Rust f64::asinh

Average Error: 45.6 → 29.5

Time: 6.3s

Precision: binary64

Cost: 111944

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_1 := 1 + \left|x\right|\\ \mathbf{if}\;t_0 \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x} - \left(x - \left|x\right|\right)\right), x\right)\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{copysign}\left(0.5 \cdot \frac{{x}^{2}}{t_1} + \left(\log t_1 + \left(\frac{3}{t_1} + \frac{3}{{t_1}^{2}}\right) \cdot \left({x}^{4} \cdot -0.041666666666666664\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \]

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))
        (t_1 (+ 1.0 (fabs x))))
   (if (<= t_0 -10.0)
     (copysign (log (- (/ -0.5 x) (- x (fabs x)))) x)
     (if (<= t_0 5e-5)
       (copysign
        (+
         (* 0.5 (/ (pow x 2.0) t_1))
         (+
          (log t_1)
          (*
           (+ (/ 3.0 t_1) (/ 3.0 (pow t_1 2.0)))
           (* (pow x 4.0) -0.041666666666666664))))
        x)
       (copysign (log (+ (fabs x) (+ x (/ 0.5 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 t_1 = 1.0 + fabs(x);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = copysign(log(((-0.5 / x) - (x - fabs(x)))), x);
	} else if (t_0 <= 5e-5) {
		tmp = copysign(((0.5 * (pow(x, 2.0) / t_1)) + (log(t_1) + (((3.0 / t_1) + (3.0 / pow(t_1, 2.0))) * (pow(x, 4.0) * -0.041666666666666664)))), x);
	} else {
		tmp = copysign(log((fabs(x) + (x + (0.5 / 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 t_1 = 1.0 + Math.abs(x);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = Math.copySign(Math.log(((-0.5 / x) - (x - Math.abs(x)))), x);
	} else if (t_0 <= 5e-5) {
		tmp = Math.copySign(((0.5 * (Math.pow(x, 2.0) / t_1)) + (Math.log(t_1) + (((3.0 / t_1) + (3.0 / Math.pow(t_1, 2.0))) * (Math.pow(x, 4.0) * -0.041666666666666664)))), x);
	} else {
		tmp = Math.copySign(Math.log((Math.abs(x) + (x + (0.5 / 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)
	t_1 = 1.0 + math.fabs(x)
	tmp = 0
	if t_0 <= -10.0:
		tmp = math.copysign(math.log(((-0.5 / x) - (x - math.fabs(x)))), x)
	elif t_0 <= 5e-5:
		tmp = math.copysign(((0.5 * (math.pow(x, 2.0) / t_1)) + (math.log(t_1) + (((3.0 / t_1) + (3.0 / math.pow(t_1, 2.0))) * (math.pow(x, 4.0) * -0.041666666666666664)))), x)
	else:
		tmp = math.copysign(math.log((math.fabs(x) + (x + (0.5 / 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)
	t_1 = Float64(1.0 + abs(x))
	tmp = 0.0
	if (t_0 <= -10.0)
		tmp = copysign(log(Float64(Float64(-0.5 / x) - Float64(x - abs(x)))), x);
	elseif (t_0 <= 5e-5)
		tmp = copysign(Float64(Float64(0.5 * Float64((x ^ 2.0) / t_1)) + Float64(log(t_1) + Float64(Float64(Float64(3.0 / t_1) + Float64(3.0 / (t_1 ^ 2.0))) * Float64((x ^ 4.0) * -0.041666666666666664)))), x);
	else
		tmp = copysign(log(Float64(abs(x) + Float64(x + Float64(0.5 / 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)))));
	t_1 = 1.0 + abs(x);
	tmp = 0.0;
	if (t_0 <= -10.0)
		tmp = sign(x) * abs(log(((-0.5 / x) - (x - abs(x)))));
	elseif (t_0 <= 5e-5)
		tmp = sign(x) * abs(((0.5 * ((x ^ 2.0) / t_1)) + (log(t_1) + (((3.0 / t_1) + (3.0 / (t_1 ^ 2.0))) * ((x ^ 4.0) * -0.041666666666666664)))));
	else
		tmp = sign(x) * abs(log((abs(x) + (x + (0.5 / 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]}, Block[{t$95$1 = N[(1.0 + N[Abs[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -10.0], 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], If[LessEqual[t$95$0, 5e-5], N[With[{TMP1 = Abs[N[(N[(0.5 * N[(N[Power[x, 2.0], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[Log[t$95$1], $MachinePrecision] + N[(N[(N[(3.0 / t$95$1), $MachinePrecision] + N[(3.0 / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[x, 4.0], $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[(x + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]]

