Hyperbolic arcsine

Average Error: 53.1 → 0.0

Time: 7.0s

Precision: binary64

Cost: 20488

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -0.023:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.026:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3} + \left(x + \left(0.075 \cdot {x}^{5} + -0.044642857142857144 \cdot {x}^{7}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -0.023)
   (- (log (- (hypot 1.0 x) x)))
   (if (<= x 0.026)
     (+
      (* -0.16666666666666666 (pow x 3.0))
      (+ x (+ (* 0.075 (pow x 5.0)) (* -0.044642857142857144 (pow x 7.0)))))
     (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 <= -0.023) {
		tmp = -log((hypot(1.0, x) - x));
	} else if (x <= 0.026) {
		tmp = (-0.16666666666666666 * pow(x, 3.0)) + (x + ((0.075 * pow(x, 5.0)) + (-0.044642857142857144 * pow(x, 7.0))));
	} 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 <= -0.023) {
		tmp = -Math.log((Math.hypot(1.0, x) - x));
	} else if (x <= 0.026) {
		tmp = (-0.16666666666666666 * Math.pow(x, 3.0)) + (x + ((0.075 * Math.pow(x, 5.0)) + (-0.044642857142857144 * Math.pow(x, 7.0))));
	} 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 <= -0.023:
		tmp = -math.log((math.hypot(1.0, x) - x))
	elif x <= 0.026:
		tmp = (-0.16666666666666666 * math.pow(x, 3.0)) + (x + ((0.075 * math.pow(x, 5.0)) + (-0.044642857142857144 * math.pow(x, 7.0))))
	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 <= -0.023)
		tmp = Float64(-log(Float64(hypot(1.0, x) - x)));
	elseif (x <= 0.026)
		tmp = Float64(Float64(-0.16666666666666666 * (x ^ 3.0)) + Float64(x + Float64(Float64(0.075 * (x ^ 5.0)) + Float64(-0.044642857142857144 * (x ^ 7.0)))));
	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 <= -0.023)
		tmp = -log((hypot(1.0, x) - x));
	elseif (x <= 0.026)
		tmp = (-0.16666666666666666 * (x ^ 3.0)) + (x + ((0.075 * (x ^ 5.0)) + (-0.044642857142857144 * (x ^ 7.0))));
	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, -0.023], (-N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 0.026], N[(N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(N[(0.075 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-0.044642857142857144 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Target

Original	53.1
Target	45.5
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 < -0.023

