Rust f64::asinh

Average Error: 45.4 → 29.1

Time: 10.3s

Precision: binary64

Cost: 124744

Math

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

↓

\[\begin{array}{l} t_0 := \log \left(1 + \left|x\right|\right)\\ t_1 := 2 \cdot t_0\\ t_2 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_2 \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left(-x\right) + \left|x\right|\right) - \frac{0.5}{x}\right), x\right)\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{copysign}\left(\left(\frac{{x}^{2}}{2 + 2 \cdot \left|x\right|} + \left(t_1 + t_1\right)\right) - \left(t_0 + t_1\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + \left|x\right|\right)\right), x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (log (+ 1.0 (fabs x))))
        (t_1 (* 2.0 t_0))
        (t_2 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_2 -5.0)
     (copysign (log (- (+ (- x) (fabs x)) (/ 0.5 x))) x)
     (if (<= t_2 5e-11)
       (copysign
        (-
         (+ (/ (pow x 2.0) (+ 2.0 (* 2.0 (fabs x)))) (+ t_1 t_1))
         (+ t_0 t_1))
        x)
       (copysign (log (+ (/ 0.5 x) (+ x (fabs 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 = log((1.0 + fabs(x)));
	double t_1 = 2.0 * t_0;
	double t_2 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_2 <= -5.0) {
		tmp = copysign(log(((-x + fabs(x)) - (0.5 / x))), x);
	} else if (t_2 <= 5e-11) {
		tmp = copysign((((pow(x, 2.0) / (2.0 + (2.0 * fabs(x)))) + (t_1 + t_1)) - (t_0 + t_1)), x);
	} else {
		tmp = copysign(log(((0.5 / x) + (x + fabs(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.log((1.0 + Math.abs(x)));
	double t_1 = 2.0 * t_0;
	double t_2 = Math.copySign(Math.log((Math.abs(x) + Math.sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_2 <= -5.0) {
		tmp = Math.copySign(Math.log(((-x + Math.abs(x)) - (0.5 / x))), x);
	} else if (t_2 <= 5e-11) {
		tmp = Math.copySign((((Math.pow(x, 2.0) / (2.0 + (2.0 * Math.abs(x)))) + (t_1 + t_1)) - (t_0 + t_1)), x);
	} else {
		tmp = Math.copySign(Math.log(((0.5 / x) + (x + Math.abs(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.log((1.0 + math.fabs(x)))
	t_1 = 2.0 * t_0
	t_2 = math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x)
	tmp = 0
	if t_2 <= -5.0:
		tmp = math.copysign(math.log(((-x + math.fabs(x)) - (0.5 / x))), x)
	elif t_2 <= 5e-11:
		tmp = math.copysign((((math.pow(x, 2.0) / (2.0 + (2.0 * math.fabs(x)))) + (t_1 + t_1)) - (t_0 + t_1)), x)
	else:
		tmp = math.copysign(math.log(((0.5 / x) + (x + math.fabs(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 = log(Float64(1.0 + abs(x)))
	t_1 = Float64(2.0 * t_0)
	t_2 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	tmp = 0.0
	if (t_2 <= -5.0)
		tmp = copysign(log(Float64(Float64(Float64(-x) + abs(x)) - Float64(0.5 / x))), x);
	elseif (t_2 <= 5e-11)
		tmp = copysign(Float64(Float64(Float64((x ^ 2.0) / Float64(2.0 + Float64(2.0 * abs(x)))) + Float64(t_1 + t_1)) - Float64(t_0 + t_1)), x);
	else
		tmp = copysign(log(Float64(Float64(0.5 / x) + Float64(x + abs(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 = log((1.0 + abs(x)));
	t_1 = 2.0 * t_0;
	t_2 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + 1.0)))));
	tmp = 0.0;
	if (t_2 <= -5.0)
		tmp = sign(x) * abs(log(((-x + abs(x)) - (0.5 / x))));
	elseif (t_2 <= 5e-11)
		tmp = sign(x) * abs(((((x ^ 2.0) / (2.0 + (2.0 * abs(x)))) + (t_1 + t_1)) - (t_0 + t_1)));
	else
		tmp = sign(x) * abs(log(((0.5 / x) + (x + abs(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[Log[N[(1.0 + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = 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$2, -5.0], N[With[{TMP1 = Abs[N[Log[N[(N[((-x) + N[Abs[x], $MachinePrecision]), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$2, 5e-11], N[With[{TMP1 = Abs[N[(N[(N[(N[Power[x, 2.0], $MachinePrecision] / N[(2.0 + N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + t$95$1), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 + t$95$1), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(N[(0.5 / x), $MachinePrecision] + N[(x + N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]]]

