Hyperbolic arcsine

Average Error: 53.6 → 0.0

Time: 7.2s

Precision: binary64

Cost: 13768

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0076:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.0068:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3} + \left(x + 0.075 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{1}{x + \mathsf{hypot}\left(1, x\right)}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -0.0076)
   (- (log (- (hypot 1.0 x) x)))
   (if (<= x 0.0068)
     (+ (* -0.16666666666666666 (pow x 3.0)) (+ x (* 0.075 (pow x 5.0))))
     (- (log (/ 1.0 (+ 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 <= -0.0076) {
		tmp = -log((hypot(1.0, x) - x));
	} else if (x <= 0.0068) {
		tmp = (-0.16666666666666666 * pow(x, 3.0)) + (x + (0.075 * pow(x, 5.0)));
	} else {
		tmp = -log((1.0 / (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 <= -0.0076) {
		tmp = -Math.log((Math.hypot(1.0, x) - x));
	} else if (x <= 0.0068) {
		tmp = (-0.16666666666666666 * Math.pow(x, 3.0)) + (x + (0.075 * Math.pow(x, 5.0)));
	} else {
		tmp = -Math.log((1.0 / (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 <= -0.0076:
		tmp = -math.log((math.hypot(1.0, x) - x))
	elif x <= 0.0068:
		tmp = (-0.16666666666666666 * math.pow(x, 3.0)) + (x + (0.075 * math.pow(x, 5.0)))
	else:
		tmp = -math.log((1.0 / (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 <= -0.0076)
		tmp = Float64(-log(Float64(hypot(1.0, x) - x)));
	elseif (x <= 0.0068)
		tmp = Float64(Float64(-0.16666666666666666 * (x ^ 3.0)) + Float64(x + Float64(0.075 * (x ^ 5.0))));
	else
		tmp = Float64(-log(Float64(1.0 / 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 <= -0.0076)
		tmp = -log((hypot(1.0, x) - x));
	elseif (x <= 0.0068)
		tmp = (-0.16666666666666666 * (x ^ 3.0)) + (x + (0.075 * (x ^ 5.0)));
	else
		tmp = -log((1.0 / (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, -0.0076], (-N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 0.0068], N[(N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(0.075 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Log[N[(1.0 / N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]]

Target

Original	53.6
Target	45.9
Herbie	0.0

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

Derivation

1. Split input into 3 regimes

2. if x < -0.00759999999999999998

   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)))))): 3 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, 3 points decrease in error

   3. Applied egg-rr 62.5

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

   4. Simplified 0.1

      \[\leadsto \log \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)} \]
      Proof
      (log.f64 (/.f64 1 (-.f64 (hypot.f64 1 x) x))): 0 points increase in error, 0 points decrease in error
      (log.f64 (/.f64 1 (Rewrite=> sub-neg_binary64 (+.f64 (hypot.f64 1 x) (neg.f64 x))))): 14 points increase in error, 0 points decrease in error
      (log.f64 (/.f64 1 (Rewrite=> +-commutative_binary64 (+.f64 (neg.f64 x) (hypot.f64 1 x))))): 0 points increase in error, 14 points decrease in error
      (log.f64 (/.f64 1 (+.f64 (Rewrite=> neg-sub0_binary64 (-.f64 0 x)) (hypot.f64 1 x)))): 14 points increase in error, 0 points decrease in error
      (log.f64 (/.f64 1 (Rewrite<= associate--r-_binary64 (-.f64 0 (-.f64 x (hypot.f64 1 x)))))): 0 points increase in error, 0 points decrease in error
      (log.f64 (/.f64 1 (Rewrite<= neg-sub0_binary64 (neg.f64 (-.f64 x (hypot.f64 1 x)))))): 0 points increase in error, 10 points decrease in error
      (log.f64 (/.f64 1 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (-.f64 x (hypot.f64 1 x)))))): 10 points increase in error, 4 points decrease in error
      (log.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 1 -1) (-.f64 x (hypot.f64 1 x))))): 14 points increase in error, 0 points decrease in error
      (log.f64 (/.f64 (Rewrite=> metadata-eval -1) (-.f64 x (hypot.f64 1 x)))): 0 points increase in error, 10 points decrease in error
      (log.f64 (/.f64 (Rewrite<= metadata-eval (-.f64 0 1)) (-.f64 x (hypot.f64 1 x)))): 0 points increase in error, 0 points decrease in error
      (log.f64 (/.f64 (-.f64 (Rewrite<= +-inverses_binary64 (-.f64 (*.f64 x x) (*.f64 x x))) 1) (-.f64 x (hypot.f64 1 x)))): 0 points increase in error, 1 points decrease in error
      (log.f64 (/.f64 (Rewrite<= associate--r+_binary64 (-.f64 (*.f64 x x) (+.f64 (*.f64 x x) 1))) (-.f64 x (hypot.f64 1 x)))): 11 points increase in error, 3 points decrease in error
      (log.f64 (/.f64 (-.f64 (*.f64 x x) (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 x x)))) (-.f64 x (hypot.f64 1 x)))): 4 points increase in error, 10 points decrease in error
      (log.f64 (Rewrite=> div-sub_binary64 (-.f64 (/.f64 (*.f64 x x) (-.f64 x (hypot.f64 1 x))) (/.f64 (+.f64 1 (*.f64 x x)) (-.f64 x (hypot.f64 1 x)))))): 10 points increase in error, 4 points decrease in error

