{blue}{{x}^{5} \cdot 0.075 + \left({x}^{3} \cdot -0.16666666666666666 + \left(x + -0.044642857142857144 \cdot {x}^{7}\right)\right)} \]
      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_best_oopsla_all_46_json_45_simplify-82 [=>] 0.2	\[ \color{blue}{0.075 \cdot {x}^{5} + \left(-0.16666666666666666 \cdot {x}^{3} + \left(-0.044642857142857144 \cdot {x}^{7} + x\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.2	\[ \color{blue}{{x}^{5} \cdot 0.075} + \left(-0.16666666666666666 \cdot {x}^{3} + \left(-0.044642857142857144 \cdot {x}^{7} + x\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.2	\[ {x}^{5} \cdot 0.075 + \left(\color{blue}{{x}^{3} \cdot -0.16666666666666666} + \left(-0.044642857142857144 \cdot {x}^{7} + x\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.2	\[ {x}^{5} \cdot 0.075 + \left({x}^{3} \cdot -0.16666666666666666 + \color{blue}{\left(x + -0.044642857142857144 \cdot {x}^{7}\right)}\right) \]

   4. Applied egg-rr 0.2

      \[\leadsto \color{blue}{0 - \left({x}^{3} \cdot 0.16666666666666666 - \left({x}^{5} \cdot 0.075 + \left(x + -0.044642857142857144 \cdot {x}^{7}\right)\right)\right)} \]

   5. Simplified 0.2

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

      [Start] 0.2	\[ 0 - \left({x}^{3} \cdot 0.16666666666666666 - \left({x}^{5} \cdot 0.075 + \left(x + -0.044642857142857144 \cdot {x}^{7}\right)\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-36 [=>] 0.2	\[ \color{blue}{\left({x}^{5} \cdot 0.075 + \left(x + -0.044642857142857144 \cdot {x}^{7}\right)\right) - \left({x}^{3} \cdot 0.16666666666666666 - 0\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-82 [=>] 0.2	\[ \color{blue}{\left(x + \left({x}^{5} \cdot 0.075 + -0.044642857142857144 \cdot {x}^{7}\right)\right)} - \left({x}^{3} \cdot 0.16666666666666666 - 0\right) \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.2	\[ \color{blue}{\left(\left({x}^{5} \cdot 0.075 + -0.044642857142857144 \cdot {x}^{7}\right) + x\right)} - \left({x}^{3} \cdot 0.16666666666666666 - 0\right) \]
      rational_best_oopsla_all_46_json_45_simplify-81 [=>] 0.2	\[ \left(\left({x}^{5} \cdot 0.075 + -0.044642857142857144 \cdot {x}^{7}\right) + x\right) - \color{blue}{{x}^{3} \cdot 0.16666666666666666} \]
      rational_best_oopsla_all_46_json_45_simplify-107 [=>] 0.2	\[ \color{blue}{x + \left(\left({x}^{5} \cdot 0.075 + -0.044642857142857144 \cdot {x}^{7}\right) - {x}^{3} \cdot 0.16666666666666666\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.2	\[ x + \left(\left(\color{blue}{0.075 \cdot {x}^{5}} + -0.044642857142857144 \cdot {x}^{7}\right) - {x}^{3} \cdot 0.16666666666666666\right) \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.2	\[ x + \left(\color{blue}{\left(-0.044642857142857144 \cdot {x}^{7} + 0.075 \cdot {x}^{5}\right)} - {x}^{3} \cdot 0.16666666666666666\right) \]

   if 1.05000000000000004 < x

   1. Initial program 32.4

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

   2. Taylor expanded in x around inf 0.3

      \[\leadsto \log \color{blue}{\left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\log \left(0.125 \cdot \frac{1}{{x}^{3}} - \left(0.5 \cdot \frac{1}{x} + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;x + \left(\left(-0.044642857142857144 \cdot {x}^{7} + 0.075 \cdot {x}^{5}\right) - {x}^{3} \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	20488

\[\begin{array}{l} t_0 := 0.5 \cdot \frac{1}{x}\\ \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\log \left(0.125 \cdot \frac{1}{{x}^{3}} - t_0\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;x + \left(\left(-0.044642857142857144 \cdot {x}^{7} + 0.075 \cdot {x}^{5}\right) - {x}^{3} \cdot 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 \cdot x + t_0\right)\\ \end{array} \]

Alternative 2
Error	0.3
Cost	13768

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.02:\\ \;\;\;\;{x}^{5} \cdot 0.075 + \left(x + {x}^{3} \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right)\\ \end{array} \]

Alternative 3
Error	0.3
Cost	13768

\[\begin{array}{l} t_0 := 0.5 \cdot \frac{1}{x}\\ \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\log \left(0.125 \cdot \frac{1}{{x}^{3}} - t_0\right)\\ \mathbf{elif}\;x \leq 1.02:\\ \;\;\;\;{x}^{5} \cdot 0.075 + \left(x + {x}^{3} \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 \cdot x + t_0\right)\\ \end{array} \]

Alternative 4
Error	0.4
Cost	7240

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

Alternative 5
Error	0.4
Cost	7048

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

Alternative 6
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 7
Error	26.5
Cost	6724

\[\begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 1\right)\\ \end{array} \]

Alternative 8
Error	15.8
Cost	6724

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

Alternative 9
Error	30.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023090 
(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)))))