Hyperbolic arcsine

Average Error: 53.3 → 0.1

Time: 9.5s

Precision: binary64

Cost: 13320

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0024385085958784306:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 4.312812659238603 \cdot 10^{-8}:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -0.0024385085958784306)
   (- (log (- (hypot 1.0 x) x)))
   (if (<= x 4.312812659238603e-8)
     (+ x (* (* x x) (* x -0.16666666666666666)))
     (log (+ x (hypot 1.0 x))))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -0.0024385085958784306) {
		tmp = -log((hypot(1.0, x) - x));
	} else if (x <= 4.312812659238603e-8) {
		tmp = x + ((x * x) * (x * -0.16666666666666666));
	} else {
		tmp = log((x + hypot(1.0, x)));
	}
	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 <= -0.0024385085958784306) {
		tmp = -Math.log((Math.hypot(1.0, x) - x));
	} else if (x <= 4.312812659238603e-8) {
		tmp = x + ((x * x) * (x * -0.16666666666666666));
	} else {
		tmp = Math.log((x + Math.hypot(1.0, x)));
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -0.0024385085958784306:
		tmp = -math.log((math.hypot(1.0, x) - x))
	elif x <= 4.312812659238603e-8:
		tmp = x + ((x * x) * (x * -0.16666666666666666))
	else:
		tmp = math.log((x + math.hypot(1.0, 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 <= -0.0024385085958784306)
		tmp = Float64(-log(Float64(hypot(1.0, x) - x)));
	elseif (x <= 4.312812659238603e-8)
		tmp = Float64(x + Float64(Float64(x * x) * Float64(x * -0.16666666666666666)));
	else
		tmp = log(Float64(x + hypot(1.0, 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 <= -0.0024385085958784306)
		tmp = -log((hypot(1.0, x) - x));
	elseif (x <= 4.312812659238603e-8)
		tmp = x + ((x * x) * (x * -0.16666666666666666));
	else
		tmp = log((x + hypot(1.0, 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, -0.0024385085958784306], (-N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 4.312812659238603e-8], N[(x + N[(N[(x * x), $MachinePrecision] * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Target

Original	53.3
Target	46.0
Herbie	0.1

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

Derivation

1. Split input into 3 regimes

2. if x < -0.0024385085958784306

   1. Initial program 62.5

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

   2. Simplified 62.5

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
      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 62.0

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

   4. Applied egg-rr 0.0

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

   if -0.0024385085958784306 < x < 4.31281265923860322e-8

   1. Initial program 59.6

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

   2. Simplified 59.6

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
      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)))))): 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 0.0

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

   4. Applied egg-rr 0.1

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

   5. Applied egg-rr 0.0

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

   if 4.31281265923860322e-8 < x

   1. Initial program 32.1

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

   2. Simplified 0.3

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0024385085958784306:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 4.312812659238603 \cdot 10^{-8}:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	13320

\[\begin{array}{l} \mathbf{if}\;x \leq -1.665929492288052:\\ \;\;\;\;-\log \left(\frac{-0.5}{x} + x \cdot -2\right)\\ \mathbf{elif}\;x \leq 4.312812659238603 \cdot 10^{-8}:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternative 2
Error	0.3
Cost	7044

\[\begin{array}{l} \mathbf{if}\;x \leq -1.665929492288052:\\ \;\;\;\;-\log \left(\frac{-0.5}{x} + x \cdot -2\right)\\ \mathbf{elif}\;x \leq 0.7428427558371782:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \]

Alternative 3
Error	0.4
Cost	6856

\[\begin{array}{l} \mathbf{if}\;x \leq -1.665929492288052:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.7428427558371782:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \]

Alternative 4
Error	15.4
Cost	6724

\[\begin{array}{l} \mathbf{if}\;x \leq -1.665929492288052:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	30.1
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022311 
(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)))))