   5. Applied egg-rr 0.1

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

   6. Simplified 0.1

      \[\leadsto \color{blue}{-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)} \]
      Proof
      (neg.f64 (log.f64 (-.f64 (hypot.f64 1 x) x))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-lft-identity_binary64 (+.f64 0 (neg.f64 (log.f64 (-.f64 (hypot.f64 1 x) x))))): 2 points increase in error, 0 points decrease in error

   if -0.00759999999999999998 < x < 0.00679999999999999962

   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)))))): 3 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, 3 points decrease in error

   3. Taylor expanded in x around 0 0.0

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

   if 0.00679999999999999962 < x

   1. Initial program 33.3

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

   2. Simplified 0.0

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
      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)))))): 3 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, 3 points decrease in error

   3. Applied egg-rr 62.4

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

   4. Simplified 62.4

      \[\leadsto \log \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)} \]
      Proof
      (log.f64 (/.f64 1 (-.f64 (hypot.f64 1 x) x))): 0 points increase in error, 0 points decrease in error
      (log.f64 (/.f64 1 (Rewrite=> sub-neg_binary64 (+.f64 (hypot.f64 1 x) (neg.f64 x))))): 14 points increase in error, 0 points decrease in error
      (log.f64 (/.f64 1 (Rewrite=> +-commutative_binary64 (+.f64 (neg.f64 x) (hypot.f64 1 x))))): 0 points increase in error, 14 points decrease in error
      (log.f64 (/.f64 1 (+.f64 (Rewrite=> neg-sub0_binary64 (-.f64 0 x)) (hypot.f64 1 x)))): 14 points increase in error, 0 points decrease in error
      (log.f64 (/.f64 1 (Rewrite<= associate--r-_binary64 (-.f64 0 (-.f64 x (hypot.f64 1 x)))))): 0 points increase in error, 0 points decrease in error
      (log.f64 (/.f64 1 (Rewrite<= neg-sub0_binary64 (neg.f64 (-.f64 x (hypot.f64 1 x)))))): 0 points increase in error, 10 points decrease in error
      (log.f64 (/.f64 1 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (-.f64 x (hypot.f64 1 x)))))): 10 points increase in error, 4 points decrease in error
      (log.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 1 -1) (-.f64 x (hypot.f64 1 x))))): 14 points increase in error, 0 points decrease in error
      (log.f64 (/.f64 (Rewrite=> metadata-eval -1) (-.f64 x (hypot.f64 1 x)))): 0 points increase in error, 10 points decrease in error
      (log.f64 (/.f64 (Rewrite<= metadata-eval (-.f64 0 1)) (-.f64 x (hypot.f64 1 x)))): 0 points increase in error, 0 points decrease in error
      (log.f64 (/.f64 (-.f64 (Rewrite<= +-inverses_binary64 (-.f64 (*.f64 x x) (*.f64 x x))) 1) (-.f64 x (hypot.f64 1 x)))): 0 points increase in error, 1 points decrease in error
      (log.f64 (/.f64 (Rewrite<= associate--r+_binary64 (-.f64 (*.f64 x x) (+.f64 (*.f64 x x) 1))) (-.f64 x (hypot.f64 1 x)))): 11 points increase in error, 3 points decrease in error
      (log.f64 (/.f64 (-.f64 (*.f64 x x) (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 x x)))) (-.f64 x (hypot.f64 1 x)))): 4 points increase in error, 10 points decrease in error
      (log.f64 (Rewrite=> div-sub_binary64 (-.f64 (/.f64 (*.f64 x x) (-.f64 x (hypot.f64 1 x))) (/.f64 (+.f64 1 (*.f64 x x)) (-.f64 x (hypot.f64 1 x)))))): 10 points increase in error, 4 points decrease in error

   5. Applied egg-rr 62.4

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

   6. Simplified 62.4

      \[\leadsto \color{blue}{-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)} \]
      Proof
      (neg.f64 (log.f64 (-.f64 (hypot.f64 1 x) x))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-lft-identity_binary64 (+.f64 0 (neg.f64 (log.f64 (-.f64 (hypot.f64 1 x) x))))): 2 points increase in error, 0 points decrease in error

   7. Applied egg-rr 61.6

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

   8. Simplified 0.0

      \[\leadsto -\log \color{blue}{\left(\frac{1}{x + \mathsf{hypot}\left(1, x\right)}\right)} \]
      Proof
      (neg.f64 (log.f64 (-.f64 (hypot.f64 1 x) x))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-lft-identity_binary64 (+.f64 0 (neg.f64 (log.f64 (-.f64 (hypot.f64 1 x) x))))): 2 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0076:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.0068:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3} + \left(x + 0.075 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{1}{x + \mathsf{hypot}\left(1, x\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.0
Cost	13512

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0011:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.001:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{1}{x + \mathsf{hypot}\left(1, x\right)}\right)\\ \end{array} \]

Alternative 2
Error	0.2
Cost	13320

\[\begin{array}{l} \mathbf{if}\;x \leq -0.96:\\ \;\;\;\;-\log \left(x \cdot -2 + \frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.001:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternative 3
Error	0.0
Cost	13320

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0011:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.001:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternative 4
Error	0.3
Cost	7240

\[\begin{array}{l} \mathbf{if}\;x \leq -0.96:\\ \;\;\;\;-\log \left(x \cdot -2 + \frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2 + 0.5 \cdot \frac{1}{x}\right)\\ \end{array} \]

Alternative 5
Error	0.4
Cost	7048

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{0.5}{x}\right)\\ \end{array} \]

Alternative 6
Error	0.3
Cost	7048

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

Alternative 7
Error	0.6
Cost	6920

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{0.5}{x}\right)\\ \end{array} \]

Alternative 8
Error	0.6
Cost	6856

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternative 9
Error	16.0
Cost	6724

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

Alternative 10
Error	30.5
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022340 
(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)))))