Hyperbolic arcsine

Average Error: 53.4 → 0.4

Time: 7.7s

Precision: binary64

Cost: 39240

Math

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

↓

\[\begin{array}{l} t_0 := \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)\\ \mathbf{if}\;x \leq -48450.84557538221:\\ \;\;\;\;-\log \left(\left(\frac{-0.5}{x} - x\right) - x\right)\\ \mathbf{elif}\;x \leq 9.821702311257485 \cdot 10^{-8}:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;t_0 + t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (log (sqrt (+ x (hypot 1.0 x))))))
   (if (<= x -48450.84557538221)
     (- (log (- (- (/ -0.5 x) x) x)))
     (if (<= x 9.821702311257485e-8)
       (+ x (* -0.16666666666666666 (pow x 3.0)))
       (+ t_0 t_0)))))

C

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

↓

double code(double x) {
	double t_0 = log(sqrt((x + hypot(1.0, x))));
	double tmp;
	if (x <= -48450.84557538221) {
		tmp = -log((((-0.5 / x) - x) - x));
	} else if (x <= 9.821702311257485e-8) {
		tmp = x + (-0.16666666666666666 * pow(x, 3.0));
	} else {
		tmp = t_0 + t_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 t_0 = Math.log(Math.sqrt((x + Math.hypot(1.0, x))));
	double tmp;
	if (x <= -48450.84557538221) {
		tmp = -Math.log((((-0.5 / x) - x) - x));
	} else if (x <= 9.821702311257485e-8) {
		tmp = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	} else {
		tmp = t_0 + t_0;
	}
	return tmp;
}

Python

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

↓

def code(x):
	t_0 = math.log(math.sqrt((x + math.hypot(1.0, x))))
	tmp = 0
	if x <= -48450.84557538221:
		tmp = -math.log((((-0.5 / x) - x) - x))
	elif x <= 9.821702311257485e-8:
		tmp = x + (-0.16666666666666666 * math.pow(x, 3.0))
	else:
		tmp = t_0 + 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 = log(sqrt(Float64(x + hypot(1.0, x))))
	tmp = 0.0
	if (x <= -48450.84557538221)
		tmp = Float64(-log(Float64(Float64(Float64(-0.5 / x) - x) - x)));
	elseif (x <= 9.821702311257485e-8)
		tmp = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)));
	else
		tmp = Float64(t_0 + 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 = log(sqrt((x + hypot(1.0, x))));
	tmp = 0.0;
	if (x <= -48450.84557538221)
		tmp = -log((((-0.5 / x) - x) - x));
	elseif (x <= 9.821702311257485e-8)
		tmp = x + (-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = t_0 + 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[Log[N[Sqrt[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -48450.84557538221], (-N[Log[N[(N[(N[(-0.5 / x), $MachinePrecision] - x), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 9.821702311257485e-8], N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + t$95$0), $MachinePrecision]]]]

Target

Original	53.4
Target	45.7
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 < -48450.8455753822127

   1. Initial program 63.6

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

   2. Simplified 63.6

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
      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)))))): 32 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 63.1

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

   4. Applied egg-rr 0.1

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

   5. Taylor expanded in x around -inf 0.1

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

   6. Simplified 0.1

      \[\leadsto -\log \left(\color{blue}{\left(\frac{-0.5}{x} - x\right)} - x\right) \]
      Proof
      (-.f64 (/.f64 -1/2 x) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (Rewrite<= metadata-eval (neg.f64 1/2)) x) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 1/2 x))) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (neg.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 1/2 1)) x)) x): 0 points increase in error, 0 points decrease in error
      (-.f64 (neg.f64 (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 1 x)))) x): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (neg.f64 (*.f64 1/2 (/.f64 1 x))) (neg.f64 x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (neg.f64 (*.f64 1/2 (/.f64 1 x))) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 x))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 x) (neg.f64 (*.f64 1/2 (/.f64 1 x))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= sub-neg_binary64 (-.f64 (*.f64 -1 x) (*.f64 1/2 (/.f64 1 x)))): 0 points increase in error, 0 points decrease in error

   if -48450.8455753822127 < x < 9.821702311257485e-8

   1. Initial program 59.0

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

   2. Simplified 59.0

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
      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)))))): 32 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 0.7

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

   if 9.821702311257485e-8 < x

   1. Initial program 32.6

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

   2. Simplified 0.2

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
      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)))))): 32 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 0.2

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -48450.84557538221:\\ \;\;\;\;-\log \left(\left(\frac{-0.5}{x} - x\right) - x\right)\\ \mathbf{elif}\;x \leq 9.821702311257485 \cdot 10^{-8}:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right) + \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	13576

\[\begin{array}{l} \mathbf{if}\;x \leq -48450.84557538221:\\ \;\;\;\;-\log \left(\left(\frac{-0.5}{x} - x\right) - x\right)\\ \mathbf{elif}\;x \leq 9.821702311257485 \cdot 10^{-8}:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + -1\right)\\ \end{array} \]

Alternative 2
Error	0.4
Cost	13320

\[\begin{array}{l} \mathbf{if}\;x \leq -48450.84557538221:\\ \;\;\;\;-\log \left(\left(\frac{-0.5}{x} - x\right) - x\right)\\ \mathbf{elif}\;x \leq 9.821702311257485 \cdot 10^{-8}:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternative 3
Error	0.5
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -48450.84557538221:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.028846596609468356:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array} \]

Alternative 4
Error	0.5
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -48450.84557538221:\\ \;\;\;\;-\log \left(\left(\frac{-0.5}{x} - x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.028846596609468356:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array} \]

Alternative 5
Error	0.6
Cost	7048

\[\begin{array}{l} \mathbf{if}\;x \leq -48450.84557538221:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.028846596609468356:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \]

Alternative 6
Error	0.7
Cost	6856

\[\begin{array}{l} \mathbf{if}\;x \leq -48450.84557538221:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.028846596609468356:\\ \;\;\;\;\left(x \cdot \left(x + -2\right)\right) \cdot \frac{1}{x + -2}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \]

Alternative 7
Error	15.6
Cost	6724

\[\begin{array}{l} \mathbf{if}\;x \leq -48450.84557538221:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	30.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022300 
(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)))))