Target

Original	45.4
Target	0.1
Herbie	29.1

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

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) < -5

   1. Initial program 31.8

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

   2. Taylor expanded in x around -inf 0.3

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + -1 \cdot x\right) - 0.5 \cdot \frac{1}{x}\right)}, x\right) \]

   3. Simplified 0.3

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left(-x\right) + \left|x\right|\right) - \frac{0.5}{x}\right)}, x\right) \]
      Proof

      [Start] 0.3	\[ \mathsf{copysign}\left(\log \left(\left(\left|x\right| + -1 \cdot x\right) - 0.5 \cdot \frac{1}{x}\right), x\right) \]
      rational.json-simplify-41 [=>] 0.3	\[ \mathsf{copysign}\left(\log \left(\color{blue}{\left(-1 \cdot x + \left|x\right|\right)} - 0.5 \cdot \frac{1}{x}\right), x\right) \]
      rational.json-simplify-39 [=>] 0.3	\[ \mathsf{copysign}\left(\log \left(\left(\color{blue}{x \cdot -1} + \left|x\right|\right) - 0.5 \cdot \frac{1}{x}\right), x\right) \]
      rational.json-simplify-72 [=>] 0.3	\[ \mathsf{copysign}\left(\log \left(\left(\color{blue}{\left(-x\right)} + \left|x\right|\right) - 0.5 \cdot \frac{1}{x}\right), x\right) \]
      rational.json-simplify-20 [=>] 0.3	\[ \mathsf{copysign}\left(\log \left(\left(\left(-x\right) + \left|x\right|\right) - \color{blue}{\frac{1 \cdot 0.5}{x}}\right), x\right) \]
      metadata-eval [=>] 0.3	\[ \mathsf{copysign}\left(\log \left(\left(\left(-x\right) + \left|x\right|\right) - \frac{\color{blue}{0.5}}{x}\right), x\right) \]

   if -5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x) < 5.00000000000000018e-11

   1. Initial program 59.2

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

   2. Applied egg-rr 59.2

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

   3. Taylor expanded in x around 0 58.8

      \[\leadsto \mathsf{copysign}\left(\frac{1}{\frac{1}{\color{blue}{0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|} + \log \left(1 + \left|x\right|\right)}}}, x\right) \]

   4. Applied egg-rr 58.8

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(\frac{{x}^{2}}{2 + 2 \cdot \left|x\right|} + \left(2 \cdot \log \left(1 + \left|x\right|\right) + 2 \cdot \log \left(1 + \left|x\right|\right)\right)\right) - \left(\log \left(1 + \left|x\right|\right) + 2 \cdot \log \left(1 + \left|x\right|\right)\right)}, x\right) \]

   if 5.00000000000000018e-11 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x)

   1. Initial program 33.0

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

   2. Taylor expanded in x around inf 2.1

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(0.5 \cdot \frac{1}{x} + \left(\left|x\right| + x\right)\right)}, x\right) \]

   3. Simplified 2.1

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{0.5}{x} + \left(x + \left|x\right|\right)\right)}, x\right) \]
      Proof

