Rust f64::asinh

Average Error: 45.3 → 0.3

Time: 3.7s

Precision: binary64

Cost: 19780

Math

\[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00098:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -0.00098)
   (copysign (log (/ 1.0 (- (hypot 1.0 x) x))) x)
   (if (<= x 1.26)
     (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
     (copysign (log (+ x x)) x))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -0.00098) {
		tmp = copysign(log((1.0 / (hypot(1.0, x) - x))), x);
	} else if (x <= 1.26) {
		tmp = copysign((x + (-0.16666666666666666 * pow(x, 3.0))), x);
	} else {
		tmp = copysign(log((x + x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	return Math.copySign(Math.log((Math.abs(x) + Math.sqrt(((x * x) + 1.0)))), x);
}

↓

public static double code(double x) {
	double tmp;
	if (x <= -0.00098) {
		tmp = Math.copySign(Math.log((1.0 / (Math.hypot(1.0, x) - x))), x);
	} else if (x <= 1.26) {
		tmp = Math.copySign((x + (-0.16666666666666666 * Math.pow(x, 3.0))), x);
	} else {
		tmp = Math.copySign(Math.log((x + x)), x);
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -0.00098:
		tmp = math.copysign(math.log((1.0 / (math.hypot(1.0, x) - x))), x)
	elif x <= 1.26:
		tmp = math.copysign((x + (-0.16666666666666666 * math.pow(x, 3.0))), x)
	else:
		tmp = math.copysign(math.log((x + x)), x)
	return tmp

Julia

function code(x)
	return copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
end

↓

function code(x)
	tmp = 0.0
	if (x <= -0.00098)
		tmp = copysign(log(Float64(1.0 / Float64(hypot(1.0, x) - x))), x);
	elseif (x <= 1.26)
		tmp = copysign(Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0))), x);
	else
		tmp = copysign(log(Float64(x + x)), x);
	end
	return tmp
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(log((abs(x) + sqrt(((x * x) + 1.0)))));
end

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.00098)
		tmp = sign(x) * abs(log((1.0 / (hypot(1.0, x) - x))));
	elseif (x <= 1.26)
		tmp = sign(x) * abs((x + (-0.16666666666666666 * (x ^ 3.0))));
	else
		tmp = sign(x) * abs(log((x + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]

↓

code[x_] := If[LessEqual[x, -0.00098], N[With[{TMP1 = Abs[N[Log[N[(1.0 / N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.26], N[With[{TMP1 = Abs[N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Target

Original	45.3
Target	0.1
Herbie	0.3

\[\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \frac{\left|x\right|}{\mathsf{hypot}\left(1, \frac{1}{\left|x\right|}\right) + \frac{1}{\left|x\right|}}\right), x\right) \]

Derivation

1. Split input into 3 regimes

2. if x < -9.7999999999999997e-4

   1. Initial program 31.1

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]

   2. Simplified 0.1

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
      Proof
      (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (hypot.f64 1 x))) x): 0 points increase in error, 0 points decrease in error
      (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (Rewrite<= hypot-1-def_binary64 (sqrt.f64 (+.f64 1 (*.f64 x x)))))) x): 71 points increase in error, 0 points decrease in error
      (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 x x) 1))))) x): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 62.3

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

   4. Simplified 0.1

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)}, x\right) \]
      Proof
      (/.f64 1 (-.f64 (hypot.f64 1 x) x)): 0 points increase in error, 0 points decrease in error
      (/.f64 1 (Rewrite=> sub-neg_binary64 (+.f64 (hypot.f64 1 x) (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (/.f64 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 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 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 1 (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 x (hypot.f64 1 x))))): 0 points increase in error, 0 points decrease in error
      (/.f64 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=> associate-/r*_binary64 (/.f64 (/.f64 1 -1) (-.f64 x (hypot.f64 1 x)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> metadata-eval -1) (-.f64 x (hypot.f64 1 x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= metadata-eval (-.f64 0 1)) (-.f64 x (hypot.f64 1 x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (Rewrite<= +-inverses_binary64 (-.f64 (*.f64 x x) (*.f64 x x))) 1) (-.f64 x (hypot.f64 1 x))): 34 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate--r+_binary64 (-.f64 (*.f64 x x) (+.f64 (*.f64 x x) 1))) (-.f64 x (hypot.f64 1 x))): 0 points increase in error, 27 points decrease in error
      (/.f64 (-.f64 (*.f64 x x) (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 x x)))) (-.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 (+.f64 1 (*.f64 x x)) (-.f64 x (hypot.f64 1 x))))): 1 points increase in error, 1 points decrease in error

   if -9.7999999999999997e-4 < x < 1.26000000000000001

   1. Initial program 58.9

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]

   2. Simplified 58.9

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
      Proof
      (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (hypot.f64 1 x))) x): 0 points increase in error, 0 points decrease in error
      (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (Rewrite<= hypot-1-def_binary64 (sqrt.f64 (+.f64 1 (*.f64 x x)))))) x): 71 points increase in error, 0 points decrease in error
      (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 x x) 1))))) x): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 58.8

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

   4. Simplified 58.8

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

   5. Taylor expanded in x around 0 0.1

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

   if 1.26000000000000001 < x

   1. Initial program 32.5

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]

   2. Simplified 0.2

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
      Proof
      (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (hypot.f64 1 x))) x): 0 points increase in error, 0 points decrease in error
      (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (Rewrite<= hypot-1-def_binary64 (sqrt.f64 (+.f64 1 (*.f64 x x)))))) x): 71 points increase in error, 0 points decrease in error
      (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 x x) 1))))) x): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in x around inf 0.7

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left|x\right| + x\right)}, x\right) \]

   4. Simplified 0.7

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x + x\right)}, x\right) \]
      Proof
      (+.f64 x x): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= unpow1_binary64 (pow.f64 x 1)) x): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> sqr-pow_binary64 (*.f64 (pow.f64 x (/.f64 1 2)) (pow.f64 x (/.f64 1 2)))) x): 146 points increase in error, 13 points decrease in error
      (+.f64 (Rewrite<= fabs-sqr_binary64 (fabs.f64 (*.f64 (pow.f64 x (/.f64 1 2)) (pow.f64 x (/.f64 1 2))))) x): 0 points increase in error, 0 points decrease in error
      (+.f64 (fabs.f64 (Rewrite<= sqr-pow_binary64 (pow.f64 x 1))) x): 13 points increase in error, 146 points decrease in error
      (+.f64 (fabs.f64 (Rewrite=> unpow1_binary64 x)) x): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00098:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	45892

\[\begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{copysign}\left(-\mathsf{log1p}\left(\left(\mathsf{hypot}\left(1, x\right) + -1\right) - x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 2
Error	0.3
Cost	19716

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00086:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 3
Error	0.4
Cost	13512

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 4
Error	0.4
Cost	13512

\[\begin{array}{l} \mathbf{if}\;x \leq -0.95:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2 - \frac{0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 5
Error	11.3
Cost	13320

\[\begin{array}{l} \mathbf{if}\;x \leq -3.2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 6
Error	0.6
Cost	13320

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 7
Error	22.5
Cost	13124

\[\begin{array}{l} \mathbf{if}\;x \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]

Alternative 8
Error	26.2
Cost	13060

\[\begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]

Alternative 9
Error	30.4
Cost	6528

\[\mathsf{copysign}\left(x, x\right) \]

Reproduce

herbie shell --seed 2022328 
(FPCore (x)
  :name "Rust f64::asinh"
  :precision binary64

  :herbie-target
  (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))) x)

  (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))