Hyperbolic arcsine

Average Error: 52.9 → 0.2

Time: 9.8s

Precision: binary64

Cost: 13576

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1582.755949319897:\\ \;\;\;\;\frac{-0.25}{x \cdot x} + \log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.00018058092518157055:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + -1\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -1582.755949319897)
   (+ (/ -0.25 (* x x)) (log (/ -0.5 x)))
   (if (<= x 0.00018058092518157055)
     (+ x (* (* x x) (* x -0.16666666666666666)))
     (+ 1.0 (+ (log (+ x (hypot 1.0 x))) -1.0)))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -1582.755949319897) {
		tmp = (-0.25 / (x * x)) + log((-0.5 / x));
	} else if (x <= 0.00018058092518157055) {
		tmp = x + ((x * x) * (x * -0.16666666666666666));
	} else {
		tmp = 1.0 + (log((x + hypot(1.0, x))) + -1.0);
	}
	return tmp;
}

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 <= -1582.755949319897) {
		tmp = (-0.25 / (x * x)) + Math.log((-0.5 / x));
	} else if (x <= 0.00018058092518157055) {
		tmp = x + ((x * x) * (x * -0.16666666666666666));
	} else {
		tmp = 1.0 + (Math.log((x + Math.hypot(1.0, x))) + -1.0);
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -1582.755949319897:
		tmp = (-0.25 / (x * x)) + math.log((-0.5 / x))
	elif x <= 0.00018058092518157055:
		tmp = x + ((x * x) * (x * -0.16666666666666666))
	else:
		tmp = 1.0 + (math.log((x + math.hypot(1.0, x))) + -1.0)
	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 <= -1582.755949319897)
		tmp = Float64(Float64(-0.25 / Float64(x * x)) + log(Float64(-0.5 / x)));
	elseif (x <= 0.00018058092518157055)
		tmp = Float64(x + Float64(Float64(x * x) * Float64(x * -0.16666666666666666)));
	else
		tmp = Float64(1.0 + Float64(log(Float64(x + hypot(1.0, x))) + -1.0));
	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 <= -1582.755949319897)
		tmp = (-0.25 / (x * x)) + log((-0.5 / x));
	elseif (x <= 0.00018058092518157055)
		tmp = x + ((x * x) * (x * -0.16666666666666666));
	else
		tmp = 1.0 + (log((x + hypot(1.0, x))) + -1.0);
	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_] := If[LessEqual[x, -1582.755949319897], N[(N[(-0.25 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00018058092518157055], N[(x + N[(N[(x * x), $MachinePrecision] * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Target

Original	52.9
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 < -1582.7559493198969

   1. Initial program 63.4

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

   2. Simplified 63.4

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
      Proof
      (log.f64 (+.f64 x (hypot.f64 1 x))): 0 points increase in error, 0 points decrease in error
      (log.f64 (+.f64 x (Rewrite<= hypot-1-def_binary64 (sqrt.f64 (+.f64 1 (*.f64 x x)))))): 31 points increase in error, 0 points decrease in error
      (log.f64 (+.f64 x (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 x x) 1))))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in x around -inf 0.2

      \[\leadsto \color{blue}{\left(\log \left(\frac{-1}{x}\right) + \log 0.5\right) - 0.25 \cdot \frac{1}{{x}^{2}}} \]

   4. Simplified 0.2

      \[\leadsto \color{blue}{\log \left(\frac{-1}{x}\right) + \left(\log 0.5 + \frac{\frac{-0.25}{x}}{x}\right)} \]
      Proof
      (+.f64 (log.f64 (/.f64 -1 x)) (+.f64 (log.f64 1/2) (/.f64 (/.f64 -1/4 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 x)) (+.f64 (log.f64 1/2) (/.f64 (/.f64 (Rewrite<= metadata-eval (neg.f64 1/4)) x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 x)) (+.f64 (log.f64 1/2) (Rewrite<= associate-/r*_binary64 (/.f64 (neg.f64 1/4) (*.f64 x x))))): 3 points increase in error, 5 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 x)) (+.f64 (log.f64 1/2) (/.f64 (neg.f64 1/4) (Rewrite<= unpow2_binary64 (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 x)) (+.f64 (log.f64 1/2) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 1/4 (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 x)) (+.f64 (log.f64 1/2) (neg.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 1/4 1)) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 x)) (+.f64 (log.f64 1/2) (neg.f64 (Rewrite<= associate-*r/_binary64 (*.f64 1/4 (/.f64 1 (pow.f64 x 2))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (/.f64 -1 x)) (Rewrite<= sub-neg_binary64 (-.f64 (log.f64 1/2) (*.f64 1/4 (/.f64 1 (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (log.f64 (/.f64 -1 x)) (log.f64 1/2)) (*.f64 1/4 (/.f64 1 (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 0.0

      \[\leadsto \color{blue}{\left(\frac{-0.25}{x \cdot x} + \log \left(\frac{-1}{x} \cdot 0.5\right)\right) \cdot 1} \]

   6. Taylor expanded in x around 0 0.0

      \[\leadsto \left(\frac{-0.25}{x \cdot x} + \log \color{blue}{\left(\frac{-0.5}{x}\right)}\right) \cdot 1 \]

   if -1582.7559493198969 < x < 1.80580925181570551e-4

   1. Initial program 58.9

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

   2. Simplified 58.9

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
      Proof
      (log.f64 (+.f64 x (hypot.f64 1 x))): 0 points increase in error, 0 points decrease in error
      (log.f64 (+.f64 x (Rewrite<= hypot-1-def_binary64 (sqrt.f64 (+.f64 1 (*.f64 x x)))))): 31 points increase in error, 0 points decrease in error
      (log.f64 (+.f64 x (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 x x) 1))))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in x around 0 0.4

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

   4. Applied egg-rr 0.5

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

   5. Applied egg-rr 0.4

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

   if 1.80580925181570551e-4 < x

   1. Initial program 30.7

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
      Proof
      (log.f64 (+.f64 x (hypot.f64 1 x))): 0 points increase in error, 0 points decrease in error
      (log.f64 (+.f64 x (Rewrite<= hypot-1-def_binary64 (sqrt.f64 (+.f64 1 (*.f64 x x)))))): 31 points increase in error, 0 points decrease in error
      (log.f64 (+.f64 x (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 x x) 1))))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 0.1

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

   4. Applied egg-rr 0.1

      \[\leadsto \color{blue}{\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + -1\right) + 1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1582.755949319897:\\ \;\;\;\;\frac{-0.25}{x \cdot x} + \log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.00018058092518157055:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + -1\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	13320

\[\begin{array}{l} \mathbf{if}\;x \leq -1582.755949319897:\\ \;\;\;\;\frac{-0.25}{x \cdot x} + \log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.00018058092518157055:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternative 2
Error	0.4
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -1582.755949319897:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.02162174461487409:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array} \]

Alternative 3
Error	0.4
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -1582.755949319897:\\ \;\;\;\;\frac{-0.25}{x \cdot x} + \log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.02162174461487409:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array} \]

Alternative 4
Error	0.5
Cost	6856

\[\begin{array}{l} \mathbf{if}\;x \leq -1582.755949319897:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.02162174461487409:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \]

Alternative 5
Error	15.5
Cost	6724

\[\begin{array}{l} \mathbf{if}\;x \leq -1582.755949319897:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	30.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022312 
(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)))))