   1. Initial program 62.7

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

   2. Applied egg-rr 62.7

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

   3. Simplified 0.1

      \[\leadsto \log \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)} \]
      Proof
      (/.f64 1 (-.f64 (hypot.f64 1 x) x)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= metadata-eval (+.f64 1 0)) (-.f64 (hypot.f64 1 x) x)): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 1 (Rewrite<= +-inverses_binary64 (-.f64 (*.f64 x x) (*.f64 x x)))) (-.f64 (hypot.f64 1 x) x)): 31 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 1 (*.f64 x x)) (*.f64 x x))) (-.f64 (hypot.f64 1 x) x)): 0 points increase in error, 28 points decrease in error
      (/.f64 (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 x x) 1)) (*.f64 x x)) (-.f64 (hypot.f64 1 x) x)): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (Rewrite<= fma-udef_binary64 (fma.f64 x x 1)) (*.f64 x x)) (-.f64 (hypot.f64 1 x) x)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> sub-neg_binary64 (+.f64 (fma.f64 x x 1) (neg.f64 (*.f64 x x)))) (-.f64 (hypot.f64 1 x) x)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> +-commutative_binary64 (+.f64 (neg.f64 (*.f64 x x)) (fma.f64 x x 1))) (-.f64 (hypot.f64 1 x) x)): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite=> neg-sub0_binary64 (-.f64 0 (*.f64 x x))) (fma.f64 x x 1)) (-.f64 (hypot.f64 1 x) x)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate--r-_binary64 (-.f64 0 (-.f64 (*.f64 x x) (fma.f64 x x 1)))) (-.f64 (hypot.f64 1 x) x)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (-.f64 (*.f64 x x) (fma.f64 x x 1)))) (-.f64 (hypot.f64 1 x) x)): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (-.f64 (*.f64 x x) (fma.f64 x x 1))) (Rewrite=> sub-neg_binary64 (+.f64 (hypot.f64 1 x) (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (-.f64 (*.f64 x x) (fma.f64 x x 1))) (+.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (hypot.f64 1 x)))) (neg.f64 x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (-.f64 (*.f64 x x) (fma.f64 x x 1))) (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 (neg.f64 (hypot.f64 1 x)) x)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (-.f64 (*.f64 x x) (fma.f64 x x 1))) (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 x (neg.f64 (hypot.f64 1 x)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (-.f64 (*.f64 x x) (fma.f64 x x 1))) (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 x (hypot.f64 1 x))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (-.f64 (*.f64 x x) (fma.f64 x x 1)))) (neg.f64 (-.f64 x (hypot.f64 1 x)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 -1 (-.f64 (*.f64 x x) (fma.f64 x x 1))) (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (-.f64 x (hypot.f64 1 x))))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> times-frac_binary64 (*.f64 (/.f64 -1 -1) (/.f64 (-.f64 (*.f64 x x) (fma.f64 x x 1)) (-.f64 x (hypot.f64 1 x))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite=> metadata-eval 1) (/.f64 (-.f64 (*.f64 x x) (fma.f64 x x 1)) (-.f64 x (hypot.f64 1 x)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> *-lft-identity_binary64 (/.f64 (-.f64 (*.f64 x x) (fma.f64 x x 1)) (-.f64 x (hypot.f64 1 x)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> div-sub_binary64 (-.f64 (/.f64 (*.f64 x x) (-.f64 x (hypot.f64 1 x))) (/.f64 (fma.f64 x x 1) (-.f64 x (hypot.f64 1 x))))): 5 points increase in error, 0 points decrease in error

   4. Applied egg-rr 0.1

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

   5. 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))))): 0 points increase in error, 0 points decrease in error

   if -0.023 < x < 0.0259999999999999988

   1. Initial program 59.0

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

   2. Taylor expanded in x around 0 0.0

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

   3. Applied egg-rr 0.0

      \[\leadsto -0.16666666666666666 \cdot {x}^{3} + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.075, {x}^{5}, -0.044642857142857144 \cdot {x}^{7}\right), 1, x\right)} \]

   4. Applied egg-rr 0.0

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

   5. Taylor expanded in x around 0 0.0

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

   if 0.0259999999999999988 < x

   1. Initial program 32.4

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

   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)))))): 40 points increase in error, 1 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 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.023:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.026:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3} + \left(x + \left(0.075 \cdot {x}^{5} + -0.044642857142857144 \cdot {x}^{7}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.0
Cost	20488

\[\begin{array}{l} \mathbf{if}\;x \leq -0.023:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.026:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(x + -0.044642857142857144 \cdot {x}^{7}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternative 2
Error	0.1
Cost	13444

\[\begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-6}:\\ \;\;\;\;\left(1 - \log \left(\mathsf{hypot}\left(1, x\right) - x\right)\right) + -1\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternative 3
Error	0.2
Cost	13320

\[\begin{array}{l} \mathbf{if}\;x \leq -0.95:\\ \;\;\;\;-\log \left(x \cdot -2 - \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.00105:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternative 4
Error	0.1
Cost	13320

\[\begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-6}:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternative 5
Error	0.1
Cost	13320

\[\begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-6}:\\ \;\;\;\;\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternative 6
Error	0.3
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array} \]

Alternative 7
Error	0.3
Cost	7112

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

Alternative 8
Error	0.4
Cost	7048

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

Alternative 9
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 10
Error	15.4
Cost	6724

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

Alternative 11
Error	30.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022331 
(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)))))