Rust f64::asinh

Average Error: 45.2 → 0.5

Time: 2.8s

Precision: binary64

Cost: 71880

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -10.0)
     (copysign (log (+ (fabs x) (hypot 1.0 x))) x)
     (if (<= t_0 4e-12)
       (copysign x x)
       (copysign (log (+ x (hypot 1.0 x))) x)))))

C

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

↓

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = copysign(log((fabs(x) + hypot(1.0, x))), x);
	} else if (t_0 <= 4e-12) {
		tmp = copysign(x, x);
	} else {
		tmp = copysign(log((x + hypot(1.0, 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 t_0 = Math.copySign(Math.log((Math.abs(x) + Math.sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = Math.copySign(Math.log((Math.abs(x) + Math.hypot(1.0, x))), x);
	} else if (t_0 <= 4e-12) {
		tmp = Math.copySign(x, x);
	} else {
		tmp = Math.copySign(Math.log((x + Math.hypot(1.0, 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):
	t_0 = math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x)
	tmp = 0
	if t_0 <= -10.0:
		tmp = math.copysign(math.log((math.fabs(x) + math.hypot(1.0, x))), x)
	elif t_0 <= 4e-12:
		tmp = math.copysign(x, x)
	else:
		tmp = math.copysign(math.log((x + math.hypot(1.0, 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)
	t_0 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	tmp = 0.0
	if (t_0 <= -10.0)
		tmp = copysign(log(Float64(abs(x) + hypot(1.0, x))), x);
	elseif (t_0 <= 4e-12)
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float64(x + hypot(1.0, 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)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + 1.0)))));
	tmp = 0.0;
	if (t_0 <= -10.0)
		tmp = sign(x) * abs(log((abs(x) + hypot(1.0, x))));
	elseif (t_0 <= 4e-12)
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x + hypot(1.0, 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_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, -10.0], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 4e-12], N[With[{TMP1 = Abs[x], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Target

Original	45.2
Target	0.0
Herbie	0.5

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

Derivation

1. Split input into 3 regimes

2. if (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x) < -10

   1. Initial program 31.6

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

   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): 0 points increase in error, 2 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

   if -10 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x) < 3.99999999999999992e-12

   1. Initial program 59.2

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

   2. Simplified 59.2

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
      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): 0 points increase in error, 2 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 59.3

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

   4. Simplified 59.2

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)}, x\right) \]
      Proof
      (copysign.f64 (log.f64 (/.f64 1 (-.f64 (hypot.f64 1 x) x))) x): 0 points increase in error, 0 points decrease in error
      (copysign.f64 (log.f64 (/.f64 1 (Rewrite=> sub-neg_binary64 (+.f64 (hypot.f64 1 x) (neg.f64 x))))) x): 12 points increase in error, 0 points decrease in error
      (copysign.f64 (log.f64 (/.f64 1 (+.f64 (hypot.f64 1 x) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 x))))) x): 0 points increase in error, 16 points decrease in error
      (copysign.f64 (log.f64 (/.f64 1 (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 -1 x) (hypot.f64 1 x))))) x): 4 points increase in error, 12 points decrease in error
      (copysign.f64 (log.f64 (/.f64 1 (+.f64 (Rewrite=> mul-1-neg_binary64 (neg.f64 x)) (hypot.f64 1 x)))) x): 0 points increase in error, 0 points decrease in error
      (copysign.f64 (log.f64 (/.f64 1 (+.f64 (Rewrite=> neg-sub0_binary64 (-.f64 0 x)) (hypot.f64 1 x)))) x): 12 points increase in error, 0 points decrease in error
      (copysign.f64 (log.f64 (/.f64 1 (Rewrite<= associate--r-_binary64 (-.f64 0 (-.f64 x (hypot.f64 1 x)))))) x): 0 points increase in error, 16 points decrease in error
      (copysign.f64 (log.f64 (/.f64 1 (Rewrite<= neg-sub0_binary64 (neg.f64 (-.f64 x (hypot.f64 1 x)))))) x): 16 points increase in error, 0 points decrease in error
      (copysign.f64 (log.f64 (/.f64 1 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (-.f64 x (hypot.f64 1 x)))))) x): 0 points increase in error, 16 points decrease in error
      (copysign.f64 (log.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 1 -1) (-.f64 x (hypot.f64 1 x))))) x): 16 points increase in error, 0 points decrease in error
      (copysign.f64 (log.f64 (/.f64 (Rewrite=> metadata-eval -1) (-.f64 x (hypot.f64 1 x)))) x): 0 points increase in error, 16 points decrease in error
      (copysign.f64 (log.f64 (/.f64 (Rewrite<= metadata-eval (-.f64 0 1)) (-.f64 x (hypot.f64 1 x)))) x): 4 points increase in error, 12 points decrease in error
      (copysign.f64 (log.f64 (/.f64 (-.f64 (Rewrite<= +-inverses_binary64 (-.f64 (*.f64 x x) (*.f64 x x))) 1) (-.f64 x (hypot.f64 1 x)))) x): 0 points increase in error, 0 points decrease in error
      (copysign.f64 (log.f64 (/.f64 (Rewrite<= associate--r+_binary64 (-.f64 (*.f64 x x) (+.f64 (*.f64 x x) 1))) (-.f64 x (hypot.f64 1 x)))) x): 12 points increase in error, 4 points decrease in error
      (copysign.f64 (log.f64 (/.f64 (-.f64 (*.f64 x x) (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 x x)))) (-.f64 x (hypot.f64 1 x)))) x): 3 points increase in error, 13 points decrease in error
      (copysign.f64 (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)))))) x): 13 points increase in error, 3 points decrease in error