      [Start] 2.1	\[ \mathsf{copysign}\left(\log \left(0.5 \cdot \frac{1}{x} + \left(\left|x\right| + x\right)\right), x\right) \]
      rational.json-simplify-20 [=>] 2.1	\[ \mathsf{copysign}\left(\log \left(\color{blue}{\frac{1 \cdot 0.5}{x}} + \left(\left|x\right| + x\right)\right), x\right) \]
      metadata-eval [=>] 2.1	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{0.5}}{x} + \left(\left|x\right| + x\right)\right), x\right) \]
      rational.json-simplify-41 [=>] 2.1	\[ \mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \color{blue}{\left(x + \left|x\right|\right)}\right), x\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 29.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left(-x\right) + \left|x\right|\right) - \frac{0.5}{x}\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{copysign}\left(\left(\frac{{x}^{2}}{2 + 2 \cdot \left|x\right|} + \left(2 \cdot \log \left(1 + \left|x\right|\right) + 2 \cdot \log \left(1 + \left|x\right|\right)\right)\right) - \left(\log \left(1 + \left|x\right|\right) + 2 \cdot \log \left(1 + \left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + \left|x\right|\right)\right), x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	29.1
Cost	85192

\[\begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_1 := 1 + \left|x\right|\\ \mathbf{if}\;t_0 \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left(-x\right) + \left|x\right|\right) - \frac{0.5}{x}\right), x\right)\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{copysign}\left(0.5 \cdot \frac{{x}^{2}}{t_1} + \log t_1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + \left|x\right|\right)\right), x\right)\\ \end{array} \]

Alternative 2
Error	28.8
Cost	59720

\[\begin{array}{l} t_0 := 1 + \left|x\right|\\ t_1 := \log t_0\\ t_2 := 2 \cdot t_1\\ \mathbf{if}\;x \leq -0.92:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left(-x\right) + \left|x\right|\right) - \frac{0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;\mathsf{copysign}\left(\frac{1}{\frac{1}{0.5 \cdot \frac{{x}^{2}}{t_0} + \left(\left(t_1 + t_2\right) - t_2\right)}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + \left|x\right|\right)\right), x\right)\\ \end{array} \]

Alternative 3
Error	28.9
Cost	26632

\[\begin{array}{l} \mathbf{if}\;x \leq -1000000:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left(-x\right) + \left|x\right|\right) - \frac{0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 500:\\ \;\;\;\;\mathsf{copysign}\left(\frac{1}{\frac{1}{\log \left(\sqrt{x \cdot x + 1} + \left|x\right|\right)}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + \left|x\right|\right)\right), x\right)\\ \end{array} \]

Alternative 4
Error	29.1
Cost	26376

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+158}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(-x\right) + \left|x\right|\right), x\right)\\ \mathbf{elif}\;x \leq 500:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + \left|x\right|\right)\right), x\right)\\ \end{array} \]

Alternative 5
Error	30.0
Cost	19908

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left(-x\right) + \left|x\right|\right) - \frac{0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + \left|x\right|\right)\right), x\right)\\ \end{array} \]

Alternative 6
Error	30.0
Cost	19844

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(-x\right) + \left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \left(x + \left|x\right|\right)\right), x\right)\\ \end{array} \]

Alternative 7
Error	30.1
Cost	19652

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(-x\right) + \left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + x\right), x\right)\\ \end{array} \]

Alternative 8
Error	40.6
Cost	19588

\[\begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + 1\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + x\right), x\right)\\ \end{array} \]

Alternative 9
Error	51.4
Cost	19456

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

Alternative 10
Error	52.1
Cost	13252

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-307}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{1}{x}\right), x\right)\\ \end{array} \]

Alternative 11
Error	58.0
Cost	13120

\[\mathsf{copysign}\left(-\log \left(\frac{-1}{x}\right), x\right) \]

Reproduce

herbie shell --seed 2023066 
(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))