Target

Original	45.6
Target	0.0
Herbie	29.5

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

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 32.0

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

   2. Taylor expanded in x around -inf 0.1

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

   3. Simplified 0.1

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

      [Start] 0.1	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \left(-1 \cdot x - 0.5 \cdot \frac{1}{x}\right)\right), x\right) \]
      rational_best-simplify-1 [=>] 0.1	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \left(\color{blue}{x \cdot -1} - 0.5 \cdot \frac{1}{x}\right)\right), x\right) \]
      rational_best-simplify-10 [=>] 0.1	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \left(\color{blue}{\left(-x\right)} - 0.5 \cdot \frac{1}{x}\right)\right), x\right) \]
      rational_best-simplify-55 [=>] 0.1	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \left(\left(-x\right) - \color{blue}{1 \cdot \frac{0.5}{x}}\right)\right), x\right) \]
      rational_best-simplify-1 [=>] 0.1	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \left(\left(-x\right) - \color{blue}{\frac{0.5}{x} \cdot 1}\right)\right), x\right) \]
      rational_best-simplify-7 [=>] 0.1	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \left(\left(-x\right) - \color{blue}{\frac{0.5}{x}}\right)\right), x\right) \]

   4. Applied egg-rr 0.1

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

   if -10 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x) < 5.00000000000000024e-5

   1. Initial program 58.6

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

   2. Taylor expanded in x around 0 58.3

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-0.041666666666666664 \cdot \left(\left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right) \cdot {x}^{4}\right) + \left(0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|} + \log \left(1 + \left|x\right|\right)\right)}, x\right) \]

   3. Simplified 58.3

      \[\leadsto \mathsf{copysign}\left(\color{blue}{0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|} + \left(\log \left(1 + \left|x\right|\right) + \left(\frac{3}{1 + \left|x\right|} + \frac{3}{{\left(1 + \left|x\right|\right)}^{2}}\right) \cdot \left({x}^{4} \cdot -0.041666666666666664\right)\right)}, x\right) \]
      Proof

      [Start] 58.3	\[ \mathsf{copysign}\left(-0.041666666666666664 \cdot \left(\left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right) \cdot {x}^{4}\right) + \left(0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|} + \log \left(1 + \left|x\right|\right)\right), x\right) \]
      rational_best-simplify-3 [=>] 58.3	\[ \mathsf{copysign}\left(-0.041666666666666664 \cdot \left(\left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right) \cdot {x}^{4}\right) + \color{blue}{\left(\log \left(1 + \left|x\right|\right) + 0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|}\right)}, x\right) \]
      rational_best-simplify-47 [=>] 58.3	\[ \mathsf{copysign}\left(\color{blue}{0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|} + \left(\log \left(1 + \left|x\right|\right) + -0.041666666666666664 \cdot \left(\left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right) \cdot {x}^{4}\right)\right)}, x\right) \]
      rational_best-simplify-1 [=>] 58.3	\[ \mathsf{copysign}\left(0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|} + \left(\log \left(1 + \left|x\right|\right) + -0.041666666666666664 \cdot \color{blue}{\left({x}^{4} \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right)\right)}\right), x\right) \]
      rational_best-simplify-50 [=>] 58.3	\[ \mathsf{copysign}\left(0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|} + \left(\log \left(1 + \left|x\right|\right) + \color{blue}{\left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right) \cdot \left({x}^{4} \cdot -0.041666666666666664\right)}\right), x\right) \]

   if 5.00000000000000024e-5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x)

   1. Initial program 32.9

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

   2. Taylor expanded in x around inf 1.0

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

   3. Simplified 1.0

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

      [Start] 1.0	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \left(0.5 \cdot \frac{1}{x} + x\right)\right), x\right) \]
      rational_best-simplify-3 [=>] 1.0	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\left(x + 0.5 \cdot \frac{1}{x}\right)}\right), x\right) \]
      rational_best-simplify-55 [=>] 1.0	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \color{blue}{1 \cdot \frac{0.5}{x}}\right)\right), x\right) \]
      rational_best-simplify-1 [=>] 1.0	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \color{blue}{\frac{0.5}{x} \cdot 1}\right)\right), x\right) \]
      rational_best-simplify-7 [=>] 1.0	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \color{blue}{\frac{0.5}{x}}\right)\right), x\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 29.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x} - \left(x - \left|x\right|\right)\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{copysign}\left(0.5 \cdot \frac{{x}^{2}}{1 + \left|x\right|} + \left(\log \left(1 + \left|x\right|\right) + \left(\frac{3}{1 + \left|x\right|} + \frac{3}{{\left(1 + \left|x\right|\right)}^{2}}\right) \cdot \left({x}^{4} \cdot -0.041666666666666664\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	29.5
Cost	85192

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

Alternative 2
Error	29.6
Cost	78472

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

Alternative 3
Error	30.6
Cost	19844

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-308}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \]

Alternative 4
Error	30.6
Cost	19844

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-308}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x} - \left(x - \left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \]

Alternative 5
Error	41.3
Cost	13188

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-308}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \]

Alternative 6
Error	30.7
Cost	13188

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-308}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternative 7
Error	52.3
Cost	13124

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

Alternative 8
Error	58.2
Cost	12928

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

Reproduce

herbie shell --seed 2023100 
(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))