   5. Taylor expanded in x around 0 0.6

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

   if 3.99999999999999992e-12 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x)

   1. Initial program 31.6

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

   2. Simplified 0.7

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
      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): 0 points increase in error, 2 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 0.7

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

   4. Simplified 0.7

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)}, x\right) \]
      Proof
      (copysign.f64 (log.f64 (/.f64 1 (-.f64 (hypot.f64 1 x) x))) x): 0 points increase in error, 0 points decrease in error
      (copysign.f64 (log.f64 (/.f64 1 (Rewrite=> sub-neg_binary64 (+.f64 (hypot.f64 1 x) (neg.f64 x))))) x): 12 points increase in error, 0 points decrease in error
      (copysign.f64 (log.f64 (/.f64 1 (+.f64 (hypot.f64 1 x) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 x))))) x): 0 points increase in error, 16 points decrease in error
      (copysign.f64 (log.f64 (/.f64 1 (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 -1 x) (hypot.f64 1 x))))) x): 4 points increase in error, 12 points decrease in error
      (copysign.f64 (log.f64 (/.f64 1 (+.f64 (Rewrite=> mul-1-neg_binary64 (neg.f64 x)) (hypot.f64 1 x)))) x): 0 points increase in error, 0 points decrease in error
      (copysign.f64 (log.f64 (/.f64 1 (+.f64 (Rewrite=> neg-sub0_binary64 (-.f64 0 x)) (hypot.f64 1 x)))) x): 12 points increase in error, 0 points decrease in error
      (copysign.f64 (log.f64 (/.f64 1 (Rewrite<= associate--r-_binary64 (-.f64 0 (-.f64 x (hypot.f64 1 x)))))) x): 0 points increase in error, 16 points decrease in error
      (copysign.f64 (log.f64 (/.f64 1 (Rewrite<= neg-sub0_binary64 (neg.f64 (-.f64 x (hypot.f64 1 x)))))) x): 16 points increase in error, 0 points decrease in error
      (copysign.f64 (log.f64 (/.f64 1 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (-.f64 x (hypot.f64 1 x)))))) x): 0 points increase in error, 16 points decrease in error
      (copysign.f64 (log.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 1 -1) (-.f64 x (hypot.f64 1 x))))) x): 16 points increase in error, 0 points decrease in error
      (copysign.f64 (log.f64 (/.f64 (Rewrite=> metadata-eval -1) (-.f64 x (hypot.f64 1 x)))) x): 0 points increase in error, 16 points decrease in error
      (copysign.f64 (log.f64 (/.f64 (Rewrite<= metadata-eval (-.f64 0 1)) (-.f64 x (hypot.f64 1 x)))) x): 4 points increase in error, 12 points decrease in error
      (copysign.f64 (log.f64 (/.f64 (-.f64 (Rewrite<= +-inverses_binary64 (-.f64 (*.f64 x x) (*.f64 x x))) 1) (-.f64 x (hypot.f64 1 x)))) x): 0 points increase in error, 0 points decrease in error
      (copysign.f64 (log.f64 (/.f64 (Rewrite<= associate--r+_binary64 (-.f64 (*.f64 x x) (+.f64 (*.f64 x x) 1))) (-.f64 x (hypot.f64 1 x)))) x): 12 points increase in error, 4 points decrease in error
      (copysign.f64 (log.f64 (/.f64 (-.f64 (*.f64 x x) (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 x x)))) (-.f64 x (hypot.f64 1 x)))) x): 3 points increase in error, 13 points decrease in error
      (copysign.f64 (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)))))) x): 13 points increase in error, 3 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 4 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	19784

\[\begin{array}{l} \mathbf{if}\;x \leq -0.8:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{x \cdot -2 + \frac{-0.5}{x}}\right), x\right)\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 2
Error	0.5
Cost	13576

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 0.8:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + x\right)\right), x\right)\\ \end{array} \]

Alternative 3
Error	0.5
Cost	13576

\[\begin{array}{l} \mathbf{if}\;x \leq -0.8:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{x \cdot -2 + \frac{-0.5}{x}}\right), x\right)\\ \mathbf{elif}\;x \leq 0.8:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + x\right)\right), x\right)\\ \end{array} \]

Alternative 4
Error	0.6
Cost	13320

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 5
Error	12.0
Cost	13188

\[\begin{array}{l} \mathbf{if}\;x \leq -0.7:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]

Alternative 6
Error	22.6
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 7
Error	26.4
Cost	13060

\[\begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]

Alternative 8
Error	30.6
Cost	6528

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

Reproduce

herbie shell --seed 2022343 
(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))